View Is Initiated [2] and Proved That Maximally 휓-Epistemic (M휓E) Model Is Incompatible with QM

Abstract Booklet

International Conference on Quantum Frontiers and Fundamentals

(30th April - 4th May, 2018)

Hosted by Raman Research Institute, Bengaluru, India

Sponsored by John Templeton Foundation, USA

Synopsis

Remarkable developments have occurred in recent times concerning investigations of fundamental aspects of Quantum Mechanics involving an intimate link between foundational ideas and experimental studies, in particular, stemming from striking progress in the relevant experimental technologies. This includes the coherent preparation and manipulation of quantum systems such as photons, electrons, neutrons, ions, and molecules. These rapid strides have enabled comprehensive empirical probing of profound quantum features like quantum superposition, wave particle duality, nonlocality and contextuality, shedding light on a wide range of fundamental issues. Facets of quantum entanglement have also been revealed starting from dense coding, teleportation and entanglement swapping to quantum steering. This, in turn, has given rise to a rich interplay between fundamental aspects of Quantum Mechanics and information theoretic studies, for instance, providing insights into the nature of quantum correlations as resource for information theoretic tasks. Novel ideas of generalized quantum measurement, including that of weak measurement, have been developed with foundational implications and empirical ramifications. Probing fundamental quantum features for continuous variable systems and higher dimensional quantum systems are two other important areas of increasing current interest, as well as in recent years considerable progress has been made in the area of testing fundamental aspects of quantum mechanics and the notion of macrorealism in the macroscopic domain. Finally, in light of all these developments, revisiting the interpretational aspects of quantum mechanics for newer insights into foundational issues is a challenging endeavour. Against this backdrop, the proposed conference intends to bring together the relevant experimentalists and theorists for critically deliberating upon the state of play and prospects in the areas mentioned above.

Contents

INVITED TALKS ...... 4 1. Aditi Sen ...... 5 2. Alok Kumar Pan ...... 5 3. Anil Kumar ...... 6 4. Anupam Garg ...... 6 5. Apoorva Patel ...... 7 6. Archan Majumdar ...... 7 7. Arun Kumar Pati ...... 8 8. Arvind ...... 8 9. Costantino Budroni ...... 9 10. Debasis Sarkar ...... 9 11. D. Sokolovski ...... 10 12. Guruprasad Kar ...... 11 13. Howard M. Wiseman ...... 12 14. Keiichi Edamastsu ...... 12 15. Krister Shalm ...... 13 16. Lorenzo Maccone ...... 13 17. Marco Genovese ...... 14 18. N.D. Hari Dass ...... 15 19. N. Mukunda ...... 16 20. Pankaj Agarwal ...... 16 21. Partha Ghose ...... 16 22. Perola Milman ...... 17 23. Peter P Rohde ...... 17 24. Piotr Kolenderski ...... 17 25. Prasanta K. Panigrahi ...... 18 26. R. Srikanth ...... 18 27. R. Usha Devi ...... 19 28. Saikat Ghosh ...... 20 29. Sai Vinjanampathy ...... 20 30. Sibasish Ghosh ...... 21 31. S. M. Roy ...... 21

32. Sougato Bose ...... 23 33. Subhash Chaturvedi ...... 24 34. Thomas Jennewein ...... 24 35. Xavier Oriols ...... 24 36. Yaron Kedem ...... 25 37. Yutaka Shikano ...... 26 38. Yves Caudano ...... 26 CONTRIBUTED TALKS ...... 28 39. Arun Sehrawat ...... 29 40. Atul Mantri ...... 32 41. Chiranjib Mukhopadhyay ...... 35 42. C. Jebaratnam ...... 37 43. Debashis Saha...... 39 44. Debmalya Das ...... 42 45. George Thomas ...... 43 46. Kumar Shivam ...... 44 47. Marko Toros ...... 45 48. Nicolò Lo Piparo ...... 46 49. R. Prabhu ...... 49 50. S. Aravinda ...... 51 51. Saubhik Sarkar ...... 53 52. Shiladitya Mal ...... 55 53. Siddhant Das ...... 58 54. Sk Sazim ...... 59 55. Suman Chand ...... 62 56. Tabish Qureshi ...... 64 57. Tanumoy Pramanik ...... 66 58. Vivek M. Vyas ...... 67 POSTERS ...... 69 59. Aiman Khan ...... 70 60. Akshay Gaikwad ...... 72 61. Amandeep Singh...... 75 62. Anirudh Reddy ...... 78 63. Anjali K ...... 79 64. Ashutosh Singh ...... 80

65. Asmita Kumari ...... 83 66. Álvaro Mozota Frauca ...... 85 67. B Sharmila ...... 85 68. Bikash K. Behera ...... 87 69. Chandan Datta ...... 89 70. Debarshi Das ...... 91 71. Devashish Pandey ...... 93 72. Gautam Sharma ...... 96 73. Kaushik Joarder ...... 98 74. Mitali Sisodia ...... 99 75. Mohd Asad Siddiqui ...... 101 76. Mushtaq B. Shah ...... 104 77. Muthuganesan R ...... 106 78. Nasir Alam ...... 108 79. Natasha Awasthi ...... 110 80. Rajendra Singh Bhati ...... 112 81. Ravi Kamal Pandey ...... 114 82. Rengaraj ...... 117 83. S Lakshmibala ...... 118 84. Saptarshi Roy ...... 120 85. Seeta Vasudevrao ...... 122 86. Shamiya Javed ...... 124 87. Som Kanjilal ...... 126 88. Souradeep Sasmal ...... 127 89. Subhajit Bhar ...... 129 90. Surya Narayan Sahoo ...... 130 91. Syed Raunaq Ahmed ...... 131 92. U. Shrikant ...... 134

INVITED TALKS

Frozen quantum correlations

Aditi Sen Harish Chandra Research Institute, Allahabad

Abstract: Characterizing correlations between different subsystems of a composite quantum system has been an important field of research in quantum information. This is due to the fact that quantum correlations, in the form of entanglement, is shown to be significantly more useful for performing communication and computational tasks over their classical counterparts. A main obstacle encountered in realizations of quantum information protocols is the rapid decay of quantum correlations with time in multiparty quantum systems exposed to environments. I will present scenarios involving physical systems that are realizable in the laboratory, and a specific model of environment, in which entanglement of the system, even when exposed to the environment, remains constant for a finite interval of time at the beginning of the dynamics. We call this phenomena as freezing of entanglement.

Ontological models, universal contextuality and communication game

Alok Kumar Pan National Institute of Technology Patna, India

Abstract: In recent times, the ontological model framework for an operational theory has received much interest. An elegant mathematical and conceptual framework of ontological models is introduced [1] which provides a precise approach to examine the question concerning reality of quantum states. Two major distinctions have been proposed, referred to as 휓-onticand 휓-epistemic interpretation of quantum state. While the former interprets the quantum state as a part of reality, the latter is in favour of interpreting the quantum states as merely the knowledge of the reality but not the reality itself. By considering the degree of epistemicity, a stronger notion of epistemic view is initiated [2] and proved that maximally 휓-epistemic (M휓E) model is incompatible with QM. It is also shown [2] that how this stronger notion of quantum state epistemicity connects to other no-go theorems, particularly the notion of generalized framework of non-contextuality. In this talk, I shall first provide a simpler proof to show connection between M휓E and preparation non-contextuality devoid of the response function. I then introduce the subtleties involved in the notion of preparation contextuality. Further, I shall re-examine the role of determinism a recent proof that demonstrates the quantum violation universal non-contextual inequalities. Finally, I shall discuss how quantum preparation contextuality powers a communication game.

References: [1] N. Harrigan andR. W. Spekkens, Found. Phys. 40, 125 (2010). [2] M. Leifer andO. Maroney, Phys. Rev. Lett 110, 120401 (2013).

Quantum Information Processing by NMR

Anil Kumar Department of Physics and NMR Research Centre,Indian Institute of Science, Bangalore-560012.

Abstract: After a sedate beginning in mid1980’s, the subject of “Quantum Computation and Quantum Information Processing” has seen explosive developments in recent years. From a pessimistic view of yester years “Will a Quantum Computer be made in our life time” to a very optimistic view of today “A Quantum computer will be available in next 4-5 years, which will beat the fastest present-day Classical Computer”.

Till recently, the most successful of the various experimental techniques has been that of Nuclear Magnetic Resonance (NMR). Our research group, which started experimental work in this field in late 90’s, has performed several experiments which include; (i) preparation of pseudo-pure states, (ii) implementation of logic gates using one and two- dimensional NMR, (iii) implementation of Deutsch-Josza (DJ) and Grover’s search algorithms, (iv) use of quadrupolar and dipolar coupled nuclei oriented in liquid crystal media for quantum information processing, (v) entanglement and entanglement transfer, (vi) observation of non-adiabatic geometric phase in NMR and its use in NMR QIP, (vii) implementation of some quantum games and (viii) implementation of adiabatic quantum algorithms by NMR and the use of Strongly Modulated Pulses (SMPs) in NMR QIP. As part of “Introduction” some of these works will be highlighted.

Recent developments include, (i) Experimental proof of Quantum No-Hiding theorem. (ii)Use of Nearest Neighbor (NN) Heisenberg XY interaction for creation of entanglement in a linear chain of 3-qubits (iii) Multi-Partite Quantum Correlations Revel Frustration in a Quantum Ising Spin System and (iv) use of Genetic Algorithm in NMR QC. We have also by NMR, non-destructively distinguished Bell states and more generalized orthogonal states. Recently we have used the IBM’s 5 qubit “Quantum experience” to perform the same nondestructive discrimination. This will also be highlighted.

The quantum-classical correspondence for spin

Anupam Garg Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA

Abstract: Path integrals for spin have long been sought, not only for applications, but also for their intrinsic aesthetic value. Such path integrals are naturally written in terms of spin coherent states, but they are fraught with mathematical diﬃculties, just as are coherent state path integrals for a massive particle with momentum p and position q.

For example, the action is, naively, the integral of (푝푞̇ − 퐻(푝, 푞)) but 퐻(푝, 푞), which one would like to interpret as the classical Hamiltonian, is ambiguous.Options for 퐻(푝, 푞) include the P, Q, Weyl, (or inﬁnitely many other) symbols of the quantum mechanical Hamiltonian H. More generally, one would like a correspondence between any quantum mechanical

6 operator, not just H, and c-number functions on phase space. For massive particles, such a correspondence is provided by the Weyl-Wigner-Moyal formalism. We have recently developed a complete analog of this formalism for spin, and we can define P, Q, and Weyl symbols for any spin operator. The lattermost is particularly nice. For example, the Weyl symbol of the density matrix is the Wigner function for any system. The use of the Weyl symbol for H is especially advantageous in considering the semiclassical approximation to the path integral for spin, in that the first quantum correction to the leading term (which one may view as the eikonal approximation) then needs no Solari-Kochetov type terms to resolve the global anomaly in the fluctuation determinant. We have also found exact asymptotic relations between various symbols in the limit of large spin, and approximate relations analogous to the Moyal bracket. The talk will aim to give a broadly accessible overview of these topics, as well as provide examples of the various symbols, such as the Wigner function for the singlet state of two spin-s particles, and its limit as s → ∞.

Quantum trajectories for measurement of entangled states

Apoorva Patel IISc, Bangalore

Abstract: Quantum correlations can be described using real stochastic variables, e.g. Wigner functions, provided that the variables can take negative values. Recently, quantum trajectories obeying nonlinear stochastic evolution have been observed during weak measurements of superconducting transmon qubits. How quantum correlations are encoded in the trajectory weights will be illustrated, using weak measurements of the Bell state as an example.

Sequential sharing of nonlocal correlations

Archan Majumdar S.N. Bose National Centre for Basic Science, Kolkata

Abstract: We investigate the question as to whether the nonlocality of a single member of an entangled pair of particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. Considering a pair of spin-1/2 particles, we first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne- Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.

We next consider the steering scenario for two-qubits with two measurement settings for each observer. We show that the analogue steering inequality can be violated for an Alice and at most two Bobs on the other side. Finally, we show that increasing the number of

7 measurement settings allows for the possibility of multiple Bobs to be able to sequentially steer Alice's state.

Superposition, Quantum Addition and Quantum Coherence

Arun Kumar Pati Harish-Chandra Research Institute, HBNI, Allahabad 211019, India

Abstract: In quantum world, for pure states, it is natural that quantum coherence arises from coherent superposition of different amplitudes. However, for mixed states, there is no such notion of coherent superposition of mixed states. On the other hand, it is possible to have classical mixture of two or more mixed states. Therefore, it is a priori not clear how quantum coherence can arise from `the superposition principle' for mixed states. Here, we will argue that for mixed states, the quantum coherence may arises from the notion of quantum addition, which allows us to include an additional contribution arising from the non-commutativity of the mixed components over and above the classical mixture. In this talk, I will show that the quantum coherence can come from the discrepancy between quantum and classical addition. This shows that how quantum coherence arises for mixed states from an analogous notion of superposition, namely, the quantum addition.

Arvind Quantum Contextuality and Quantum Cryptography

Dr. Arvind IISER, Mohali

Abstract: The notion of quantum contextuality is important in distinguishing quantum behaviour from the classical one. Quantum contextuality has been posited to be a possible reason for the computational speedup achieved by quantum computing algorithms. There also have been a number of different ways of looking at quantum contextuality. In a broad sense the security of quantum key distribution protocols depends on intrinsically quantum aspects of physical systems. We have studied the role played by quantumness, as qualified by quantum contextuality, in a quantum key distribution (QKD) protocol. In the talk I will discuss our recent protocol where we develop a contextuality based QKD protocl and explicitly show that the unconditional security of the protocol by a generalized contextuality monogamy relationship based on the no-disturbance principle. This new protocol has conceptual and practical advantages over other protocols.

Memory cost of temporal correlations Costantino Budroni Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Vienna, Austria

Abstract: We investigate a possible notion of nonclassicality for single systems, based on the notion of memory cost of simulating observed probabilities obtained after a sequence of measurements. More precisely, we investigate the set of probabilities achievable with finite- state machines, i.e., machines with a finite number of perfectly-distinguishable states, output probabilities, and state-transition probabilistic rules. We discuss such models in the framework of classical, quantum, and generalized probabilistic theories (GPTs). The finite number of perfectly-distinguishable states is interpreted as a memory resource available to the machine and needed to produce correlations. With this restriction, we show how our notion allow us to distinguish classical, quantum, and GPTs correlations already in the simplest nontrivial scenario, namely, that of two inputs, two outputs, and sequences of length two. Moreover, we investigate the behavior of such models in more complex scenarios and in the asymptotic limit of measurement sequences of infinite length.

Debasis Sarkar Multipartite Entanglement in Quantum Network Scenario

Prof. Debasis Sarkar Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata- 700009 India. (email:- [email protected], [email protected])

Abstract: Entanglement is one of the most profound inventions of modern quantum theory. A bipartite quantum state without entanglement is called separable. A multipartite quantum state that is not separable with respect to any bipartition is called genuinely entangled. This type of entanglement is important not only for researches concern the foundations of quantum theory but also in various quantum information protocols and quantum information processing tasks. Apart from entanglement another remarkable feature of quantum systems is: systems that have never interacted could become entangled. This process is usually known as entanglement swapping, in which two independent pairs of entangled particles are first created and then one particle from each pair is jointly measured. As a result the other particles become entangled. In the present talk, we address basically the following questions: Consider some tripartite states whose genuine entanglement cannot be detected by applying some standard (DIEW’s), now is it possible to find some suitable entanglement swapping process, after which the genuine entanglement of swapped state can be detected by those DIEWs? We will answer this question affirmatively by framing a protocol based on entanglement swapping procedure in our talk.

What is measured by the “weak measurements”, and why “the tunneling time” may never be found?

D. Sokolovski1,3 and E. Akhmatskaya2,3

1Departmento de Quimica-Fisica, Universidad del Pais Vasco, UPV/EHU, Leioa, Spain 2Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo, 14 48009 Bilbao, Bizkaia, Spain 3IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain

Abstract: Consider a system, which can reach its final destination via several interfering (virtual) scenarios (pathways, paths). In all cases, quantum mechanics equips each pathways with a probability amplitude, say A(i). Let us add an accurate (strong) meter, giving a reading B(i) every time the system is seen to take the ith path. This will turn the virtual paths into real pathways, which can then be seen travelled with certain frequencies (probabilities). The price to pay is the perturbation incurred: the probability to reach the final state, previously given by 2 2 |∑i A(i)| , is now ∑i |A(i)| . But what if we make the meter “weak”, so as to perturb the system as little as possible? The uncertainty principle [1] tells us to treat the set of interfering scenarios as a single real pathway, and leaves the value of the functional B, associated with the transition, indeterminate. Indeed, the readings of a weak meter, now taking all possible values, will tell us little about the path chosen by the system. It is easy to check that its average reading will remain finite, given by the real (or, possibly, imaginary) part of the complex expression ∑i B(i)α(i), where α(i) ≡ A(i)/∑i A(j). The sum is often referred to as the “weak value”, w, of B [2]. “So what?” one might say. With no probabilities created by the ineptmeter, the response of the system has to be formulated in terms of the relevant probability amplitudes. We agree. The weak value of B is just a weighted sum of the (relative) amplitudes, and little else. Moreover, by changing the initial and final states of the system, we can make hBiw take an arbitrary value, large or small, real or imaginary [3]. The uncertainty principle was right all along: the only thing we can learn about interfering alternatives is that they are endowed with probability amplitudes. Since early 1930s, people have been asking a question “how long does it take for a particle to tunnel?”, to which no simple answer has been found to date. It is easy to see why. Tunneling is an interference phenomenon, where virtual Feynman paths, which spend different durations in the barrier, interfere destructively to produce a small transmission probability. One faces the same dilemma: or to measure the time accurately, thereby destroying tunneling, or to leave interference intact, and obtain only the “weak value” of the time functional, <τ>w,-the “complex time” of [4], -by means of a “weak” Larmor clock. We will show that, in the latter case, it would be imprudent to treat Re<τ>w, Im<τ>w, or|<τ>w| as a physical duration [5]. A similar difficulty occurs if one tries to relate the final position of a transmitted wave packet to the delay experienced by the particle in the barrier region. Although it could take some effort to see, hidden behind this procedure, a “weak measurement” of the spatial delay induced by the barrier [6], the verdict has to be the same. The resultant “phase time”, involving the real part of the “weak value” of the delay, is as trustworthy as the real part of the “complex time” <τ>w, and presents no challenge to Einstein’s causality, despite its being anomalously short for wide barriers.

Finally, one may ask, whether it is helpful to reduce a physical argument to a discussion about the probability amplitudes, often considered to be mere computational tools? We argue that this is the only way to proceed in a case when a quantity, well deﬁned in a classical limit,

10 is lost to quantum interference. Treating parts of complex time parameters as real time intervals, might lead to superluminal velocities [6], or durations spent in a region, which exceed the total time of motion [5]. And nobody really wants that.

References: [1] Feynman, R.P, Leighton, R., & Sands, M. The Feynman Lectures on Physics III (Dover Publications, Inc., New York, 1989), Ch.1: Quantum Behavior. [2] Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W. Colloquium: Understanding quantum weak values: basics and applications. Rev. Mod. Phys. 86, 307 (2014). [3] D. Sokolovski and E. Akhmatskaya, An even simpler understanding of quantum weakvalues. Ann. Phys. A388, 382, (2018). [4] D. Sokolovski, L. M. Baskin, Traversal time in quantum scattering. Phys. Rev. A, 36, 4604(1987). [5] D. Sokolovski, Salecker-Wigner-Peresclock, Feynmanpaths, and a tunneling time that should not exist, Phys. Rev. A96, 022120 (2018). [6] D. Sokolovski, E. Akhmatskaya, Superluminal paradox in wavepacket propagation and its quantum mechanical resolution. Ann. Phys. 339, 307 (2013).

Interplay among Nonlocality, Measurement Incompatibility and Uncertainty

Guruprasad Kar Indian Statistical Institute, Kolkata

Abstract: Quantum nonlocality is revealed through the violation of some Bell’s inequality. It is known that largest quantum violation of Bell-CHSH inequality is 2√2. Recently it has been shown that this optimal violation is deeply related to quantum (fine grained) uncertainty. On the other hand, it has also been shown that the degree of measurement incompatibility of a no-signaling theory provides an upper bound for the violation of Bell-CHSH inequality and this bound is saturated in quantum mechanics. We discuss this aspect for other no-signaling theories. Spekken’s Toy theory, in this context, is shown to be different. We argue that the degree of measurement incompatibility in Spekkens’ Toy theory is nontrivial due to existence of steering, but the presence of high uncertainty in the theory does not allow it to reach the non-trivial upper bound of Bell-CHSH expression and this is consistent with its local feature.

Howard M. Wiseman Observing Bohmian momentum transfer in double-slit “which-way” measurements

Ya Xiao,1, 2 Howard M. Wiseman,3 Jin-Shi Xu,1, 2YaronKedem,4 Chuan-Feng Li,1,2and Guang-Can Guo1, 2

1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of China 2Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China 3Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia 4Department of Physics, Stockholm University, Alba Nova University Center, 106 91 Stockholm, Sweden

Abstract: Making a “which-way” measurement (WWM) to identify which slit a particle goes through in a double-slit apparatus will reduce the visibility of interference fringes. There has been a long- standing controversy over whether this can be attributed to an uncontrollable momentum transfer. To date, no experiment has quantified the momentum transfer in a way that relates quantitatively to the loss of visibility. Here, by reconstructing the Bohmian trajectories of single photons, we experimentally obtain the distribution of momentum transfer, which is observed to be not a momentum kick that occurs at the point of the WWM, but nonclassically accumulates during the propagation of the photons. We further confirm a quantitative relation between the loss of visibility consequent on a WWM and the late-time momentum transfer. The results of our work give an intuitive picture that aids understanding of wave-particle duality and complementarity.

Keiichi Edamastsu Quantum interference and wave-particle duality of dynamically unpolarized single photons

Naofumi Abe, Yasuyoshi Mitsumori, Mark Sadgrove and Keiichi Edamastsu∗ Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan (∗[email protected])

Abstract: We generated completely mixed, dynamically unpolarized single photons from an NV center in diamond [1]. This unique photon source makes it possible to investigate non- obvious quantum properties of particles that lie in mixed states of their internal degrees of freedom. Here we demonstrate quantum interference experiments using the unpolarized single photons we generated. The interference visibility changes depending on the which-path information added by means of polarization rotation, even though the completely mixed (unpolarized) polarization state is unchanged by the rotation. We discuss how to evaluate the degree of which-path information and underlying wave-particle duality in our experiments. Also, the photon source can be used to investigate the measurement uncertainty relations of mixed states [2] in photon polarization qubits [3].

References: [1] N. Abe, Y. Mitsumori, M. Sadgrove, and K. Edamatsu, Sci. Rep. 7, 46722 (2017). [2] M. Ozawa, arXiv:1404.3388 (quant-ph) (2014). [3] K. Edamatsu, Phys. Scr. 91, 073001 (2016)

Generating randomness using fundamental tests of nature

Krister Shalm NIST, Boulder, Colorado, USA

Abstract: In 1943 Einstein wrote to Max Born saying “As I have said so many times, God doesn't play dice with the world.” This discussion with Born was just one part of a much large debate on the consequences of quantum theory on the nature of reality. In 1935 Einstein, Podolsky, and Rosen famously published a paper with the aim of showing that the wave function in quantum mechanics does not provide a complete description of reality. The gedanken experiment showed that quantum theory, as interpreted by Niels Bohr, leads to situations where distant particles, each with their own “elements of reality”, could instantaneously affect one another. Such action at a distance seemingly conflicts with relativity. The hope was that a local theory of quantum mechanics could be developed where individual particles are governed by elements of reality, even if these elements are hidden from us. In such a theory, now known as local realism, these elements of reality or hidden variables could remove the randomness inherent in quantum mechanics.

In 1964 John Bell in a startling result showed that the predictions of quantum mechanics are fundamentally incompatible with any local realistic theory. In other words, an experiment can be done that can rule out all theories based on local hidden variables. Carrying out this test has been technologically challenging. It wasn’t until 2015 when three independent groups were able to rule out local realism in experiments free of loopholes. In this talk I will discuss the loophole-free Bell test carried out at the National Institute of Standards and Technology. I will also discuss how we can use such a Bell test to build a random number generator that can be certified by quantum mechanics itself. Such a random number generator that can trace its roots back to the original Einstein thought experiments is the closest we can true randomness.

Sum uncertainty relations

Lorenzo Maccone University of Pavia, Italy

Abstract: Uncertainty relations that introduce a bound to the sum of variances have surprising advantages with respect to the usual Heisenberg-Roberston ones that put a bound to the product of variances. I will discuss some examples of specific sum uncertainty relations and discuss methods to obtain new ones.

Quantum optics as a tool for visualizing fundamental phenomena

Marco Genovese INRiM, Italy

Abstract: Quantum optical systems present several interesting properties that allow using them as a tool for visualizing physical phenomena otherwise subject of theoretical speculation only, as Bose Einstein condensation for Hawking radiation [1]. As a second example, we consider Page and Wootters [2] model. This model, considering that exist states of a system composed by entangled subsystems that are stationary but one can interpret the component subsystems as evolving, suggests that the global state of the universe can be envisaged as one of this static entangled state, whereas the state of the subsystems (us, for example) can evolve. In the first part of this talk I will briefly present two experiments addressed to visualise this phenomenon. The first is based on PDC polarisation entangled photons that allows showing with a practical example a situation where this idea works, i.e. a subsystem of an entangled state works as a "clock" of another subsystem [3]. However, the simple original Page and Wootters model needs some extension for describing several time measurements [4].

Similarly, in our first experiment the use of a two-dimensional clock implies that the time is discrete, periodic and can take only two values: 0 and 1. In a second experiment [5], that I present here for the first time, we use a continuous system (the position of a photon) to describe time, which gives us access to measurements at arbitrary times and hence arbitrary two-time correlations.

Closed Time-like Curves (CTC), one of the most striking predictions of general relativity, are another example of untestable theoretical speculation. They are notorious for generating paradoxes, such as the grandfather's paradox, but these paradoxes can be solved in a quantum network model [6], where a qubit travels back in time and interacts with its past copy. However, there is a price to pay. The resolution of the causality paradoxes requires to break quantum theory's linearity. This leads to the possibility of quantum cloning, violation of the uncertainty principle and solving NP-complete problems in polynomial time. Interestingly, violations of linearity occur even in an open time-like curve (OTC), when the qubit does not interact with its past copy, but it is initially entangled with another, chronology-respecting, qubit. The non-linearity is needed here to avoid violation of the monogamy of entanglement. To preserve linearity and avoid all other drastic consequences, we discuss how the state of the qubit in the OTC is not a density operator, but a pseudo-density operator (PDO) - a recently proposed generalisation of density operators, unifying the description of temporal and spatial quantum correlations. Here I present an experimental simulation of the OTC using polarization-entangled photons, also providing the first full quantum state tomography of the PDO describing the OTC, verifying the violation of the monogamy of entanglement induced by the chronology-violating qubit. At the same time the linearity is preserved since the PDO already contains both the spatial degrees of freedom and the linear temporal quantum evolution. These arguments also offer a possible solution to black hole entropy problem.

References: [1] J.Steinahauer et al.,Nature Physics volume12, 959–965 (2016).

[2] D.N. Page and W.K. Wootters, Phys. Rev. D 27, 2885 (1983); W.K. Wootters, Int. J. Theor. Phys. 23, 701 (1984). [3] E.Moreva, M.Gramegna, G.Brida, L.Maccone, M.Genovese, Phys. Rev. A 89, 052122 (2014). [4] V.Giovannetti, S.Lloyd, L.Maccone, Phys. Rev. D, 92, 045033 (2015). [5] E.Moreva, M.Gramegna, G.Brida, L.Maccone, M.Genovese, Phys. Rev.D in press. arXiv:1710.00707 [6] D. Deutsch, Phys. Rev. D 44, 10, 1991. [7] C.Marletto, V.Vedral, S. Virzì, E.Rebufello, A.Avella, M.Gramegna, I.P.Degiovanni, M.Genovese, in press.

Optimal weak value measurements of pure and mixed states

N.D. Hari Dass1 and R. Rajath Krishna2 1TIFR, Hyderabad 2Imperial College, London

Abstract: We apply the notion of optimality of measurements as given by Wootters and Fields, to weak value tomography of pure states, and, weak tomography (without post- selection) of mixed states. They defined measurements to be optimal if they ‘minimised’ the effects of statistical errors. They had only considered standard tomography of mixed states and they had to actually maximise the state averaged information, quantified by the negative logarithm of ‘error volume’.

In our pure state analysis, we optimise both the state averaged information as well as error volumes. We prove, for finite dimensional Hilbert spaces, that varieties of weak value measurements are optimal when the post-selected states are mutually unbiased wrt the eigenvectors of the observable being measured. We prove a number of important results about the geometry of state spaces. We derive the Kahler potential for the N-dimensional case with the help of which we give an exact treatment of the arbitrary-spin case. As a spin- off, we show that the Majorana Star construction for arbitrary spin is global.

We have extended our analysis to weak tomography of mixed states. Here, weak value tomography is problematic. We have investigated two protocols: i) using weak measurements without post-selection to measure expectation values for an optimal set of observables, and, ii) a method proposed by Lundeen and Bamber. The mixed state space is flat and geometrically not as interesting as the pure state case. The weak tomography of mixed states also requires measuring 푁 −1 observables just as in the case of their standard tomography. However, the errors are state independent, allowing one to analyse the optimality without state averaging. In the first case, we prove that measurements are optimal when the eigenvectors of the observables are mutually unbiased. In the Lundeen-Bamber case the arbitrarily selected states are shown to be mutually unbiased wrt eigenvectors of the observable measured.

We have also analysed the weak value tomography of mixed states proposed by Shengjun Wu. Though observable weak values can be introduced this way, the method is inefficient as a tomography. Also, optimality criteria are hard to find.

A 'Square Root' approach to Wigner distributions for continuous and discrete variables

N. Mukunda CHEP, IISc, Bangalore

Abstract: We present a two-step method for the introduction of classical phase space descriptions of states and observables in quantum mechanics - an elementary way to introduce phase-space variables suggested by Dirac, followed by the extraction of a square root of an integral kernel to handle traces of products of operators. In the case of continuous variables, this is shown to lead to the well known Wigner distribution description of states accompanied by the Weyl description of operators. The application of this method to finite dimensional systems is sketched, bringing out the differences between the even and odd dimensional cases. For odd dimensions, a form of Wigner distribution based on properties of finite groups of odd order and their representations is also described.

Tripartite Nonlocality

Pankaj Agarwal IOP Bhubaneswar, India

Nonlocality remains one of the most mysterious phenomenon associated with quantum systems. We explore the nonlocality of simple tripartite systems using a set of known and proposed inequalities. For certain measurement scenarios, facet inequalities are not maximally violated by maximally entangled states. However, there are inequalities which do so for simple systems. There are classes of mixed states, where new inequalities do better.

A Unified Theory of Classical and Quantum Light!

Partha Ghose The National Academy of Sciences, India, 5 Lajpatrai Road, Allahabad 211002, India.

Abstract: It is shown that the relativistic wave equation for a complex massless scalar field leads to a Schrödinger-like equation for stationary solutions, which in turn leads to the classical Helmholtz equation, revealing a hidden quantum-like structure that is both surprising and interesting.

Modular variables: from fundamental tests of quantum mechanics to quantum error correction

Perola Milman CNRS/ Univ. Paris-Diderot, France

Abstract: Modular variables were introduced in the 80’s by Y. Aharanov as a tool to describe non-locality in the Aharanov-Bohm effect. This formalism is particularly well adapted to treat systems described by continuous variables with a periodic structure. In the present contribution we go beyond this domain of application and show how modular variables can be used to generalise fundamental tests of quantum mechanics initially conceived for discrete systems to systems described by continuous variables. We also show an intimate connection between modular variables and quantum error correction in continuous variables, and argue that this formalism may be the most adapted to build a unified picture of quantum information processing in both continuous and discrete variables systems.

Peter P Rohde The Quantum Internet - Towards the Singularity

Dr. Peter P Rohde University of Technology, Sydney

Abstract: We live in an increasingly interconnected world. The classical internet enables the majority of contemporary computing applications, and it wasn’t until the advent of the internet that the truly exciting prospects for computation emerged. It is to be expected that in the future era of quantum computing, there will likewise be a desire to interconnect them to fully exploit their capabilities. I will discuss some of the exciting prospects of a ‘quantum internet’, enabling new networked approaches and applications for quantum computing. I will discuss not only the technical implications of this, but also the societal and economic ones, and how this will transform the future of computing.

Experiments on nanostructures, clock synchronization and quantum communication with single photons

Piotr Kolenderski Institute of Physics, Nicolaus Copernicus University, Ul Grudziadzka 587-100 Torun, Poland

Abstract: The single photon sources based on the process of Spontaneous Parametric Down Conversion (SPDC) are widely used for numerous applications ranging from fundamental tests of quantum theory to practical applications in quantum information processing. During the talk I will present two applications. First one is a single photon interaction with a plasmonic beam splitter, which is a two dimensional metallic nanostructure build on glass.

The second one is a method of enhancing certain quantum communication and remote clock synchronization protocols by using carefully chosen spectrally entangled photon pairs.

Exceptional points and manipulating light in PT-media

Prasanta K. Panigrahi

Professor, Department of Physical Sciences, Indian Institute of Science Education and Research, Kolkata

Abstract: Media with PT-symmetry (balanced gain and loss) allows control of light and manipulation. Designed PT-symmetric media regulates reflectivity and transmissivity reveals the unusual optical properties. This includes Exceptional points, zero width resonance and bound states in continuum.

Non-Markovian dephasing and depolarizing channels

R. Srikanth Division of Theoretical Sciences, PPISR, India

Abstract: A Pauli channel subjects a given state of a qubit to a convex combination of actions of Pauli operators [1]. Here, we construct non-Markovian extensions of some Pauli channels by generalizing the Kraus representation to a single-parameter family and studying parameter regimes for which non-Markovianity holds. Further, we quantify the degree of non-Markovianity and study implications for quantum communication. The problem of generalizing (non-)Markovianity from the classical to the quantum case isn’t straightforward, mainly because the conditional probabilities depend both on the dynamical evolution and measurement processes involved [2, 3]. A natural quantum extension of the classical divisibility criterion would be CP-divisibility, the requirement that the dynamical map be a composition of completely positive maps [4]. The type of non-Markovian extensions we consider uses this CP-divisibility definition. Other approaches include the idea a suitable distance between two initial states must be non-increasing under quantum Markovian evolution, which we have used elsewhere [5]. Specifically, here we witness the non- Markovianity of the extended dephasing and depolarizing qubit channels, by the appearance of negativity in the eigenvalue spectrum of the Choi matrix for an intermediate dynamical map. The degree of non-Markovianity is quantified by the measures of [4, 6]. The advantage or disadvantage of non-Markovianity of the extended dephasing channel is investigated via the Holevo information.

References: [1] S. Omkar, R. Srikanth, Subhashish Banerjee. Dissipative and non-dissipative single-qubit channels: dynamics and geometry. Quantum Information Processing, 12, 3725-3744 (2013). [2] Concepts of quantum non-Markovianity: a hierarchy. Li Li, Michael J. W. Hall, Howard M. Wiseman. arXiv:1712.08879.

[3] Quantum Non-Markovianity: Characterization, Quantification and Detection. Ángel Rivas, Susana F. Huelga, Martin B. Plenio. Rep. Prog. Phys. 77, 094001 (2014) [4] M. J. W. Hall, J. D. Cresser, L. Li and E. Andersson, Phys. Rev. A 89, 042120 (2014). [5] Non-Markovian evolution: a quantum walk perspective. P. Kumar, Subhashish Banerjee, R. Srikanth, Vinayak Jagadish, Francesco Petruccione. arXiv: 1711.03267 [6] Entanglement and non-Markovianity of quantum evolutions. Ángel Rivas, Susana F. Huelga, Martin B. Plenio. Phys. Rev. Lett. 105, 050403 (2010).

Violation of conditional entropic uncertainty relation

R. Usha Devi Department of Physics, Bangalore University Bengaluru-560056, India.

Abstract: One of the peculiar aspects of quantum theory is that the outcomes of measurements of non-commuting observables cannot be predicted accurately. This feature is captured quantitatively in terms of uncertainty relations, which give upper bounds on the precision with which the measurement outcomes of incompatible observables can be predicted. Massen-Uffink entropic uncertainty relation [1] constrains the sum of entropies associated with the measurements of a pair of non-commuting observables. An extended entropic uncertainty relation proposed by Berta et. al., [2] revealed that the sum of conditional entropies violate the Massen-Uffink bound, when an entangled state is considered. In this talk, we discuss the violation of conditional entropic uncertainty relations. We bring out the crucial role of incompatible measurements [3-7] in such violations.

References:

[1] H. Maassen & J. B. M. Uffink,”Generalized uncertainty relations”, Phys. Rev. Lett. 60, 1103(1988). [2] M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, “The uncertainty principle in the presence of quantum memory”, Nature Physics 6, 659 (2010). [3] H. S. Karthik, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability, steering, and entropic uncertainty”, Phys. Rev. A 91, 012115 (2015). [4] H. S. Karthik, A. R. Usha Devi, and A. K. Rajagopal, “Unsharp measurements and joint measurability”, Current Science 109, 2061 (2015) [5] H. S. Karthik, A. R. Usha Devi, J. Prabhu Tej, and A. K. Rajagopal, "Conditional entropic uncertainty and quantum correlations", Submitted to Special Virtual Issue on Quantum Correlations, Optics Communications (2018). [6] H. S. Karthik, A. R. Usha Devi, J. Prabhu Tej, and A. K. Rajagopal, “Joint measurability and temporal steering”, J. Opt. Soc. Am. B 32, A34 (2015). [7] H. S. Karthik, A. R. Usha Devi, J. Prabhu Tej, A. K. Rajagopal, Sudha, and A. Narayanan “N- term pairwise-correlation inequalities, steering, and joint measurability”, Phys. Rev. A 95, 052105 (2017).

Motion amplification with squeezed graphene resonators

Saikat Ghosh IIT Kanpur, India

Abstract : Amplifying small motion with low noise and back-action has been a fundamental goal in science along with applications in the growing field of opto-mechanics. Here we discuss our experiments demonstrating a widely tunable, broadband, and high gain all- mechanical motion amplifiers based on graphene/Silicon Nitride (SiNx) hybrids. In such an amplifier, a tiny motion of large-area Silicon Nitride membrane produces much large motion in a circular graphene drum resonator deposited over a hole in SiNx and coupled to it via van der Waals interactions. Using interferometric readout for both graphene and SiNx motion, we measure at a sensitivity of 30 fm/√퐻푧, along with a gain of 38 dB for a 15 µm diameter, suspended graphene resonator. An additional parametric amplification stage reduces noise at a specific phase or quadrature of motion, with a corresponding sensitivity at 17.15 fm/√퐻푧, close to the fundamental thermal limit for the SiNx resonator. Our results suggest that graphene like bottom-up fabricated 2-d resonators can emerge as efficient choice for tunable, quantum limited motion transducers and amplifiers at room-temperature.

Quantum Synchronisation in the Quantum Regime

Sai Vinjanampathy IIT Bombay

Abstract: It is well known that weakly coupled, self-sustained oscillators can mutually lock the phase of their oscillations in classical mechanics. Such a phenomenon is known as synchronisation. Given the experimental progress of seeing resonators in the quantum regime, there is an ongoing effort to observe synchronisation and other non-linear dynamical effects in quantum systems. It is desirable to observe such synchronisation of quantum systems not just in the classical regime, but also in the quantum regime. This quantum regime is defined by low number of excitations and a highly non-classical steady state of the self- sustained oscillator. In this talk, I will introduce quantum synchronisation and present several results relating to the synchronisation of quantum systems.

New family of bound entangled states residing on the boundary of Peres set

Sibasish Ghosh Optics & Quantum Information Group, The Institute of Mathematical Sciences Joint work with Saronath Halder and Manik Banik (arXiv: 1801.00405 (quant-ph)

Abstract: Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize all these states for an arbitrary composite quantum system. Here we present a (d−3)/2-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in ℂd⊗ℂd with d odd and d ≥5. We also provide the `tile structure' corresponding to these UPBs. The range of each such PPT-BE state is contained in a 2(d−1) dimensional entangled subspace whereas the associated UPB-subspace is of dimension (d−1)2+1. We further show that each of these PPT-BE states can be written as a convex combination of (d−1)/2 number of rank-4 PPT- BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set of all PPT states in ℂd⊗ℂd, namely the Peres set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the (d−3)/2-simplex --is sitting on the boundary between the set and the set of non- PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.

Rigorous Quantum Limits on Monitoring Higher Order Quadratures of Quantum Fields

S. M. Roy HBCSE, Tata Institute of Fundamental Research, Mumbai 400 088 (Electronic address: [email protected])

Abstract: Quantum theory sets ‘rigorous quantum limits’ (RQL) on how accurately the position of a free mass or a harmonic oscillator may be monitored [1]. In view of theoretical and experimental progress in higher order squeezing of photon ﬁelds ([2],[4]) and their potential usefulness in gravitational wave detection ([5],[7], [10],[15],[6],[9],[11]) I present ‘rigorous quantum limits’(RQL) on monitoring higher order quadratures of quantum ﬁelds . I construct higher order coherent states (subspaces), higher order maximally contractive states (subspaces) and maximally expanding states (subspaces), all of which are distinct from the corresponding states for second order quadratures.

Consider the time development of the product of integer powers of Heisenberg field operators for different modes ‘r’ of the electromagnetic field (or any Bosonic field) ℿr(a푟(t)) ≡ (Q(t) + iP(t))/√2, where the Hermitian operators Q(t), P(t) are the corresponding higher order quadratures, with Q(0) ≡ Q, P(0) ≡ P. The positivity of a matrix built out of initial expectation

† values of products of powers of ar, a s constitute the generalized uncertainty relations which are analogous in spirit ,but diﬀerent from the constraints on higher moments of Wigner distributions considered before [3].The new uncertainty relations involve expectation values of products of number operators for each mode.

The Heisenberg equations of motion express σ2(Q(t)) =< (∆Q(t))2> and σ2(P(t)) =< (∆P(t))2 > in terms of σ2(Q),σ2(P) and < {∆Q,∆P} >, where ∆Q(t) = Q(t)− < Q(t) > ,∆P(t) = P(t)− < P(t) > , < A >≡ TrρA, and ρ is the initial state density operator. I identify the higher order coherent states as those for which σ2 (Q (t)) and σ2 (P (t)) are independent of time, the higher order maximally contractive states and higher order maximally expanding states as those for which < {∆Q, ∆P} > has respectively the minimum and maximum values permitted by the generalized uncertainty relations involving expectation values of products of number operators. The higher order coherent states correspond to classical motion. Higher order squeezed states are non-classical. The higher order maximally contractive and maximally expanding states may be considered to be maximally non-classical. Keywords: Higher order quadratures, higher order coherent states, higher order contractive states, higher order squeezed states, higher order uncertainty relations.

References : [1] S. M. Roy, ’Rigorous quantum limits on monitoring free masses and harmonic oscillators’, Phys. Rev. A 97, 032108 (2018). [2] C. K. Hong and L. Mandel , ’Higher-Order Squeezing of a Quantum Field’, Phys.Rev.Lett.54.323 (1985). [3] R. Simon and N. Mukunda, ’Moments of the Wigner Distribution and a Generalized Uncertainty Principle ’, arXiv:quaant-ph/9708037;J Solomon Ivan, N Mukunda and R Simon, ’Moments of non- Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case ’, Journal of Physics A: Mathematical and Theoretical, Volume 45, Number 19 (2012). [4] Sunil Rani, Jawahar Lal, and Nafa Singh,International Journal of Optics , ’Higher-Order Amplitude Squeezing in Six-Wave Mixing Process ’, Volume 2011, Article ID 629605 (2011). [5] B. P. Abbott et al (LIGO Scientific Collaboration) and Virgo Collaboration, Phys. Rev. Lett. 116,061102 (2016). [6] E.g. K. S. Thorne, R. W. P. Drever,C. M. Caves, M. Zimmermann, and V. D. Sandberg, Phys. Rev. Lett. 40,667,(1978);R. Weiss, in Sources of Gravitational Radiation, Editor L. Smarr (Cambridge University Press, Cambridge,1979); V. B. Braginsky,Y.I. Vorontsov and K. S. Thorne, Science 209 (4456)547 (1980); B. Abbott et al (LIGO Scientific Collaboration) New J. Phys.11,073032(2009); J. Abadi et al (LIGO Scientific Collaboration) Nature Physics 7 962 (2011) : S. L. Danilishin and F. Y. Khalili, Living Rev. Relativ. 15(1)5 (2012); Y. Ma et al, Nature Physics 13,776 (2017). [7] V. B. Braginsky and Yu. I. Vorontsov, Ups. Fiz. Nauk 114,41 (1974) [Sov. Phys. Usp. 17 ,644 (1975)] [8] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52,341(1980). [9] H. P. Yuen, Phys. Rev. Letters 51, 719 (1983). [10] C. M. Caves,Phys. Rev. Letters 54, 2465 (1985). [11] M. Ozawa, Phys. Rev. Letters 60, 385 (1988). See also, M. Ozawa,Ann. Phys. (N.Y.) 311(2),350 (2004), and Current Science 109(11),2006 (2015); J. P. Gordon and W. H. Louisell,in Physics of Quantum Electronics, Ed. P. L. Kelly et al, p.833 (McGraw-Hill,N.Y.1966 ). [12] E. Kennard, Zeitschr. Phys. 44 326 (1927); H. Weyl, Gruppentheorie und quantenmechanik (Hirzel, Leipzig, 1928); H. Robertson, Phys. Rev. 34, 163 (1929) .R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); R. Simon, E.C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); R. Simon and N. Mukunda, J. Opt. Soc. Am. A 17, 2440 (2000); Arvind, B. Dutta, N. Mukunda, R. Simon, Pramana 456, 471 (1995).

[13] E. Arthurs and J. L. Kelly Jr. ,Bell System Tech. J.,44(4),725(1965); P. Busch, T. Heinonen and P. Lahti, Physics reports 452(6),155((2007); P. Busch, P. Lahti, and R. F. Werner,Rev. Mod. Phys. 86(4),1261(2014); S. M. Roy, Current Science 109(11)2029(2015); [14] D. Walls, Squeezed states of light, Nature 306, 141 (1983); M. O. Scully and M.S. Zubairy,’Quantum Optics’, Cambridge Univ. Press (1997); C. Gerry and P. L. Knight , ’Introductory Quantum Optics’ Cambridge Univ. Press (2005); R. Loudon, ’The Quantum Theory of Light’ (Oxford University Press, 2000); D. F. Walls and G.J. Milburn, ’Quantum Optics’, Springer Berlin (1994); C W Gardiner and Peter Zoller, ”Quantum Noise”, 3rd ed, Springer Berlin (2004). [15] F. Y. Khalili et al, Phys. Rev. Lett. 105,070403 (2010). [16] Grote, H.; Danzmann, K.; Dooley, K. L.; Schnabel, R.; Slutsky, J.; Vahlbruch, H. (2013). ”First Long-Term Application of Squeezed States of Light in a Gravitational-Wave Observatory” (http://arxiv.org /abs/1302.2188). Phys. Rev. Lett. 110: 181101. arXiv:1302.2188 (https://arxiv.org/abs/1302.2188) ; J. Aasi et al, ’Enhancing the sensitivity of the LIGO gravitational detector by using squeezed states of light’, arXiv:1310.0383[uant-ph] ; Markus Aspelmeyer, Pierre Meystre, and Keith Schwab,’ Quantum optomechanics’, Phys. Today 65(7), 29 (2012); Goda, K. et al. “A quantum-enhanced prototype gravitational-wave detector”. Nature Physics 4, 472-476 (2008); Abadie, J. et al. “A gravitational wave observatory operating beyond the quantum shot-noise limit”. Nature Physics, 7, 962965 (2011); Harry, G.M. et al. “Advanced LIGO: the next generation of gravitational wave detectors”. Classical Quant. Grav. 27, 084006 (2010). [17] J. von Neumann, Chap. 6 , ’ Mathematical Foundations of Quantum Mechanics’ (Princeton University, Princeton, New Jersey, (1955). [18] C. M. Caves and G. J. Milburn, Phys. Rev. A 36, 5543 (1987);M. T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301 (1990);G. J. Milburn, K. Jacobs, and D. F. Walls, Phys. Rev. A 50, 5256 (1994); H. Mabuchi, arXiv:quantph/9801039v3 (1998). [19] Priyanshi Bhasin, Ujan Chakraborty and S. M. Roy (manuscript in preparation). [20] S. M. Roy and V. Singh, Phys. Rev. 25 D,3413(1982). [21] R. Simon, E.C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987) and Phys. Lett. A 124,223(1987); Arvind, B. Dutta, N. Mukunda, R. Simon, Pramana 456, 471 (1995) and arXiv:9509002; R. Simon and N. Mukunda, J. Opt. Soc. Am. A 17, 2440 (2000) R. Simon and N. Mukunda ,quant-ph/9708037.

Sougato Bose Exploring Non-Classical Behaviour with Nano-Scale Objects

Prof. Sougato Bose Programme Tutor (Postgraduate Taught), Department of Physics and Astronomy, University College London,

Abstract: Engineering quantum behaviour for macro-objects is one of the most important targets in contemporary physics -- not only to just verify "whether" it is extendable, but also the challenge, and the ramifications in practical applications such as quantum sensing. I am going to describe experimental proposals for how nano-scale and micro-scale objects can be made to manifest the quantum superposition principle and gravitationally mediated entanglement, and the implications such experiments for evidencing the quantum coherent behaviour of gravity. Further, we are going to show how macro-objects (in the sense of various trapped well isolated nano-scale mechanical oscillators) can be made to empirically violate our everyday notion of realism, namely the existence of well defined attributes before observation. This will be a mechanism to display non-classicality for a system with a well defined classical description, namely a harmonic oscillator, solely with coarse grained position measurements, even when it is initially prepared in a coherent state -- the most classical of all quantum states.

On linking continuous and discrete quantum phase space descriptions

Subhash Chaturvedi IISER Bhopal

Abstract: We discuss some crucial ingredients that go into providing a possible link between continuous and discrete quantum phase space descriptions which have hitherto been treated separately. Once established, this link would facilitate passage from structures natural to the continuum to their analogues in the discrete and conversely. Some preliminary work on Gaussian states is discussed as an example.

The Case for Foundational Quantum Science Tests Using Quantum Communication Satellites

Thomas Jennewein University of Waterloo

Abstract: Space based quantum communication technologies such as quantum key distribution has advanced into space, and other missions are imminent. It is therefore of great interest to consider the fundamental physics questions that can be studied or addressed using these systems. Physical theories are developed to describe phenomena in particular regimes, and generally are valid only within a limited range of scales. Direct tests of quantum theory and quantum entanglement have been performed at the smallest probable scales at the Large Hadron Collider, ∼10^(−20) m, up to hundreds of kilometers, with the current limit of about 1200 km. Yet, it is important to push direct tests of quantum theory to larger length scales, approaching that of the radius of curvature of spacetime, where we begin to probe the interaction between gravity and quantum phenomena. I will discuss potential tests of fundamental physics that are conceivable with artificial satellites in Earth orbit, and in particular provide an overview and the status on the proposed Canadian mission QEYSSat (Quantum Encryption and Science Satellite) which aims to place a quantum receiver in space.

The need for a classical Bohmian pointer: proposal for measuring the electron velocity using weak values from displacement currents

Xavier Oriols Department of Electronic Engineering, Universitat Autónoma de Barcelona, 08193-Bellaterra, Barcelona, Spain

Abstract: In this conference, I will revisit the Bohmian explanation of the measurement process emphasizing the need for a classical behaviour of the measuring pointer [1,2].

Typical single-particle models for the pointer leads to unphysical results. I will show that, on the contrary, a Bohmian pointer defined as the centre of mass of a many-particle (sub) system behaves classically and the previous single-particle misleading result simply disappears [2].

From the above conclusion, I will show that the Bohmian formulation of the measurement process for the high-frequency (TeraHertz) current of electrons in quantum devices can be understood as a weak measurement [3,4].This result opens new opportunities for fundamental and applied quantum physics using state-of-the-art electronic technology [3-5]. In particular, up to now, the attempts to measure local velocities with position-momentum weak values have been experimentally developed mainly for (relativistic) photons.

Inspired by the Ramo-Shockley-Pellegrini theorem that provides a relation between displacement current and electron velocity, I will show that the total current measured in electron devices can provide either a weak measurement of the momentum or strong measurement of the position [3]. By properly combining such measurements, I will present a proposal for a multi terminal electron device that can effectively measure the Bohmian velocity of (non-relativistic) electrons.

References:

[1] A. Benseny, G. Albareda, A.S. Sanz, J. Mompart, and X. Oriols, Applied Bohmian mechanics, European Physical Journal D 68, 286 (2014). [2] X. Oriols and A. Benseny "Conditions for the classicality of the center of mass of many-particle quantum states" New Journal of Physics, 19 063031 (2017). [3] D. Marian, N. Zanghí, X. Oriols "Weak Values from Displacement Currents in Multiterminal Electron Devices " Phys. Rev. Lett. 116, 110404 (2016). [4] F. L. Traversa, G. Albareda, M. Di Ventra, and X. Oriols "Robust weak-measurement protocol for Bohmian velocities" Phys. Rev. A 87, 052124 (2013). [5] X. Oriols, "Quantum-trajectory approach to time-dependent transport in mesoscopic systems with electron-electron interactions" Phys. Rev. Lett. 98, 066803 (2007).

Trajectories of a photon from an experimental viewpoint

Yaron Kedem Center for Quantum Materials, Nordita, Sweden

Abstract: Discussions regarding the fundamentals of quantum mechanics tend to be philosophical in nature. As physicists it can be good to have in mind what can be actually observed with the currently available technology. In this talk I will present two recently performed experiments that investigated a common theme, the trajectory of a photon, from different perspective. The first one [1] is based on a new measurement protocol, utilizing the frequency degree of freedom, which yields an operational definition for the trajectory of a single photon. The resulting trajectories are discontinuous and agree with a conceptual definition coming from the two-state vector formalism. The second experiment [2] depict the Bohmian trajectory of a photon that is entangled with another photon. The results clearly show how a measurement on the additional photon instantaneously change the trajectory of the first one. Looking at the two experiments side by side we have a vivid demonstration of the hard choice that quantum mechanics forces us to make: either a realistic trajectory with

nonlocal effects or a local theory with trajectories that do not comply with an intuitive (classical) notion of what a trajectory is.

References: [1] Zong-Quan Zhou, Xiao Liu, YK, Jin-Min Cui, Zong-Feng Li, Yi-Lin Hua, Chuan-Feng Li and Guang-Can Guo, “Experimental observation of anomalous trajectories of single photons”, Physical Review A 95 (4), 042121 [2] Ya Xiao, YK, Jin-Shi Xu, Chuan-Feng Li and Guang-Can Guo “Experimental nonlocal steering of Bohmian trajectories”, Optics express 25 (13), 14463-14472

Control of Optical Propagation Distance by Post-selection

Yutaka Shikano Keio University, Japan

Abstract: In the field of optical imaging and optical measurement, it is important to observe variation of optical field along propagation direction. However, to implement the measurement, optical components or measurement apparatus must be mechanically moved, in some cases, over a long distance. In this study, we propose and experimentally confirm new method to displace the optical wave packet along its propagation direction without changing apparatus position. Our method is based on “weak measurement” scheme, in which the polarization state and the wave packet position along propagation direction are weakly interacted followed by post-selection of the polarization state. The important point is that small back-and-forth shift of interaction can induce tens of times larger displacement of wave packet along propagation direction. By using our method, three-dimensional measurement of intensity distribution or image refocusing can be compactly implemented without mechanical movement. This work is collaboration with Hirokazu Kobayashi at Kochi University of Technology.

Yves Caudano Revisiting the quantum three-box paradox using weak measurements and the Majorana representation of the states on the Bloch sphere

Mirko Cormann1, Mathilde Remy1, and Yves Caudano1. 1Physics Department, Namur Institute for Complex Systems (naXys), University of Namur, Rue de Bruxelles 61, B-5000 Namur, Belgium

Abstract: Weak measurements [1] evidence complex numbers with unusual properties, called weak [1] and modular [2] values. The real and imaginary parts of these values have attracted most of the attention due to their direct connection to many experiments (notably, by predicting the deflection of post-selected particles in a Stern-Gerlach experiment, as in the original AAV paper). Our work focuses instead on their polar representation [3, 4]. For qubit systems, we relate the modulus to the visibility in an interferometric experiment, while we link the argument to solid angles on the Bloch sphere, which are associated to a geometric phase

26 similar to the Pancharatnam phase [5]. Using the Majorana representation [6], we map the states of higher-dimensional N-level systems to states of the symmetric subspace of N − 1 qubits. This way, we factor both the modulus and the argument of an arbitrary N-level modular value into N − 1 qubit contributions, each connected to the results obtained for qubit observables. In particular, the argument derives from a sum of N − 1 solid angles on the Bloch sphere, defined by the state evolution from pre- to post-selection.

We exploit the Majorana representation to revisit a well-know paradox previously studied by weak measurements: the quantum three-box paradox [7, 8]. This paradox deals with the question of determining in which box the particles were between pre- and post-selection. The Majorana representation of the equivalent 3-level quantum system allows us to reformulate the problem in terms of the symmetric states of a pair of qubits. In our two-particle version of the paradox, the particle pairs are pre- and post-selected in classical separable states but are necessarily found in entangled intermediate states when opening one amongst two of the three boxes. In our formulation, the boxes are indeed quantum, represented by projectors on maximally entangled Bell states. The occurrence of entanglement in the bipartite system is an unavoidable feature of the Majorana representation. We also show how the negative sign of the −1 weak value associated to one of the box projectors is linked to a geometric quantum phase defined on the Bloch sphere. We are currently setting up an experiment to investigate this formulation of the paradox using weak measurements of pairs of entangled photons interacting in a Hong-Ou-Mandel interferometer.

References: [1] Y. Aharonov, D. Z. Albert, and L.Vaidman, Phys. Rev. Lett. 60, 1351–1354 (1988). [2] Y. Kedem and L. Vaidman, Phys. Rev. Lett. 105, 230401 (2010). [3] M. Cormann, M. Remy, B. Kolaric, and Y. Caudano, Phys. Rev. A 93, 042124 (2016). [4] M. Cormann and Y. Caudano, J. Phys. A: Math. Theor. 50, 305302 (2017). [5] S. Pancharatnam, Proc. Ind. Acad. Sci. A 44, 247–262 (1956). [6] E. Majorana, Nuovo Cimento 9, 43–50 (1932). [7] Y. Aharonov and L. Vaidman, J. Phys. A: Math. Gen. 24, 2315–2328 (1991). [8] K. J. Resch, J. S. Lundeen, and A. M.Steinberg, Phys. Lett. A 324, 125–131 (2004).

CONTRIBUTED TALKS

Quantum constraints stronger than uncertainty relations

Arun Sehrawat Harish-Chandra Research Institute, Allahabad, India (arXiv: 1706.09319)

Abstract: Quantum constraints—are necessary and sufficient restrictions on the expectation values—originate from the three defining properties (Hermiticity, normalization, and positivity) of a quantum state and through the Born rule.

Summary: Introduction: The state for a d-level quantum system (qudit) can be described by a statistical operator휌, on the system's Hilbert space ℋ, such that [1–3] 휌 = 휌(Hermiticity), (1) tr(휌) = 1 (normalization), (2) 0≤ 휌 (positivity). (3) It has been shown in [4, 5] that an operator 휌 fulfills (3) if and only if it obeys 0≤ 푆 = (−1) tr(휌) 푆 (4) For all 1≤ 푛 ≤ 푑, commencing with 푆 = tr(휌). It is advantageous to use inequalities (4) between real numbersthan a single operator-inequality (3). Here we consider an experimental scenario—a finite number of independent qudits are identically prepared in a quantum state 휌, and then individual qudits are measured using different settings for operators 퐴, 퐵,⋯, 퐶—where all the expectation values 〈퐴〉, 〈퐵〉,⋯, 〈퐶〉are drawn from a single 휌.The goal is to find all the necessary and sufficient constraints that these expectation values must follow. An uncertainty relation (UR) [6–10] is a necessary, but not always a sufficient, condition. If we know the density operator 휌, then we can calculate all the average values by the Born rule [11]. 〈퐴〉 = tr(휌퐴) 〈퐵〉 = tr(휌퐵) (5) ⋮ 〈퐶〉 = tr(휌퐶)

The rule does not put any restriction on the mean values. The constraints on expectation values come from the three conditions (1), (2), and (4) on a statistical operator. We call such limitations quantum constraints (QCs) and present a systematic procedure to achieve them for any set of operators.

Method and Results: To transfer the conditions from a quantum state onto the expectation values, one needs to pick an operator-basis with which one can represent every operator. All bounded operators on ℋ form a 푑 -dimensional Hilbert-Schmidt space ℬ(ℋ)endowed withthe inner product⟦퐴, 퐵⟧ = tr(퐴 퐵), where 퐴, 퐵 ∈ ℬ(ℋ). Suppose 픅 = Γ , Γ ,Γ =δ , (6) ,

29 is an orthonormal basis of ℬ(ℋ)[4, 5, 12], where δ, is the Kronecker delta function. Now we can resolve every operator 퐴 ∈ ℬ(ℋ)in basis 픅 as [2]

퐴 = a Γ (7) where a = Γ,A are complex numbers. In this way, 휌 is fully characterized by 푑 complex numbers r = Γ,ρ. Furthermore, we can express the expectation value

〈퐴〉 = tr(휌퐴) = ⟦ρ,A⟧ = r̅ a (8) as the standard inner product between two complex vectors made of r's and a's. As a result, we can combine all the equations in (5) into a single matrix equation,

〈퐴〉 a a a r̅ ⋯ ⎛〈퐵〉 ⎞ b b b r̅ = ,(9) ⎜ ⋮ ⎟ ⋮ ⋱⋮ ⋮ 〈퐶〉 cc⋯ c r̅ ⎝⎠ which is the numerical representation of Born's rule in basis (6).All the information about preparation and measurement settings go into the column R and the matrix M, respectively. The three conditions (1),(2), and (4) on 휌enter through R as described next and emerge as QCs on the expectation values in E. Condition (1) can be expressed in terms of r's as r = 〈Γ 〉 = 〈Γ〉 for all 훾 (10) For (2) and (4), we need to compute

( ) ( ) tr 휌 = ⋯ r ⋯r tr(Γ ⋯Γ ) 11

for every1≤ 푚 ≤ 푑.In the case of a standard operator-basis {|푗⟩⟨푘|},, tr(휌 ) can be achieved by multiplying the density matrix 푚 times and then taking the trace.In (11), one can observe thattr(휌)is a polynomial of degree 푚, where the expectation values (10) are variables. Consequently, 0≤ 푆[see (4)] leads to a 푛-degree QC. If 퐴, 퐵,⋯, 퐶are ℓ Hermitian operators that represent observables, then the set of expectation values [for E, see (9)] ℰ = {E| 휌 obeys (1), (2),and (4) } (12) will be a compact and convex set inℝℓ.A point outside ℰ does not come from any quantumstate, whereas every point in ℰ corresponds to at least one quantum state, so as a whole it is the only allowed region. Therefore, QCs obtained through the above formalism are optimal. One cannot achieve a region smaller than ℰ without sacrificing a subset of quantum states. For a qubit (푑 =2), ℰ is always bounded by an ellipsoid [13]. If one defines a suitable concave function on ℰ to quantify a combined uncertainty for 퐴, 퐵,⋯, 퐶, then the global minimum of the function will be at the extreme points of ℰ. Each extreme point of ℰcorresponds to a pure state that can be parameterized with 2(푑 − 1)real numbers [3].In this way, producing a tight UR becomes an optimization problem, where at most 2(푑 − 1)parameters are involved [16].

Through various examples, we have shown that a region ℛ bounded by a tight UR always contains ℰand generally bigger than ℰ. It implies that there are points in ℛ (but not in ℰ) that satisfies a tight UR but there exists no quantum state corresponding to them. In that sense URs cannot be better than the QCs that originate from (1), (2), and (4) via the matrix equation (9).

Conclusion: We provided a systematic procedure to obtain QCs for any set of operators that act on a qudit's Hilbert space. To attain QCs, one does not need any uncertainty measure such as the standard deviation, entropy, etc. One can derive them straight from “Hermiticity, normalization, and positivity” of a statistical operator through the Born rule by employing an operator-basis. Moreover, QCs do not depend on the choice of basis, so one can make a selection according to own convenience. QCs are optimal as they characterize only the allowed region ℰ in a real space for a set of observables.A point outside ℰ cannot be associated witha quantum state, even if it satisfies a tight UR. It reveals that even a tight UR is a necessary, but not generally a sufficient, condition on the expectation values. In the corresponding paper, the case of a single qubit is thoroughly investigated that includes Schrodinger's UR [7] and symmetric informationally complete positive operator valued measure (SIC-POVM) [15]. Furthermore, QCs for spin-1 operators, the Weyl operators [12], and mutually unbiased bases [14] are reported. In addition, a straightforward mechanism—that is also employed in [16]—is presented with which one can create a tight UR. Various tight URs are obtained for spin-풿observables, where 풿can be ,1, ,2,⋯.

References: [1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955). [2] U. Fano,Rev. Mod. Phys. 29, 74 (1957). [3] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006). [4] G. Kimura,Phys. Lett. A 314, 339 (2003). [5] M. S. Byrd and N. Khaneja,Phys. Rev. A 68, 062322 (2003). [6] H. P. Robertson, Phys. Rev. 34, 163 (1929). [7] E. Schrodinger, Proceedings of the Prussian Academy of Sciences XIX, 296 (1930). [8] H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988). [9] P. Busch, T. Heinonen, and P. Lahti,Phys. Rep. 452, 155 (2007). [10] P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner,Rev. Mod. Phys. 89, 015002 (2017). [11] M. Born, Z. Phys. 37, 863 (1926); English translation in J. A. Wheeler and W. H. Zurek, eds., Quantum Theory and Measurement(Princeton University Press, Princeton, 1983). [12] J. Schwinger, Proc. Natl. Acad. Sci. U. S.A. 46, 570 (1960). [13] J. Kaniewski, M. Tomamichel, and S. Wehner,Phys. Rev. A 90, 012332 (2014). [14] T. Durt, B.-G. Englert, I. Bengtsson, and K. Zyczkowski, Int. J. Quantum Inf. 8, 535 (2010). [15] J. Rehacek, B.-G. Englert, D. Kaszlikowski, Phys. Rev. A 70, 052321 (2004). [16] A. Riccardi, C. Macchiavello, and L. Maccone, Phys. Rev. A 95, 032109 (2017).

Flow ambiguity: A path towards classically driven blind quantum computation

Atul Mantri,1, 2 Tommaso F. Demarie,1, 2 Nicolas C. Menicucci,3, 4 Joseph F. Fitzsimons1, 2

1Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 2Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore 117543 3School of Science, RMIT University, Melbourne, Victoria 3001, Australia 4School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia

Abstract: In this work, we take the first step towards the construction of a blind quantum computing protocol with a completely classical client and a single quantum server.

Summary: Introduction: It is very likely that when a universal quantum computer will finally become available, it will be hosted by large institutions and accessed remotely by clients. For example, companies like Google, IBM, and Rigetti, as well as academic institutions including the University of Bristol, have begun making their quantum devices available for remote access. [1] This situation will inevitably lead to questions as to the integrity and privacy of the client's computation. In the past, quantum protocols have been proposed to address similar problems. Protocols which provide security of client's quantum computation, as well as input and output, are known as blind quantum computing protocols [2]. Similarly, protocols which capture the idea of verification of quantum computing, i.e., the ability to detect, with very high probability, any attempt by a malicious server to deviate from the computation are known as verifiable quantum computing protocols [3]. A common feature among known protocols for these tasks is that either the client require a small quantum device on their side or there must exist at least two non-communicating quantum servers. In other words, there is a requirement that two or more parties involved in the protocol possess quantum processors. Ideally, we would like to have a secure delegated quantum computing protocol between a completely classical client and a quantum server. In this work, we take the first steps towards this problem. We construct a blind quantum computing protocol which maintains security of the client's computation even against the quantum server. In the next section, we briefly describe our main ideas and results, but the full details can be found in [4].

Main Ideas and Results: Our aim is to explore the possibility of blind quantum computation with a purely classical client. We demonstrate this fact by constructing a protocol for a task we call as classically driven blind quantum computing (CDBQC) and analyse its security in the stand-alone setting. Our protocol uses measurement-based quantum computing (MBQC) [5] as the underlying principle. We show that the protocol allows a client to hide non-trivial information about their computation from the powerful quantum server by making use of a novel technique that we call flow ambiguity. In particular, we analyse the case of a single instance of the protocol and show that the amount of information obtained by the server is bounded below what is necessary to unambiguously distinguish the computation.

In the MBQC framework we denote by 훥 the computation of the client such that 훥 = {퐺, 훼, 푓}. Here 퐺 denotes the graph state, 훼 is the set of measurement angles on the graph

32 state, and 푓 represents the generalised information flow [6] which captures how angles are to be adapted based on results of previous measurements. Intuitively, generalised information flow or g-flow is used to assign a set of local corrections to a subset of unmeasured qubits to ensure deterministic computation, despite the random nature of the measurement outcomes obtained during the computation. It is important to note that for a fixed graph there exist multiple choices of the input and output vertex sets that result in deterministic measurement patterns consistent with the same fixed total ordering of vertices. Specifically, we show that the transcript of any run of the protocol is consistent with multiple non-equivalent computations. This is due to the fact that the information about the g-flow for the underlying resource state is hidden from the server. This particular ambiguity in the flow enables the classical user to hide the essential aspects of the computation.

The CDBQC protocol is interactive and proceeds as follows. Firstly, the client sends the dimension of the graph to the server to prepare the graph state |퐺⟩. At each step 푖: the client chooses a bit 푟 uniformly random. Using 푟 and the previous measurement outcome 푏 , client updates the angle 훼 to construct 훼 in the following way: 훼 = (−1) 훼 +(푟 ⊕ 푠)휋 where 푠 and 푠 denote the corrections dictated by flow based on previous measurement results. The server performs a projective measurement of 푖-th vertex in the XY-plane of the Bloch sphere, denoted 푀 = {|± ⟩⟨± |}, where {|± ⟩} = (|0⟩ ± 푒 |1⟩) and sends √ the measurement outcome 푏 to the client. The client records 푏 = 푏 ⊕ 푟 in 푏 and then updates the set (푠 , 푠 ). If the 푖-th vertex is output qubit then the bit 푏 is registered in the set 푝. The client and the server repeat this procedure for all the vertex of the graph in the given total order. The client implements the final round of corrections on the string 푝 (equivalent to 푝 in the case of honest server) to obtain the output string 푝. At the end of the protocol, the server possesses information about the angles 훼 and the measurement outcome 푏 and whereas the client's secret consists of the actual measurement angle 훼 and the flow bits 푓. We will denote the variables with the upper-case letters and particular instances of such variable with the lower-case letters.

We quantify the amount of information that on average remains hidden from the server about the client's computation at the end of the protocol. We also derive a non-trivial lower bound on the conditional entropy by calculating the total number of information flows. Note that flow is a property of the underlying graph and therefore depends on the chosen graph 퐺. We will consider the case of cluster states as they are known to be universal for quantum computation with (X, Y)-plane measurements [7]. For a generic cluster state 퐺(,) we show that there exists at least an exponential number (in the dimension of the graph) of information flows corresponding to a cluster state for a given total order of measurements.

Therefore, the conditional entropy is given by 퐻(퐴, 퐹|퐵, 퐴) ≥ 1.388푁. where 푁 = 푛푚. To demonstrate this we take a simple example of a 2×2 cluster state 퐺(퐼, 푂)(×) in the Figure 1. The figure shows 9 possible open graphs compatible with the flow conditions. In general different flows correspond to different computations.

FIG. 1. All the possible 퐺(퐼,푂), combinations that satisfy g-flow conditions for the 2×2 cluster state are shown. The arrows indicate the direction of the quantum information flow. Note that overlapping input and output sets are allowed. All patterns implement unitary embeddings on the input state..

This shows that it is indeed possible for a client to hide their chosen computation, by using the ambiguity in the flow of information, from a quantum server. Importantly, we show that it is not possible for the quantum server to guess the client's computation perfectly, since a large number of other computations are still compatible with the information server receives. For more details, we refer to the published version of this work [4] and references therein.

References: [1] https://en.wikipedia.org/wiki/Cloud-based quantum computing [2] Fitzsimons, J. F. (2017). Private quantum computation: an introduction to blind quantum computing and related protocols. npj Quantum Information, 3(1), 23. [3] Gheorghiu, A., Kapourniotis, T., & Kashefi, E. (2017). Verification of quantum computation: An overview of existing approaches. arXiv preprint arXiv:1709.06984. [4] Mantri, A., Demarie, T. F., Menicucci, N. C., & Fitzsimons, J. F. (2017). Flow ambiguity: A path towards classically driven blind quantum computation. Physical Review X, 7(3), 031004. [5] Raussendorf, R., & Briegel, H. J. (2001). A one-way quantum computer. Physical Review Letters, 86(22), 5188. [6] Browne, D. E., Kashefi, E., Mhalla, M., & Perdrix, S. (2007). Generalized ow and determinism in measurement-based quantum computation. New Journal of Physics, 9(8), 250. [7] Mantri, A., Demarie, T. F., & Fitzsimons, J. F. (2017). Universality of quantum computation with cluster states and (X,Y)-plane measurements. Scientific Reports, 7.

Fifty shades stronger: variance based sum uncertainty and reverse uncertainty relations for arbitrary quantum states and all incompatible observables

Chiranjib Mukhopadhyay, Arun Kumar Pati

Quantum Information and Computation Group, Harish Chandra Research Institute, Homi Bhabha National Institute, Allahabad 211019, India

Abstract: Several stronger sum-uncertainty and reverse uncertainty relations are derived for qudits, and connection with other quantum resources elucidated. They imply a reverse quantum speed-limit and a lower bound for fidelity.

Summary: Triviality of uncertainty relations in the textbook Heisenberg-Robertson form, even for non-commutating observables has been addressed in several recent works, pioneered by [3]. In the present work, we provide several strengthenings to such uncertainty relations.

We introduced the quantity uncertainty matrix, a positive definite operator which is denoted as Kand defined for observables A and B as K=(C±iD)(C∓ iD), (1)

Where, 퐶 = 퐴 − ⟨퐴⟩, 퐷 = 퐵 − ⟨퐵⟩.. It turns out for the (multi-) qubit case that the purity of the uncertainty matrix captures the incompatibility of the observables A and R. The proof is analytic in the limit of highly mixed states and numerically demonstrated for arbitrary qubit states.

The very first uncertainty relation we derive is the following - (First UR)- For arbitrary pure state |휓> and observables A and B, if the corresponding uncertainty matrix is denoted by K, then the following variance based sum uncertainty relation holds for all pure states |휙 >

∆퐴 + ∆퐵 ≥ |⟨[퐴, 퐵]⟩| + (2) ⟨⟩|⟩ (( )) where 휃 is the inner product angle between|휓⟩ and |휙⟩, K휔 being the weak value of the observable K with pre-selected and post-selected states |휙⟩ and |휓⟩ respectively, and 훼 =

휆⁄휆 where 휆 and 휆 are, respectively, the maximum and minimum eigenvalues of the observable K. Optimizing over all pure states can saturate this bound.

Next we turn towards generalizing the stronger uncertainty relation derived in [3] for arbitrary mixed states. Using vectorization technique, we derive the following extension - (Second UR)- If |푘 > is a (normalized) state vector perpetidicular to the vectorization k) of a density matrix k, the following variance based sum uncertainty relation holds-

∆퐴 + ∆퐵 ≥ ±푖⟨[퐴, 퐵]⟩ + < 휌훪⨂(퐴 ± 푖퐵)| 휌 >| (3)

As another application of the vectorization technique, we derive an exact equality relation valid for arbitrary qubit states, which links quantifiers of four quintessential quantum resources, viz. coherence baumgratz, entanglement, purity [6], and imaginarity [2] - which may be of independent interest.

The stronger uncertainty relations above are all subject to optimization and hence, it would be desirable to have a stronger uncertainty relation free from such optimization. To this end, we prove the following uncertainty relation. (Third UR) - For arbitrary mixed state p and observables A and R, if the corresponding uncertainty matrix is denoted by K, then the following optimization-free variance based sum uncertainty relation holds -

∆퐴2 + ∆퐵2 > |⟨[퐴, 퐵]⟩| + 푆(휌) − 푙푛 푡푟 (푒 − 퐾) , (4)

Where, S(휌) is the von Neumann entropy of the quantum state p. When we consider the uncertainty relations as a pillar of quantum mechanics, it is desirable to separate out explicitly two contributions to the uncertainty one due to classical stochasticity, the other due to intrinsic quantum fluctuations. The uncertainty lower bound in the (4) above can be thought of as sum of three distinct contributions. The first being that due to inherent non-commutativity of quantum mechanics, the second entirely due to the classical randomness introduced via mixing, and the third representing, in some sense, a state- independent contribution to the uncertainty. While uncertainty relations guarantee the existence of intrinsic fluctuations in quantum theory, they usually do not let us estimate an upper bound on such fluctuations. Thus, the problem of devising reverse uncertainty relations is one of considerable theoretical interest [5,4]. Below we formulate such a reverse uncertainty relation for observables. Other such relations have also been found [1]. (Reverse UR)-For arbitrary mixed state p and observables A and B, if the corresponding uncertainty matrix is denoted by K =휆휎, where휆 = tr (K), the following variance based reverse sum uncertainty relations hold.

∆퐴2+∆퐵2 ≤ |⟨[퐴, 퐵]⟩| + 휆ℱ2(휌, 휎) ≤ |⟨[퐴, 퐵]⟩| + 휆 (1 — 푆( 휌 || 휎 )) (5)

Where ℱ is the fidelity between two quantum states 휌 and 휎. and S( 휌 || 휎) the relative entropy distance between them. The technique used in establishing these relationships, can now be applied in finding an upper bound (as opposed to the lower bound, i.e., quantum speed limit) on evolution time r for Markovian dynamics if certain conditions are met.

휏 ≤ [(푠푖푛퐿휏)⁄ (훬 푟푒푣푒푟푠푒)] (6)

∓ (()) Where 훬 푟푒푣푒푟푠푒 = ∫ 푑푡 [푆(휌 ) − ln 푡푟 푒 ] ,ℒT is the Bures angle between the initial and final state, and LT is the generator of the dynamics. Furthermore, based on the reverse uncertainty relations furnished here, it is possible to estimate a lower bound on fidelity between two states.

We look forward to work on tasks and applications on these uncertainty relations beyond the examples presented here [1].

References: [1] C. Mukhopadhyay, A. K. Pati, (upcoming). [2] A. Ilickey and C». Gour. Quantifying the Imaginarity of Quantum Mechanics. ArXiv e-prints, January 2018. [3] Lorenzo Maccone and Arun K. Pati. Stronger uncertainty relations for all incompatible observables. Phys. Rev.J.eff., 113:260401, Dec 2014. [4] DebasisMondal, ShrobonaBagchi, and Arun Kumar Pati. Tighter uncertainty' and reverse uncertainty' relations. Phys. Rev. A, 95:052117, May 2017. [5] Zbigniew Puchala, t.ukasz Rudnicki, Krzyszt of Chabuda, Mikolaj Paraniak, and Karol Zyczkowski. Certainty' relations, mutual entanglement, and nondisplaceable manifolds. P/zys.Rev. A, 92:032109, Sep 2015. [6] J A. Streltsov, H. Kampermann, S. Wolk, M. Gessner, and D. BruG.Maximal Coherence and the Resource Theory of Purity'.ArXiv e-prints, December 2016.

Simultaneous correlations in complementary bases as quantitative resource for quantum steering

C. Jebaratnam1, Aiman Khan2, Som Kanjilal3 and Dipankar Home3

1S. N. Bose National Centre for Basic Sciences, Kolkata 700 098, India 2 Indian Institute of Technology, Roorkee, India 3Centre for Astroparticle Physics and Space Science (CAPSS), Bose Institute, Kolkata 700 091, India (E-mail: [email protected])

Abstract: A suitable measure of simultaneous correlations in complementary bases is identified as the relevant quantitative resource for quantum steering in two- and three- settings scenarios pertaining to Bell diagonal states.

Summary: Setting the stage and key results: Einstein-Podolsky-Rosen (EPR) quantum steering has been the subject of much investigation since the reformulation of the original Schrodinger-EPR argument by Wiseman et al [1]. Besides the foundational implications of this work, steerable states have demonstrable usefulness in the quantum information processing (QIP) applications such as one-sided device independent (DI) quantum cryptography, randomness generation and subchannel discrimination [2-4], giving rise to the need for a quantitative measure of steering. For this purpose, several measures of quantum steering were suggested.

One such measure of EPR steering, derived using linear steering inequalities in the two- and three-settings scenarios, recently proposed by Costa and Angelo (CA) [5], was for the first time amenable to a closed analytical expression for all bipartite qubit states. CA also showed that the steering measure provided a lower bound to the entanglement of the state as measured by its negativity. This raises the important question as to whether a higher amount of entanglement of the quantum state necessarily guarantees a higher amount of steerability, an issue that has yet remained uninvestigated.

In this work, we first probe the above mentioned question through a comparative study of negativity versus the steerability measures for both two- and three-settings using Bell diagonal states. Interestingly, this study reveals that a higher degree of entanglement does not in general imply a higher measure of quantum steering in a given steering scenario. This in turn begs the question as to what measure of correlations inherent in the quantum state would be an appropriate quantitative resource for quantum steering.

We answer this question for the case of Bell-diagonal states by invoking the measures of complementary correlations in mutually unbiased bases that have been recently proposed in Ref. [6]. The persistence of correlations in sets of mutually unbiased bases, as measured by the Holevo quantity when the quantum state is subjected to measurement in each of these bases, is a fundamental quantum feature as emphasised in [6]. The measure itself has been quantified by choosing the bases that optimise the simultaneous correlations, and analytical expressions for the measure have been obtained for a wide class of bipartite qubit states, including Bell diagonal family of states.

For the two-setting steering scenario, we show that there exists an analytical relationship between the steering measure and a measure of simultaneous correlations in two mutually unbiased bases(MUBs), referred to as C2. For the three settings case, the steering measure bears an analytical relationship with a similarly defined measure of simultaneous correlations in three MUBs, referred to as C3. To summarise, the significant features of this work are as follows: ● While it has been argued that the violation of steering inequalities quantifies certified entanglement in terms of a lower bound to negativity [5], we find that the amount of quantum steering quantified in terms of the violation of the steering inequalities is not a monotonic function of negativity, thereby showing that entanglement is not an appropriate quantitative resource for steering even though it is necessary to ensure steerability of states. ● Next, we identify an appropriate quantitative resource for quantum steering by demonstrating that the steering quantifier is analytically and monotonically related to the amount of simultaneous correlations in mutually unbiased bases as quantified by C2 and C3 (defined in [6]). ● Thus, this identification of what aspect of correlations embodied in the bipartite quantum state plays a key role in quantum steering would facilitate exploring its foundational implications as well as its applications in quantum information processing.

References:

[1] Wiseman, H. M., Jones, S. J., & Doherty, A. C. (2007). Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Physical review letters, 98(14), 140402.

[2] C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani and H. M. Wiseman, One-sided device- independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301(R) (2012). [3] Y. Z. Law, L. P. Thinh, J.-D. Bancal, and V. Scarani, Quantum randomness extraction for various levels of characterization of the devices, J. Phys. A 47, 424028 (2014). [4] M. Piani and J. Watrous, Necessary and Sufficient Quantum Information Characterization of Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 114, 060404 (2015). [5] A. C. S. Costa and R. M. Angelo, “Quantification of Einstein-Podolski-Rosen steering for two- qubit states,” Phys. Rev. A 93, 020103 (2016). [6] Y Guo and S Wu, “Quantum correlation exists in any nonproduct state,” Sci. Rep 4, 7179 (2014), Matthew F. Pusey, “Negativity and steering: A stronger Peres’ conjecture,” Phys. Rev. A 88, 032313 (2013).

Preparation contextuality as the fundamental resource in quantum communication

Debashis Saha1, Anubhav Chaturvedi1, Pawel Horodecki2, Marcin Pawlowski1

1Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland 2Faculty of Applied Physics and Mathematics, National Quantum Information Center, Gdansk University of Technology, 80-233 Gdansk, Poland

Abstract: Any advantage in oblivious communication over classical channel implies preparation contextuality. We show almost all quantum communication complexity advantage and nonlocality attribute to advantage in oblivious communication.

Summary: Oblivious communication (OC) task:---The task involves a sender (Alice) and a receiver (Bob).In each round of the one-way communication tasks Alice receives an input x from a set {1,...,|x|} and send a message (classical or quantum) to Bob. Bob receives another input y from a set {1,...,|y|} and returns an outcome z which is supposed to equal to f(x, y) whose range is binary {0,1}. The obtained inputs x, y are uniformly distributed. Here, the Alice's input x comes through a local channel described by the conditional probability p(x|w) where w is the input variable of that channel. The communication is unbounded. The only condition is the information about w, referred as the oblivious variable, should not be revealed in the communication including Bob. Let p(z|x, y) represents the probability of giving an output z for given input x, y. Any figure of merit (the success probability of guessing f(x, y)) of the communication problem can be considered as any linear function of these observed probabilities. It is more convenient to define such that it takes value within [0, 1]. Hence, the figure of merit is considered as the average success probability S with some weightage. Now, the optimal value of S depends on the communication channel whether it is classical or quantum. We provide a methodology to obtain the optimal classical bound of S.

Fig 1. Oblivious communication problem between a sender (Alice) and a receiver (Bob) who receive some input x and y respectively.

The classical message can take arbitrary large distinct levels. Let 푝(휏|푥) is the probability of sending level 휏 for input x describing Alice's encoding strategy. The oblivious condition implies Alice's encoding strategy should be such that for all 휏, 푝(휏|푤) is independent of w. While for quantum channel, Alice sends a quantum state which belongs to some Hilbert space of any dimension. The oblivious condition demands the effective quantum state given w is same for all w.

Advantage in oblivious communication over classical channel implies preparation contextuality:---The notion of preparation contextuality (PC) was proposed by Spekkens[1]. We point out that the classical bound for any oblivious communication problem is same as for any preparation noncontextual (PNC) theory. Thus S <= Sc is PNC inequality where Sc is the optimal classical value. This generalizes the result obtained for parity oblivious multiplexing [2].

Quantum advantage in communication complexity implies advantage in OC:---Quantum resources and strategies have extensive implications and applications in the field of communication complexity (CC) [6]. We construct oblivious communication (OC) tasks tailored to one-way CC problems. The maximal classical success probability in the OC tasks forms our PNC inequalities. We show that, given a set of states and measurements which provide an advantage in general CC problems, the dual states and measurements (or the states with their orthogonal mixtures and the same measurements) yield an advantage in OC tasks and violation of the PNC inequalities. We find a criterion for unbounded violation of the obtained PNC inequalities and demonstrate the same for two widely studied CC problems.

Any quantum violation of a Bell inequality implies advantage in a OC task:---While Bell- inequality violation (BV) which employ space like separation between Alice and Bob reveal Bell non-locality (NL) of the underlying ontology [7] the BV that employ time-like separation reveal the impossibility of a non-retro causal time-symmetric ontology (NT) given there is an operational time-symmetry [8]. Heuristically, for any Bell experiment an OC task can be constructed porting BV to an advantage in the OC task. For the space-like separated scenario the collapsed state on Bob's end is prepared and sent in the OC task and for the time- like separated case the pre-measurement state at Bob's end is prepared and sent in the OC task. This would make all BV operationally reveal PC. However, while deterministic encoding strategies yield the bound on Bell inequalities, the PNC bound on the success parameter of the OC task might spring from probabilistic encoding schemes [3]. A rather inadequate attempt to prove the above thesis was made in [5] as it explicitly assumed deterministic encoding schemes for the constructed OC task. We provide the complete proof for the thesis.

Conclusions:--- The main results are depicted in the figure below. In addition to these implications we know other strong evidence to support the tentative assertion that PC forms the most fundamental non-classical feature underlying quantum advantage.

Fig 2.A: pictorial depiction of our operational (the dotted box and small arrows) and ontological (outside the dotted box and large arrows) implications. Entanglement assisted, prepare and measure quantum communication complexity advantage (CC) implies advantage in oblivious communication task OC subject to some condition. The ontic features of non-locality NL and impossibility of a non- retro causal time-symmetric ontology NT are revealed by spatial and temporal quantum Bell inequality violation BV respectively, which strictly implies an advantage in OC tasks. The advantage in OC tasks reveals PC. These implications inherently assume operational quantum formalism.

The abstract is based two papers [3, 4].

References: [1] R. W. Spekkens, Phys. Rev. A 71, 052108 (2005). [2] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009). [3] D. Saha, P. Horodecki, and M. Pawlowski, arXiv:1708.04751 [quant-ph]. [4] D. Saha, and A. Chaturvedi, coming soon. [5] A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane, Phys. Rev. Lett. 119, 220402 (2017).[6] G. Brassard, Foundations of Physics 33, 1593 (2003). [7] J. S. Bell,Rev. Mod. Phys. 38, 447 (1966). [8] M.S. Leifer, and M. Pusey, Proc. Roy. Soc. A. 473, 2202 (2017).

Debmalya Das Quantum heat engine using Stirling cycle and resources of ignorance

George Thomas*, Debmalya Das** and Sibasish Ghosh*

*Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India **Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract: We use a modified quantum Stirling cycle to extract work at the Carnot-efficiency in a limiting case with distinguishable particles, fermions and bosons as working medium, without gaining information about the particles’ position.

Summary: Background: The idea of the Maxwell’s demon occupies a central position in the understanding of thermodynamics and information.

It was introduced in a thought experiment that envisaged a situation in which there could be a possible violation of the second law of thermodynamics [1]. A classical analysis of the Maxwell demon was first developed in [2], in the form of the Szilard engine. In a Szilard engine, which is an enclosed chamber with a single gas molecule, a partition is introduced in the middle. The demon measures the molecule to the left or to the right of the partition. The partition is then allowed to move in the opposite direction of the molecule isothermally while in equilibrium with a heat bath at a temperature T. This results in an isothermal expansion of the gas. The presence of the demon therefore enables the Szilard engine to perform work of magnitude kTln2 with a decrease of the entropy of the bath given by kln2. This apparently leads to a violation of the second law of thermodynamics as an equivalent increase of entropy is required somewhere in the global system. In [2] it was suggested that the same amount of work is expended in the measurement of the position of the gas molecule which saves the second law. Finally in [3] it was shown that the work done in the erasure of information in the demon’s memory must be taken into consideration and no role was played by measurement. This idea was supported and further strengthened in [4].

The quantum analysis of Szilard engine was carried out in [5]. It was shown that unlike its classical counterpart, the introduction and the removal of the partition involves certain amount of work and heat exchange. This immediately sets a quantum Szilard engine different from the classical one. Compared to the compression of the particle to the left (or right) side of the box, the insertion of the of the barrier needs less amount of work. This is because in an insertion scenario, the position of the particle is unknown. One has to perform a measurement after the insertion to find the position of the particle. Therefore, the state of the system after compression is equivalent to the state of the system, after insertion followed by the projective measurement. Similarly, during the removal process, the particle is delocalized due to tunneling, a factor that does not come into play during expansion. Hence the extractable work during the removal process is less compared to that obtained during expansion. There is an element of lack of knowledge due to degeneracy and tunneling which causes a difference in the amount of work. The work extracted from a quantum Szilard engine was calculated for an arbitrary number of particles. It was shown that for different working medium such as fermions, bosons and classical particles, the work extracted is different. The performances

42 were also compared in the low temperature limit and the high temperature limit. The method was further used in analyzing Otto cycle with single atom heat engines [6].

Motivation: The main motivation for this work is to design a quantum heat engine which works exclusively on quantum features. In earlier works, a measurement is needed to extract work. Our analysis shows an effective way of converting lack of information to useful work without a measurement. The amount of extractable work depends on the nature of the working medium such as classical particles, bosons and fermions. The cycle we use is a modified version of the Stirling cycle. Quantum versions of Stirling engines have been studied in recent past [7-10]. High temperature limit of a Stirling engine whose working medium is harmonic oscillator have been studied [8].

Results: We analyze the cases of the heat engine using modified Stirling cycle, with single and multiple particles and calculate the corresponding the work extracted. The effects of using different working media like classical particles, fermions and bosons are also studied. We further explore the cases in which multiple partitions are inserted and removed, as a part of the cycle and calculate the work extracted. Our study of the modified Stirling engine in the low temperature limit shows that it operates at the Carnot efficiency. In the high temperature limit, the engine produces very low extractable work.

References: [1] J. C. Maxwell, Life and Scientific Work of Peter Guthrie Tait, edited by C. G. Knott (Cambridge University Press, London, 1911). [2] L. Szilard, Zeitschrift für Physik 53, 840 (1929). [3] R. Landauer, IBM Journal of Research and Development 5, 183 (1961). [4] C. H. Bennett, International Journal of Theoretical Physics 21, 905 (1982). [5] S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys. Rev. Lett. 106, 070401 (2011). [6] D. Gelbwaser-Klimovsky, A. Bylinskii, D. Gangloff, R. Islam, A. Aspuru-Guzik, and V. Vuletic, ArXiv e-prints (2017), arXiv:1705.11180 [quant-ph]. [7] H. Saygin and A. Şişman, Journal of Applied Physics 90, 3086 (2001). [8] G. S. Agarwal and S. Chaturvedi, Phys. Rev. E 88, 012130 (2013). [9] X.-L. Huang, X.-Y. Niu, X.-M. Xiu, and X.-X. Yi, The European Physical Journal D 68, 32 (2014). [10] G. Thomas, M. Banik, and S. Ghosh, Entropy 19 (2017).

Coupled systems as thermodynamic machines: efficiency, work and quantum correlations

George Thomas, Manik Banik and Sibasish Ghosh

Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, C. I. T. Campus, Taramani, Chennai 600113, India

Abstract: We study coupled spins and coupled oscillators as quantum thermodynamic machines. Their performances are compared. The bounds of the efficiency and the role of quantum correlations in the optimal work are also analyzed.

Summary: Extending thermodynamics to quantum regime, widely known as quantum thermodynamics, has attracted wide attention in the recent years. Quantum thermodynamic machines are found to be efficient probes to study in such direction. Here we study coupled spins and coupled oscillators as quantum heat engines and refrigerators [1]. Under a phase space transformation, the coupled system can be expressed as the composition of two independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. Further, we study two explicit examples; coupled spin-1/2 systems and coupled quantum oscillators with analogous interactions.

Interestingly, for particular kind of interactions, the efficiency of the coupled oscillators outperforms that of the coupled spin-1/2 systems when they work as heat engines. However, for the same interaction, the coefficient of performance behaves in a reverse manner, while the systems work as the refrigerator. Thus, the same coupling can cause opposite effects in the figures of merit of heat engine and refrigerator. We also show that quantum correlations are not helpful to extract optimum work from a heat engine.

References: [1] G. Thomas, M. Banik and S. Ghosh, Implications of Coupling in Quantum Thermodynamic Machines, Entropy 2017, 19(9), 44

Kumar Shivam Lorentzian Geometry for detecting qubit entanglement

Joseph Samuel, Kumar Shivam, Supurna Sinha

Theoretical Physics Department, Raman Research Institute, Bangalore

Abstract: We study the relation between qubit entanglement and Lorentzian geometry. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition.

Summary: Detecting entanglement is one of the outstanding problems in Quantum Information Theory. In two qubit systems, the Positive Partial Transpose (PPT) criterion [1, 2, and 3] gives a simple, computable criterion for detecting entanglement. The criterion gives a necessary and sufficient condition for a state to be separable.

In this talk, I will talk about a new test for entanglement based on partial Lorentz transformation (PLT) of individual qubits. It turns out that like the positive partial transpose test (PPT), the PLT criterion is necessary and sufficient in the two qubit case. We give the PLT test as a recipe that could be directly used by those who want to apply the test. We also describe the theoretical framework behind the PLT test. In addition to showing why the test works, the Lorentzian approach yields an explicit separable form of the density matrix, when such a form exists. It also permits a complete elucidation of the state space using a Lorentzian version of the Singular Value Decomposition. The PLT test uses ideas borrowed from the

44 space-time physics of Special Relativity. There appears to be a rich Lorentzian structure hidden within the theory of quantum entanglement. The relation is probably best appreciated using spinors, which have been studied by relativists like Penrose, Newman [4] etc. A surprising feature is the natural emergence of “Energy conditions” used in Relativity. All states satisfy a “Dominant Energy Condition” (DEC) and separable states satisfy the Strong Energy Condition (SEC), while entangled states violate the SEC. This approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.

References: [1] A. Peres, Separability criterion for Density matrices Phys. Rev. Lett. 77, 1413 (1996). [2] M. Horodecki, P. Horodecki, and R. Horodecki, Physics Letters A 223(1), 1 (1996), ISSN 03759601. [3] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement Cambridge University Press, 2007, ISBN 9781139453462. [4] R. Penrose and W. Rindler, The geometry of world vectors and spin-vectors Cambridge University Press, 1984 Cambridge Monographs on Mathematical Physics of vol. 1, p. 167.

Detection and control of optically levitated particles: probing the quantum-to-classical boundary

Marko Toros University of Southampton, UK

Abstract: In the first part we discuss detection and control in levitated optomechanics. Specifically, we consider anisotropic particles trapped by an elliptically polarized focused Gaussian laser beam [1]. We obtain the full rotational and translational dynamics, as well as, the measured photo-current in a general-dyne detection. As an example, we discuss homodyne detection, and how it can be used for Wigner function reconstruction [2]. In addition, we introduce feedback terms in the dynamics and discuss cooling of the translational motions of the nanoparticle using a Kalman filter [3].

In the second part we discuss how optically levitated systems can be used to test theoretical models. In particular, we consider conventional models (classical and quantum) as well as non-unitary modifications of the quantum dynamics [4-7]. We also briefly discuss the relation of these models with Bohmian Mechanics [8]. At the end, we present a general method of model selection from the experimentally recorded time-trace data [9], which is an alternative to state-reconstruction based statistical tests.

References: [1] Toroš, Marko, Muddassar Rashid, and Hendrik Ulbricht. "Detection of anisotropic particles in levitated optomechanics." arXiv preprint arXiv:1804.01150 (2018). [2] Rashid, Muddassar, Marko Toroš, and Hendrik Ulbricht. "Wigner Function Reconstruction in Levitated Optomechanics." Quantum Measurements and Quantum Metrology 4.1 (2017): 17-25. [3] Setter, Ashley J., MarkoToroš, Jason F. Ralph, and Hendrik Ulbricht. "Real-time Kalman filter: Cooling of an optically levitated nanoparticle." Physical Review A 97.3 (2018): 033822. [4] Toroš, Marko, Giulio Gasbarri, and Angelo Bassi. "Colored and dissipative continuous

45 spontaneous localization model and bounds from matter-wave interferometry." Physics Letters A 381.47 (2017): 3921-3927. [5] Toroš, Marko, and Angelo Bassi. "Bounds on quantum collapse models from matter-wave interferometry: calculational details." Journal of Physics A: Mathematical and Theoretical 51.11 (2018): 115302. [6] Gasbarri, Giulio, Marko Toroš, Sandro Donadi, and Angelo Bassi. "Gravity induced wave function collapse." Physical Review D 96.10 (2017): 104013. [7] Gasbarri, Giulio, Marko Toroš, and Angelo Bassi. "General Galilei Covariant Gaussian Maps." Physical review letters 119.10 (2017): 100403. [8] Toroš, Marko, Sandro Donadi, and Angelo Bassi. "Bohmian mechanics, collapse models and the emergence of classicality." Journal of Physics A: Mathematical and Theoretical 49.35 (2016): 355302. [9] Ralph, Jason F., Simon Maskell, Kurt Jacobs, Muddassar Rashid, Marko Toroš, Ashley J. Setter, and Hendrik Ulbricht. "Dynamical model selection for quantum optomechanical systems." arXiv preprint arXiv:1711.09635 (2017).

Single photon entanglement distribution scheme applied to a purification protocol

Nicolò Lo Piparo1, William J. Munro1, 2, 3, and Kae Nemoto

1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda, Tokyo 101-0003, Japan 2NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa, 243-0198, Japan. 3NTT Research Center for Theoretical Quantum Physics, NTT Corporation, Atsugi, Kanagawa, 243- 0198, Japan.

Abstract: Entanglement distribution is a fundamental step for several applications in quantum information. Here, we present a new method that requires one photon for creating an arbitrary number of entangled pairs.

Summary: Entanglement distribution plays a fundamental role in several quantum information applications, such as quantum cryptography [a], quantum teleportation [b] and quantum key distribution [c]. A typical entanglement distribution protocol requires two quantum memories (QMs), each for one user (Alice and Bob), separated by a distance L, in order to store the entangled qubits, which then can be used for quantum information purposes. Some protocols require that multiple entangled pairs are created at the same time, for instance, in a conventional purification protocol [d]. In this case, Alice and Bob must be provided with a larger number of QMs and single photon sources (SPSs). Besides, the time for entangling all pairs will increase with the number of qubits. Here, we propose a method that reduces the time of entangling several QMs separated by a distance L. Remarkably, with this method, by using one photon, we can ideally entangle an arbitrary number of qubits.

Figure 1: In (a), the sawing procedure for entangling N pairs separated by a distance L. In (b), required steps for entangling four qubits by using the sawing procedure.

Our entangling distribution method relies on the double-encoding (DE) procedure used in [4], in which the polarization components of a photon are entangled with the electron spin state of an NV center embedded in a one-sided cavity. After this photon interacts with another NV center through the same DE procedure, it will ideally entangle the spins of the two NV centers. More specifically, the final state will be given by the sum of two different Bell states coupled with the two components of the polarized photon. This procedure can be extended to an arbitrary number of consecutive NV centers by encoding the photon with extra degrees of freedom (DOFs), such as time-bins, which corresponds to adding multiple modes to the photon. Therefore, the final state will be given by the sum of the tensor products of two different Bell states, with each term of the sum coupled with a different mode of the photon. In this way, we can create a series of two consecutive entangled pairs, as show in the left- hand side of Fig. 1(a).

We call this step the sawing procedure as the photon passing through the NV centers saws them by creating two consecutive entangled pairs. However, in order to create entangled pairs separated by a distance L, we couple the photon modes with different terms of the total state, by applying optical switches. This corresponds to entanglement swapping operations, which will create the pairs of right-hand side of Fig. 1(a). Figure 1(b) shows this procedure for the four qubits case. Here, the photon interacts through the DE procedure with the first NV center (Qubit1). Then, an extra DOF (time-bin encoding) is added to the photon, creating, thus, four different modes. Two of the four modes are switched (switching) and then the photon interacts with Qubit 3 and Qubit4 through the DE procedure. After that, another switching will entangle Qubit 1 with Qubit 2 and Qubit 3 with Qubit 4, respectively. All the DOFs of the photon are finally measured, projecting the two pairs in a specific Bell state. We optimize the Deutch et al. purification protocol [5] by applying the sawing procedure to it. In [5], a high fidelity entangled pair is obtained from lower fidelity pairs through local operations and classical communication. In this protocol, Alice and Bob must communicate twice: first to acknowledge the successful entanglement creation and then to share the results of the purification step. In our case, we combine these two operations into one since now, once the photon reaches Bob’s side, Alice and Bob perform the local operations on the matter qubits and the results are all communicated at the same time. Furthermore, we can consider the extra degrees of freedom of the photon as matter qubits on which perform the same local operations of the Deutsch et al. protocol. In this way, we can also reduce the number of matter qubits. Motivated by the lower number of resources needed in our scheme, we define a new figure of merit that takes into account the number of matter qubits and the average number of single photons needed to perform k purification rounds. For the standard Deutsch et al protocol and in our optimized case, we then look at the rate of generating an entangled pair after k purification rounds normalized over the total number of resources. To address the different impact on the protocol derived from the matter qubits as well as from the photon sources, we multiply these numbers by two generic cost functions, CM and Cm, respectively. We calculate the ratio, r, of the normalized rates for both protocols in the ideal case and when the optical switches are not perfect. In the latter case, we model this imperfection as a loss event with transmission probability = 99%. We, also, determine the effects on r when CM=Cm 2 [10 3; 103]:

1 T = 1 ms (a) 1 (b) (c) 8 10

0.8 9 7 k=2 8

0.6 6 7 r 6 5 Fidelity0.4 5 3 qubits error 4 correction 4 0.2 protocol 3 3 k=1 2 0 0 0 20 40 60 0 20 40 60 10 C /C Distance, L (km) M m Figure 2(a) and (b) show the ratio, r; versus the distance of the normalized rate of the optimized purification protocol with that of the conventional one up to two purification rounds and for the three qubits error correction protocol for _t = 1 and _t = 0:85; respectively. In both cases, r >1; i.e., our optimized scheme outperforms the standard one.

This is due to two reasons: first, in our scheme we do not wait until multiple pairs are entangled, therefore, we can neglect the average waiting time for entangling different pairs, which is proportional to (3=2)k: Secondly, we acknowledge in one step both the entanglement distribution and the purification of the pairs. Figure2(c) shows the ratio between the cost functions CM and Cm at a distance L = 30 km. For CM

References: [a] A. Ekert, Phys. Rev. Lett. 67, 661 (1991). [b] F. Deng, C. Li, Y. Li, H. Zhou, and Y. Wang, Phys. Rev. A 72, 022338 (2005). [c] L.-M. Duan, M. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001). [?] N. Lo Piparo, M. Razavi, and W. J. Munro, Phys. Rev. A 95, 022338 (2017). [d] D. Deutsch, A. Ekert, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996).

R. Prabhu Dynamics and steady state properties of entanglement in periodically driven Ising spin-chain

Utkarsh Mishra1, R. Prabhu2, and Debraj Rakshit3

1Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea 2Department of Physics, Indian Institute of Technology Dharwad, Dharwad-580011, Karnataka, India 3Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warszawa, Poland

Abstract: We study the dynamics of microscopic quantum correlations, viz., bipartite entanglement and quantum discord, in Ising spin chain with periodically varying external magnetic field along the transverse direction.

Summary: Quantum correlations are key resources for various quantum information processing tasks and communication protocols in many-body systems. We study the dynamics of microscopic quantum correlations, i.e., quantum correlations between two-sites, in a periodic driven Ising model, where periodic driving is implemented via external magnetic field. More specifically, starting from a close to zero-temperature initial state, the dynamics is generated by repeated application of the unitary evolution, obtained in one complete driving period 휏, for reaching at the desired time 푡 = 푛휏, where 푛 denotes the number of applied pulses and 휏 is the time-period between successive pulses.

We study the relaxation of bipartite entanglement, measured by concurrence, and quantum discord between two nearest-neighbor sites as a function of the driving cycles 푛 for different choices of driving frequencies, 휔 = 2휋/휏. We find that both concurrence and quantum discord tend to saturate to steady-state values after sufficient number of driving cycles 푛.

Figure 1: Concurrence (a) and quantum discord (b) approaching the steady-state values after applying 푛 cycles of external magnetic field. Desired system parameters and units are considered.

We also calculate the distance between the density matrix after n driving cycles and the density matrix corresponding the steady state obtained by taking the asymptotic limit 푛 →∞. The distance measure, which goes to zero in the asymptotic limit, obeys power law scaling with respect to 푛.

Figure 2: Trace distance as a function driving cycle of square pulsed magnetic field.

Next, for various choices of initial states, we look into steady-state (푛 →∞) quantum correlations, and monitor their variance as a function of 휏. The steady-state quantum correlations are characterized by presence of sharp peaks or kinks, which can be further understood by looking into the band structure of the Floquet Hamiltonian, 퐻,.

Since the local quantum correlations in the long-time steady-state may survive with finite values, it is fundamentally interesting to see if the final steady state value corresponds to a canonical Gibbs ensemble. If such canonical Gibbs states exits, the quantity is termed as canonical ergodic. We consider two distinct cases depending on the choice of driving pathway: (i) repeated driving across the critical point and (ii) repeated driving within a single phase. For case (i), when the initial state is chosen from the ordered phase, the quantum correlations always remain ergodic. On the contrary, and more interestingly, quantum correlations may undergo ergodic to non-ergodic transitions in frequency domain if the system is initialized in the disordered phase. For case (ii), we find situations, where quantum correlations originating from entanglement-separability paradigm, i.e., concurrence, exhibit completely different ergodic behavior than the information-theoretic quantum correlation, i.e., quantum discord. For such cases, although quantum discord is characterized with ergodic to non-ergodic transitions in the frequency domain, concurrence always remains ergodic.

Figure 3: Statistical behavior of concurrence and quantum discord under periodic driving across phase transition.

Figure 4: Statistical behavior of concurrence and quantum discord under periodic driving within same phase.

The work presented here is relevant to current experimental set-ups for studying Floquet dynamics, particularly via ultracold atoms in optical lattice. Many interesting directions emerging from this work require independent attention, particularly, in context of the correspondence between Floque band gap and peaks of the quantum correlations. This is beyond the scope of current work, and detailed attention will be paid in our future works. Moreover, there is a lot of scope to further explore the dynamical and steady behavior of the quantities studied in this work. It will also be interesting to conduct analogous studies in non- integrable models and higher dimensional systems.

References: [1] Extended version of this work with more technical details and references is available as: U. Mishra, R. Prabhu, and D Rakshit, arXiv:1711.07769 [quant-ph]. [2] Any communications can be addressed to [email protected]

Geometric construction of ontological models for single systems in the convex Framework

S. Aravinda1, R. Srikant2, A. Pathak3

1Institute of Mathematical Sciences,CIT Campus, Taramani, Chennai, 600113, India 2Poornaprajna Institute of Scientific Research, Sadashivnagar, Bengaluru- 560080, India 3Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India.

Abstract: In the framework of certain general probability theories of single systems, we identify various nonclassical features with the non-simpliciality of the state space. This is shown to naturally suggest an underlying simplex as an ontological model. Contextuality turns out to be an independent nonclassical feature, arising from the intransitivity of compatibility.

Summary: What minimal assumptions suffice to guarantee the nonclassical properties in QM is an important question. The general probability theories, or convex framework [1–3], provides a natural approach to this problem. In the context of multi-partite systems, it is known [4] that non-classical properties like no-cloning, intrinsic randomness and uncertainty derive from two basic assumptions: no-signaling and nonlocality. An extension of that work, obtaining these nonclassical properties even by relaxing the no-signaling condition, is discussed in Refs. [5, 6] Both these works derive monopartite nonclassical properties from assumptions concerning bipartite systems, and thus do not provide an axiomatic approach to single-systems per se. In the inverse direction, bounds on nonlocality in no-signaling theories have been derived by assumptions about monopartite system properties like uncertainty [7] and complementarity [8].

A more elementary question asks what axioms lead to nonclassicality for single systems, so that they may be distinguished from other nonclassical features that appear in multipartite systems (mentioned above) and are associated with the way individual systems are combined. In this work, we were motivated to investigate the origin of nonclassicality in finite- dimensional monopartite systems in the convex framework [9]. We identified two axioms, and two associated levels, that capture a host of nonclassical features: (a) Pairwise incongruence among m (m >2) observables, with the geometric consequence of nonsimpliciality of the state space; (b) Prohibition of m-congruence associated with an intransitive congruence structure. The first axiom guarantees many nonclassical features such as multiple pure-state decompositions, measurement disturbance, no-cloning, impossibility to jointly distinguish all pure states, unconditional but not device-independent cryptographic security, uncertainty (effectively), preparation contextuality and the lack of certain universal operations, the last two with some ontological considerations. Axiom (a), which we identify with the base-level nonclassicality, does not allow us to derive Kochen-Specker (KS) contextuality, for which axiom (b), corresponding to the higher-level nonclassicality, is needed.

Here we show that this approach naturally suggests a geometric construction for the ontological model of any operational theory: as an exponentially higher dimensional underlying simplex. The function that maps the ontological model to the operational theory can be expressed as the composition휑∗ = 휑∗ ∘ 휑∗ , where 휑∗ is a compressive map introducing nonsimpliciality, while 휑∗ is a “crumpling” operation, that introduces uncertainty. The map φ is not unique, and we show that preparation contextuality and impossibility of certain operational coherent transformations depend on the choice of map φ.

References: [1] L. Hardy (2001), quant-ph/0101012. [2] P. Mana (2003), quant-ph/0305117. [3] J. Barrett, Phys. Rev. A 75, 032304 (2007). [4] L. Masanes, A. Acin, and N. Gisin, Phys. Rev. A 73 (2006). [5] S. Aravinda and R. Srikanth, Quant. Inf. Comp. 15, 308 (2015). [6] S. Aravinda and R. Srikanth, International Journal of Theoretical Physics pp. 1–10 (2015). [7] J. Oppenheim and S. Wehner, Science 330, 1072 (2010). [8] M. Banik, M. R. Gazi, S. Ghosh, and G. Kar, Phys. Rev. A 87, 052125 (2013). [9] S Aravinda , R. Srikanth and A. Pathak, J. Phys. A: Math. Theor. 50 (2017) 465303 (33pp).

Saubhik Sarkar Disorder and noise induced dynamics in Fermi-Hubbard systems

Dr. Saubhik Sarkar1, Henrik P. Lüschen2, Pranjal Bordia2, Sean S. Hodgman2, Michael Schreiber2, Andrew J. Daley3, Mark H. Fischer4, Ehud Altman5, Immanuel Bloch2, Ulrich Schneider6

1Post Doctoral Fellow, HRI (QIC group), Allahabad, India 2Max-Planck-Institut für Quantenoptik, Garching, Germany 3University of Strathclyde, Glasgow, UK 4ETH Zurich, Switzerland 5University of California, Berkeley, USA 6University of Cambridge, Cambridge, UK

Abstract: A microscopic model of the underlying physics of decoherence is developed to understand the dynamics of the many-body localised phase experimentally observed in an open and interactive two-species Fermi-Hubbard system.

Summary:

Ergodicity breaking in many-body systems in the presence of interactions has recently been observed experimentally by realizing many-body localisation (MBL) in a system of cold atoms in optical lattices [2]. An interacting version of Aubry-André model [3], which describes two (spin) species fermions on a tight-binding lattice with onsite interaction and nearest neighbour tunneling and a quasi-periodic potential, giving rise to the disorder, is realized in this experiment. The memory of the initial state which is taken to be a charge density wave is found to persist locally for intermediate to long time scales for strong enough disorder.

In a perfectly closed quantum system this persistence of MBL would be for infinite time but in real experiments a coupling to the environment is always present which would destroy the MBL eventually. In our work the susceptibility to dissipative coupling is examined via controlled spontaneous emission events. To understand the decohering mechanism a microscopic model of scattering events in a single atom picture is developed which is

53 depicted in the schematic figure above. The atom is initially in a superposition of localised Wannier states and after an event of spontaneous emission goes to an incoherent mixture of Wannier states centred at different sites, localised by the scattered photon’s wavelength.

The probability for a particular site is given by the overlap of the corresponding Wannier state to the initial state. This is a dephasing channel that effectively measures the atom’s position. The other dissipation channel is atom loss which happens due to atoms going to higher bands that are not trapped in an event of spontaneous emission. This microscopic model enables us to quantify the relative rates of the two dissipation mechanisms and incorporate them in a rate model that has previously been devised [4] to study the dynamics of such systems.

The comparison between the experimental result and the theoretical prediction is shown in the figure on the next page where imbalance of the initial charge density wave, which is taken to be the measure of MBL, shows the expected stretched exponential decay. The total number of atoms suffers an exponential decay due to atom loss. In the non-interacting case the susceptibility to the scattering process is found to be not profound in localised phases as the atoms only get more localised by the dephasing process and loss processes are equally probable to occur at any site. In the interacting case however, both these processes result in affecting any sites nearby.

For moderately localised cases, where finite overlap with adjacent sites is implied, this results in stronger influence on the susceptibility.

This study helps us to quantify the effect of the dissipative processes in experiments and to determine the susceptibility to the scattering events. The susceptibility is found out to be not sensitive to interactions in deep localised phase but showed a steep rise with increase in interactions near the MBL transition. There is potential scope of research in devising an analytical model to capture this interesting behaviour qualitatively and compute the atom loss dynamics numerically.

Note: I am currently in the process of preparing a publication on some new results on a quantum information perspective of Fermi-Hubbard Hamiltonian with the corresponding dynamical study of an open system. This work is in collaboration with Dr. Ujjwal Sen and Dr. Aditi Sen De, in HRI, Allahabad and I would like to include this topic in my talk as well, as I strongly expect to complete this work in about two weeks.

References: [1] Phys. Rev. X, 7:011034, 2017 [2] Science, 349(6250):842–845, 2015 [3] Ann. Israel Phys. Soc., 3(133), 1980 [4] Phys. Rev. Lett., 116:160401, 2016

Temporal correlations and device-independent randomness

Shiladitya Mal1,2, Manik Banik3, Sujit K Choudhary4

1S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, India. 2Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India. 3Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India. 4Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, Orissa, India.

[Quantum Inf. Process (2016) 15: 2993]

Abstract: We derive Leggett-Garg inequality (LGI) from the assumptions of predictability and no signalling in time. As a novel implication of this derivation, we find application of LGI in device-independent randomness certification.

Summary: Randomness is a valuable resource for various important tasks ranging from cryptographic applications to numerical simulations such as Monte Carlo method (a useful technique which finds application in computational Physics, Statistical Physics, Physical Chemistry, Computational Biology, Computer Graphics, Finance and many other areas). For various such tasks, the genuineness of the used randomness is of primary concern. Thus, device independent certification and generation of randomness is very important from a practical point of view. Motivated by the work of Pironio and co-workers' [1] many interesting results have been obtained, in recent times, in the field of DI certification and generation of randomness. All such methods use nonlocal correlations among spatially separated parties (which is guaranteed by Bell type inequalities violation [2]) to certify randomness. In this work, we have shown that non-classical temporal correlations, i.e., correlations which violate Leggett-Gerg inequality (LGI), can also be used to certify randomness by employing derivation of LGI from new set of assumptions, predictability and no signalling in time.

It would be worth mentioning here that like Bell's scenario, in the case of temporal correlations too, we need some amount of seed randomness at the input. This is needed for freely choosing the measurement times. Thus our work provides an important information theoretic application of LGI which can be implemented in laboratory with the present day's technology.

For testing the existence of superposition of macroscopically distinct quantum states, Leggett and Garg [3] put forward the notion of macrorealism. This notion is compatible with classical physics and states that,

(A1) A macroscopic system with two or more macroscopically distinct states available to it will at all times be in one or the other of these states [Macroscopic realism per se (MR)]. (A2) It is possible, at least in principle, to determine which of these states the system is in, without any effect on the state itself or on its subsequent dynamics [Noninvasive measurability (NIM)].

The conjunction of these two assumptions, namely the assumptions of MR and NIM, together with the postulate of Induction [4] give rise to Leggett-Garg inequalities. In the present context, postulate of induction says that properties of an ensemble are determined exclusively by initial conditions and cannot be affected by final conditions. Past few years have witnessed a range of new experimental proposals and a number of actual experiments to test LGI in a variety of physical systems such as super conducting qubits, nuclear spin and photons [5, 6].

In the present work, we give an alternative derivation of LGI by using different set of assumptions: the assumption of no signalling in time (NSIT) and predictability. The assumption of NSIT, described in [7], says that a measurement does not change the outcome statistics of a later measurement, whereas predictability is the assumption that one can predict the outcomes of all possible measurements to be performed on a system [8]. These assumptions involve only measurement statistics and hence are directly testable in experiments. This derivation of LGI, therefore, allows us to conclude that in a situation where NSIT is satisfied, the violation of LGI will imply the presence of certifiable randomness. It is known that macrorealism implies both LGI as well as NSIT. But, the assumption of NSIT, alone, does not imply LGI [5]. However, as we shown in our work, it together with the assumption of predictability yields LGI. Imagine now a situation where LGI is violated but NSIT is satisfied. It would be worth mentioning here that NSIT is experimentally testable. In such situations, we can say that the model cannot be predictable. Thus LGI finds an important practical implication in a useful information theoretic task.

From algorithmic information theory it is known that randomness cannot be certified by any mathematical procedure [9]. The generation of randomness, therefore, must be based on unpredictability of some physical phenomena, so that the randomness is guaranteed by the inherent uncertain nature of the physical theory. There is no such thing as true randomness in classical world as any classical phenomenon, can, in principle, be predicted. They appear random to us due to lack of our knowledge and control of all the relevant degrees of freedom. Measurement on a quantum particle, on the other hand, is postulated to give intrinsically random results. The quantum measurements, therefore, can be used to generate true randomness [10, 11]. But, for the reliability of the randomness thus generated, one needs to trust the devices which prepare and measure the quantum states. Can randomness be certified in a Device-Independent way, i.e., can it be certified even without knowing the details of the devices used in its generation? This is a topic of current research interest [1, 12]. . In a standard Leggett-Garg test binary observables are measured on a single system at two different times sequentially. In device independent scenario, one does not have detailed knowledge about the experimental apparatuses and hence the experimental setup is like a black-box with inputs and outputs. At the input probe, one can change the parameters of measurement setup and thus can choose different measurements.

Measurements are performed at different instants of time on the system and the of occurrence of a given pair of outcomes for each pair of different time measurements are collected at the

56 output probe either by repeating the experiment many times or by employing an array of many identical systems. The other joint probabilities involved in Leggett-Garg inequalities are calculated likewise to observe their violations. These probabilities are also analyzed to see whether NSIT is obeyed. In fact, in Ref. [7], it has been shown that there exists probability distribution which violates LGI but satisfy NSIT. As we now know that such distribution cannot be predictable and therefore some randomness is associated with it. The associated randomness can be quantified by min-entropy [13] which is a statistical measure of the amount of randomness that a particular distribution contains.

Though Quantum Theory postulates to have random measurement outcomes, it does not deny for a finer theory where the measurement outcomes are only apparently random. In fact, there exists ontological models which, in principle, can predict the outcomes of each individual measurement on a single particle [2, 14]. But, the situation becomes subtler when one considers the scenarios which either involves different measurements on two correlated particles or which involve measurements on one and the same particle at different times. In the former case, it has been shown that it is not possible to predict the outcomes of an experiment which violates Bell's inequality because of the impossibility of communicating superluminally. For the latter case, our analysis shows that the observed frequency of the joint conditional outcomes of measurements performed on a single particle at two different times is unpredictable if the two-time correlations thus obtained violate LGI but satisfy NSIT. However, unlike no-superluminal signalling, NSIT is not at variance with special relativity. From the perspective of experimental implementation the LGI-based DI randomness certification seems more feasible than its spatial analogue as it does not require entanglement. Moreover, various successful experimental tests of LGI violation also give rise to the possibility towards further experiments with more macroscopicity involved. The present work also shows potential usefulness of such non-classical macroscopic systems.

References: [1] Pironio S et al. (2010) Nature (London) 464, 1021. [2] Bell J S (1964) Physics 1, 195-200; Bell J S (1966) Rev. Mod. Phys. 38, 447. [3] Leggett A J and Garg A (1985) Phys. Rev. Lett. 54, 857. [4] Leggett A J (2008) Rep. Prog. Phys. 71, 022001. [5] Williams N S and Jordan A N (2008) Phys. Rev. Lett. 100, 026804. [6] Dressel J, Broadbent C, Howell J, and Jordan A (2011) Phys. Rev. Lett. 106 040402 [7] Kofler J and Brukner C (2013) Phys. Rev. A 87, 052115. [8] Cavalcanti E G and Wiseman H M (2012) Found. Phys. 42, 1329. [9] Knuth D (1981) The Art of Computer Programming Vol. 2, Semi-numerical Algorithms, Addison Wesley. [10] Jennewein T, Achleitner U, Weihs G, Weinfurter H, and Zeilinger A (2000) Rev. Sci. Instrum. 71, 1675; [11] Stefanov A, Gisin N, Guinnard O, Guinnard L, and Zbinden H (2000) J. Mod. Opt. 47, 595. [12] Acin A, Massar S, and Pironio S (2012) Phys. Rev. Lett. 108, 100402. [13] Koenig R, Renner R, and Schaffner C (2009) IEEE Trans. Inf. Theory 55, 4337. [14] Kochen S and Specker E (1967) J. Math. Mech. 17, 59-87.

Arrival Time Distributions of Spin-1/2 Particles

Siddhant Das and Detlef Dürr

Mathematisches Institut, Ludwig-Maximilians-Universitat München, Theresienstr. 39, D-80333 München, Germany ([email protected], [email protected])

Abstract: The arrival time statistics of spin-1/2 particles governed by Pauli’s equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual quantum flux or from semiclassical considerations suggest testing these notable deviations in an experimentum crucis of the validity of the Bohmian predictions for spin-1/2 particles. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.

Summary: The prediction of arrival times of a quantum particle has a long history [1,2]. Although it has often been observed that there is no time observable in quantum mechanics, it is commonly accepted by now that measurements of time are clearly possible [3]. The time taken for a particle to arrive on a detector is not well posed within the orthodox (or Copenhagen) interpretation of quantum mechanics, therefore it has long been realized that the most direct approach to the problem of arrival times comes from quantum theories comprising of deterministic particle trajectories, such as de Broglie-Bohm pilot wave theory, or Bohmian Mechanics. Bohmian mechanics is empirically equivalent to standard quantum mechanics, in the sense that the two theories make the same predictions for any experiment that is well posed in the orthodox theory. However, time measurements are clearly different. We take this opportunity to propose a realistic experiment involving a single spin-1/2 particle moving in a waveguide, which can be achieved with the present-day experimental technology. A first principles analysis of our experiment, based on the dynamical equations of Bohmian mechanics reveals drastic and unexpected properties when control parameters are changed.

Our suggested experimental setup is this: A spin-½ particle of mass m is constrained to move within a semi-infinite cylindrical waveguide. Initially, it is trapped between the end face of the waveguide and an impenetrable potential barrier placed at a distance d. At the start of the experiment, the particle is prepared in a state 훹, then the barrier at d is suddenly switched off, allowing the particle to propagate freely within the waveguide. The arrival surface is the plane situated at distance L (> d) from the end face of the waveguide. We then compute numerically from the Bohmian equations of motion how long it takes for the particle to arrive at L, and determine the empirical arrival time distribution 훱(휏). We find that:

(1) If the initial wave function of the particle has only one non-zero component -- a ‘spin- up’ or ‘spin-down’ wave function1, the arrival time distribution 훱(휏) coincides with

the quantum flux expression ( ), where is the quantum ∫ 푱(풓, 휏). 풅풔 ≡ 휫풒풇 흉 퐽(푟, 푡) flux (or probability current density) of the Pauli wave equation. Note well that by the

very meaning of the quantum flux, 훱풒풇(흉) is a natural guess for the arrival time distribution from the point of view of standard quantum mechanics [4] as well. But

(1 Here, spin-up is w.r.t the waveguide axis.) 58

note also that 훱풒풇(흉) makes sense only if the surface integral is positive, which need not be the case. However, in our case it is non-negative for the aforementioned wave functions. We find that this result is also independent of the exact shape of the waveguide potential. (2) If the initial wave function is an equal superposition of the spin-up and spin-down wave functions, the arrival time distribution pinches off at a maximum arrival time

휏, i.e., no particle arrivals occur for 휏 > 휏. In other words, the arrival time distribution in this case has no tail, unlike in (1). (3) The above results are rather generic, and are not manifestations of any special choice

of the initial wave function 훹. We have proposed a simple experiment, the results of which orthodox quantum mechanics provides no unique way of predicting [1,2]. This is a rather remarkable situation that could be tested soon. Our results demonstrate that the distribution of first arrival times of a spin-1/2 particle bears clear signatures of Bohm’s guidance law.

References: [1] J. G. Muga, R. S. Mayato, and ´I. L. Egusquiza, eds., Time in Quantum Mechanics, 2nd ed., Lect. Notes Phys. 734, Vol. 1 (Springer, Berlin Heidelberg, 2008). [2] J. G. Muga and C. R. Leavens, Phys. Rep. 338, 353 (2000). [3] T. Zimmermann et al., Phys. Rev. Lett. 116, 233603 (2016) [4] P. Blanchard and J. Fröhlich, eds., The Message of Quantum Science: Attempts Towards a Synthesis, Lect. Notes Phys. 899 (Springer, Berlin Heidelberg, 2015) Chap. 5.

Sk Sazim Coherence makes quantum systems magical

Chiranjib Mukhopadhyay, Sk Sazim, and Arun Kumar Pati

Quantum Information and Computation Group, Harish Chandra Research Institute, HBNI, Allahabad 211019, India

Abstract: What makes quantum technologies more powerful than their classical counterparts? We argue that the linear superposition principle, implying the existence of quantum coherence, can be intuitively thought of as the driving agent behind any quantum advantage.

Summary: Two primary facets of quantum technological advancement that holds great promise are quantum communication and quantum computation. For quantum communication, the canonical resource is entanglement. For quantum gate implementation, the resource is ‘magic’ in an auxiliary system. It has already been shown that quantum coherence is the fundamental resource for the creation of entanglement. We argue on the similar spirit that quantum coherence is the fundamental resource when it comes to the creation of magic. This unifies the two strands of modern development in quantum technology under the common underpinning of existence of quantum superposition, quantified by the coherence in quantum theory.

We also attempt to obtain magic monotones inspired from coherence monotones and vice versa. We further study the interplay between quantum coherence and magic in a qutrit system and that between quantum entanglement and magic in a qutrit-qubit setting.

Linking the resource theories of coherence and magic: The resource theories of coherence and magic seem quite disjoint. But are they really so? This is the question we seek to address. Specifically, we demonstrate how the presence or lack of quantum coherence in systems constrains the amount of magic in the system. In doing so, we reveal that quantum coherence can be quantified by the maximum amount of magic generated through incoherent operation on arbitrary quantum states.

Resource theory Free states Free operations Coherence The diagonal density matrices in preferred basis Incoherent operations Magic States inside polytope accessible via Clifford Stabilizer operations unitary rotation of computational basis

We now state our first result -

Result 1: For any distance based coherence quantifier and corresponding magic quantifier, the amount of magic generated through incoherent operations on a quantum state is upper bounded by the amount of coherence originally present in that state.

This result will immediately lead us to define new set of coherence monotones based on distance based magic monotones.

Result 2: The following quantifier can be considered as valid magic monotone 푀(휌) = 푚푖푛훴푝훱퐶(휓), where ‘k’ is over all the chosen stabilizer basis directions and C is the coherence monotone along the ‘k’ direction.

Clearly, for any pure stabilizer state, the corresponding coherence element is zero and hence the magic of formation vanishes. For any mixed state within the stabilizer polytope, the magic of formation is similarly zero.

Concrete results for qutrits: Proposition.1 - Strange states are more robust under mixture with maximally incoherent, i.e. white noise than Norrell states. Proposition.2 - The Norrell state above is more robust under the admixture of colored coherent noise than the strange state above. Proposition.3 - The following condition on quantum coherence, quantified via the 푙-norm, and magic, quantified by the sum negativity (mana), holds for qutrit pure states푀(휓) ≥ 휂1− 휂, where 휂 = 퐶 (휓).

Proposition – For qubit-qutrit states, the negativity of entanglement 퐸 and the mana satisfies the following trade off relation, 16퐸(휌) +푀(휌) <4.

Illustration1: Pictorial representation of free states in resource theories of magic and coherence in the qubit case. The stabilizer polytope is an octahedron within the Bloch sphere. Any qubit state represented by a point outside the octahedron is a magic state. All incoherent states in the computational basis lie on the yellow line.

Conclusion: In this work, we have indicated the link between the resource theories of coherence and magic. We demonstrated that quantum coherence in a state is ultimately the currency for creation of magic through incoherent operations. We proposed a measure of magic in prime dimensional quantum states through product of coherence quantifiers with respect to mutually unbiased bases and, furthermore, we derived another full coherence monotone from the discrete Wigner function representation of quantum states in discrete phase space. We also proposed several sub-classes of quantum operations as free operations if coherence and magic are to be simultaneously considered as resources. We also investigated the link between coherence and magic in the concrete scenario of a qutrit system.

References: [1] C Mukhopadhyay, Sk Sazim, A K Pati, arXiv: 1801.04807. [2] Streltsov A, Singh U, Dhar H S, Bera M N and Adesso G, 2015, Phys. Rev. Lett. 115 020403. [3] Veitch V, Hamed Mousavian S A, Gottesman D and Emerson J, 2014, New Journal of Physics 16 013009.

Quantum Heat Machines with Trapped Ions

Suman Chand*, Dr. Asoka Biswas Department of Physics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India; (* Email: [email protected])

Abstract: We show how one can implement the reciprocating heat cycle of a quantum heat machine by using trapped ions and an always-on thermal environment.

Summary: Introduction:- Recently, quantum heat machine (QHM) has become an area of growing interest. QHMs utilize a quantum system as the working fluid that provides large work efficiency, beyond the limit of classical heat engines [1]. Existing proposals for implementing quantum heat engines require that the system interacts with the hot bath and the cold bath (both modeled as a classical system) in an alternative fashion and therefore assumes the ability to switch off and on the interaction with the bath during a certain stage of the heat- cycle. However, it is not possible to decouple a quantum system from its 'always-on interaction' [2] with the bath without the use of complex pulse sequences. It is also hard to identify two different baths at two different temperatures in the quantum domain that sequentially interact with the system. Here, we demonstrate how the reciprocating heat cycle of a QHM can be implemented using trapped ions as the working substance, in presence of a thermal bath. The electronic states of the ions act like a working substance, while the vibrational mode is modeled as the cold bath. In this context, we propose the use of the projective measurement of the electronic states of the ions in suitable basis, which leads to an effective heat exchange with the cold bath. Further, the suitable choice of projected states can lead to either a heat engine or a refrigerator cycle.

Model:- We consider two trapped two-level ions with the lowest-lying electronic states |푒 , 푔⟩as the relevant energy levels. These internal states of the ions interact with a common vibrational mode a. The Hamiltonian that describes this system can be written as (in the unit of Planck's constant ℏ=1).

H =H +H +H

Where, 62

() () () () () () 퐻 = 퐽휎 휎 + 휎 휎 + 퐵휎 + 휎 퐻ℎ = ω푎 푎 () () () () 퐻 = 푘푎 휎 + 휎 푎 + 푘푎 휎 + 휎 푎 Here 퐻 represents the unperturbed Hamiltonian of two ions, which interact with each other with the corresponding coupling constant퐽, 퐻ℎis the energy of the vibrational mode with frequency ω, and 퐻 defines the interaction between the internal and the vibrational degrees of freedom of the ion. The interaction strength between the electronic transitions of the 푖ℎion and the vibrational mode is given by푘, (푖 = 1,2). A magnetic field of strength 퐵is applied along the quantization axis.

Implementation:- We have considered a four-stroke Quantum Otto engine [3-5]. The electronic states of the trapped ions act like a working substance, while the internal vibrational degree of freedom works a cold bath and the thermal environment works as a hot bath. The protocol of our machine starts with some initial state {1} and it is thermalized [T] through Markovian evolution by the thermal environment (Ignition Strokes). After that, the cycle consists of two adiabatic processes (Expansion stroke and Compression stroke) interrupted by the measurement [M] (Exhaust stroke). Ignition [T] Expansion Exhaust[M] Compression 1 2  3  4  1 Conclusions:- We find that by suitable choice of the projective measurement of the electronic states of the ions, one can lead to either heat release to the cold bath or heat absorption from the cold bath, thereby operating like a heat engine or a refrigerator. Speaking specifically, projection onto the ground state of the system Hamiltonian results in a heat engine, while that onto the other states leads to refrigeration. The performance of the heat machine depends upon the interaction strength between the ions, the magnetic fields, and the measurement cost. Our model is an interesting toy model [6,7]for the characterization of the quantum engine or refrigerator. Mimicking the cold bath by the post-selective measurement (or the feedback cooling) should be easier than preparing a real cold bath for the experimentalists. As the bath is always on, our proposed model of quantum heat machine is experimentally easier to implement. Acknowledgements:- Authors would like to acknowledge IIT Ropar for making this research possible. MHRD for providing research fund, and lastly Dr. Shubhrangshu Dasgupta for fruitful discussions.

References: [1] M. O. Scully and all., Science 299, 862 (2003). [2] D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, Advances In Atomic, Molecular, and Optical Physics 64, 329 (2015). [3] H. T. Quan, Y.-x. Liu, C. P. Sun, and F. Nori, Phys. Rev. E 76, 031105 (2007). [4] George Thomas and Ramandeep S Johal, Phys. Rev. E 83, 031135 (2011). [5] F. Atlintas and all., Phys. Rev. E 90, 032102 (2014). [6] Suman Chand and Asoka Biswas, EPL 118, 60003 (2017) [7] Suman Chand and Asoka Biswas, Phys. Rev. E 95, 032111 (2017).

Two-Slit Interference of Entangled Photons

Tabish Qureshi1, Ananya Paul2

1Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 2Department of Physics, Jamia Millia Islamia, New Delhi

Abstract: It is shown that, under suitable conditions, two entangled photons separated by a polarizing beam splitter, passing through two different double-slits, can behave like a biphoton of wavelength λ/2.

Summary: Quantum mechanics tells us that wave nature and particle nature are two complementary aspects of the same entity. Whether we talk of massive particles or quanta of light, both can behave like particles and waves in different situations. Young’s double-slit experiment carried out with individual particles showed that a particle passes through two slits and interferes with itself. Later it was demonstrated that much larger particles such as C60 molecules can also show interference. It has been convincingly argued that instead of calling them waves or particles, such entities should be called quantons. Going beyond this, quantum mechanics also tells us that a group of entities, e.g., many photons together, can behave as a single quanton. Consequences of this on interference experiments with many particles, has only been recognized relatively recently [1].

First, we briefly explain the idea which motivated Jacobson and collaborators [1] to propose that many photons can behave as a single quanton in an interference experiment. Consider a beam of diatomic iodine molecules I2 each with mass 2m, traveling with a velocity v, passing through a double-slit. The resulting interference would be in accordance with a de Broglie wavelength λ2m = h/2mv. But suppose that the molecule dissociates on the way, and only separate iodine atoms, each of mass m, pass through the double-slit. Then the resulting interference would be in accordance with a de Broglie wavelength λm = h/mv, which shows that λ2m = λm/2. More generally, N particles with a de Broglie wavelength λ, can behave as single quanton of wavelength λ/N. The same should hold for photons too. An experiment was subsequently carried out which measured the de Broglie wavelength of a two-photon wavepacket [2].

We have a done a wave-packet analysis of two entangled photons passing through a double- slit (see figure above). We have shown that the two photons can behave like a single quanton of half the wavelength of the photons when detected in coincidence at the same position [3]. This is in agreement of the earlier analysis and experiment by Fonseca, Monken and PaÌdua [2]. More interestingly, we show that even if the two entangled photons are separated by a polarizing beam splitter, they can still behave like a biphoton of wavelength λ/2 [3]. In this modified setup (see figure below), the two separated photons passing through two different double-slits, surprisingly show an interference corresponding to a wavelength λ/2, instead of λ which is the wavelength of each photon. Thus, we have shown that the two photons can continue to behave like a single quanton even when they are widely separated in space, a highly nonlocal feature. This work extends the theoretical ideas of multiphoton wave packets to a nonlocal scenario.

Our result implies that even when the two entangled photons are separated in space, they may act like a single quanton which interferes with itself. Entangled particles show very strange and counter-intuitive properties. It has previously been shown that entangled photons can exhibit a nonlocal wave-particle duality [4].

References: [1] “Photonic de Broglie waves,” J Jacobson, G Björk, I Chuang, Y Yamamoto, Phys. Rev. Lett.74, 4835-4838 (1995). [2] “Measurement of the de Broglie wavelength of a multiphoton wave packet,” EJS Fonseca, CH Monken, S Pádua, Phys. Rev. Lett.82, 2868-2871 (1999). [3] “Biphoton Interference in a Double-Slit Experiment,” A Paul, T Qureshi, Quanta7, 1-6 (2018) [4] Siddiqui MA, Qureshi T. “A nonlocal wave-particle duality.” Quantum Studies: Mathematics and Foundations3, 115-122 (2016).

Experimental verification of hidden steerability

Tanumoy Pramanik,1*Young-Wook Cho,1 Sang-Wook Han,1 Yong-Su Kim,1, 2 and Sung Moon1

1Center for Quantum Information, Korea Institute of Science and Technology (KIST), Seoul, 02792, Republic of Korea 2Department of Nano-Materials Science and Engineering, Korea University of Science and Technology, Daejeon, 34113, Republic of Korea (* Electronic address: [email protected])

Abstract: We have studied hidden quantum steerability experimentally and compared it with hidden Bell nonlocality. We have found certain class of mixed entangled states which are unsteerable, show steerability when one applies local filtering operations on both systems.

Summary: Introduction: Correlations between two quantum systems can be either local, i.e., can be created using local operation and classical communication (LOCC) or nonlocal which cannot be generated with the help of LOCC. Nonlocal correlations further are categorized by three different forms and they are entanglement, steering and Bell nonlocal correlation. Entanglement is the weakest form of the nonlocal correlations, whereas Bell nonlocal correlation is the strongest form and steering lies between them. Two quantum systems are said to be in entangled state when the combined state cannot be written as separable state. Steerable state cannot be described by local hidden state (LHS) model and Bell nonlocal correlation cannot be explained by local hidden variable (LHV) model. In some information processing tasks, say, teleportation, cryptography, nonlocal correlations give higher efficiency than local correlation. Several experiments confirm the existence of these of nonlocal correlations in nature.

There are a class bipartite mixed entangled states (characterized by the parameters{푝, 훾}, where mixedness of the state characterized by the parameter 푝 ∈ {0,1}and 훾(훾 ≥ 1)is the pure state parameter which quantifies the ratio of |퐻퐻⟩ and |푉푉⟩ corresponding to the pure state (푝 =1)), which does not show Bell violation when single copy is used, but, Bell nonlocality of those states can be revealed by the application of local filtering operations. This phenomenon is known as hidden nonlocality.

In this work, we have introduced hidden quantum steerability, i.e., by applying local filtering operations one can activate steerability of entangled states which do not show steerability and experimentally verify hidden steerability of the class of above mixed entangled state parameterized by{푝, 훾}.

Result and Discussion: The figure 1 clearly shows that local filtering operations reveal hidden steerability of the class of mixed entangled states parameterized by{푝, 훾}. We have found that

66 filtering operations that revel hidden steerability are different form filtering operation that revel hidden Bell nonlocality.

Fig. 1: Plot of steerability against the mixedness parameter 푝 for 훾=1, and 훾 = 9.

Acknowledgement: This work was supported by the ICT RD program of MSIP/IITP (B0101- 16-1355), and the KIST research programs (2E27231, 2V05340).

Reference: [1] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007); S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007). [2] S. Popescu, Phys. Rev. Lett. 74, 2619 (1995), N. Gisin, Phys. Lett. A 210, 151 (1996).

Coherence property of Airy wavepackets

Vivek M. Vyas Theoretical Physics Group, Raman Research Institute, C. V. Raman Avenue, Sadashivnagar, Bengaluru 560080

Abstract: In this talk it will be shown that, the uniformly accelerating non-spreading wavepackets found in the nonrelativistic quantum free particle system are coherent states.

Summary: In the first quantum mechanics course, one is typically introduced to the Schrodinger equation obeyed by the wave function of the system and its probabilistic interpretation. In the discussion about the quantum mechanical free particle system, it is often argued that the free particle cannot be localised in an arbitrarily small volume at all times. This idea is conveyed by showing that a Gaussian wavepacket disperses and broadens indefinitely as the time evolution takes place. Such an exposition gives an impression that it is in general not possible to have a wavepacket evolution in the free Schrodinger equation without its dispersion and/or distortion. Subsequently one is introduced to the celebrated coherent (wavepacket) states of the harmonic oscillator problem, which are well known to evolve in time without dispersion. This gives an impression that the existence of a confining 67 potential, such that V (x) → ∞ as x → ±∞, is required to save a wavepacket from spreading indefinitely.

In 1979, Berry and Balazs found that there exists a family of Airy wavepackets which evolve as per the Schrodinger equation, but neither disperse nor distort, and above all move with a uniform acceleration [1]. This seemingly incomprehensible and contradictory wavepacket dynamics in the quantum free particle system has been confirmed in several optical experiments [2]. It may be a naïve query that, whether these wavepackets have any connection with the harmonic oscillator coherent states or not. This natural question, it appears has not been yet answered in the literature. The goal of our work is to bring to fore an interesting relation between these two wavepackets, and show that the non-spreading Airy wavepackets are actually coherent states [3].

It is well known that the harmonic oscillator coherent states are known to arise out of the algebra generated by {1, x, p}. It is found that by considering a larger algebra, consisting of {1, x, p, p2/2, p3/6}, the accelerating Airy wavepackets arise out of it as coherent states. Further it is found that these coherent states solve the eigenvalue problem for the linear combination of boost operator K and Hamiltonian H, which is responsible for their seemingly intriguing non-spreading accelerating nature. This provides one with a representation independent understanding of the origin and the nature of such accelerating coherent states. Moreover this treatment is naturally suitable for a realisation and identification of such coherent states in systems involving statistical mixtures and in open systems, wherein the quantum free particle is interacting with a larger reservoir system, while being in or away from thermal equilibrium.

References: [1] M. V. Berry and N. L. Balazs, “Nonspreading wave packets”, Am. J. Phys., 47:264–267, 1979. [2] G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams”, Phys. Rev. Lett., 99(21):213901–213901, 2007. [3] Vivek M. Vyas, “Airy wavepackets are coherent states”, quant-ph: 1710.10302.

POSTERS

Identifying and Quantifying Resource for Remote State Preparation Using Zero Discord States

Aiman Khan1, C.Jebaratnam2, Som Kanjilal3 and Dipankar Home3

1Indian Institute of Technology-Roorkee 2SN Bose Center For Basic Sciences, Kolkata. 3Bose Institute, Kolkata

Abstract: A hitherto unexplored usefulness of non-discordant states is demonstrated for remote state preparation. Simultaneous correlations of the channel in complementary measurement bases are identified as relevant quantifier of quantum resource.

Summary:

I. INTRODUCTION: The profound implications of entanglement regarding quantum foundational aspects and its applications in quantum information constitute one of the most vibrant areas of research in contemporary science. In this context, the question as to whether the paradigm of entanglement describes completely correlations inherent in the quantum system that produce essentially quantum effects that cannot be simulated classically is an open and complex problem[1][2][3]. In recent years, the study of useful correlations inherent in separable states, along with their applicability in quantum information processing (QIP) tasks, has become increasingly important, from a foundational standpoint, as well as because separable states are easier to produce, manipulate and protect against decohering effects in laboratory. To this end, various measures of "quantumness" of correlations beyond entanglement have been proposed, chief amongst them being quantum discord [4][5]. While entanglement has been established as a resource for QIP protocols using pure states, there exist instances where mixed bipartite quantum states, having no detected entanglement, are found to provide a distinct quantum advantage[1][2]. Recently, it was shown that all separable discordant states facilitate meaningful remote state preparation [3], where the efficiency of RSP was argued to be equal to the value of geometric discord for the quantum state, thus suggesting a resource beyond entanglement for the quantum RSP task. Against this backdrop, this work is motivated by the question as to whether zero discord states can also be useful for the purpose of QIP. A further motivation stems from the recently introduced measures of simultaneous correlations in mutually unbiased bases(SCMUB) inherent in quantum systems that seek to quantify "quantumness" by their persistence in incompatible bases used to measure the state[6][7].

II. KEY INGREDIENTS AND RESULTS: We first recapitulate the results obtained in [3] for the following class of Bell diagonal states:

(1)

The accuracy with which any target quantum state, specified by Bloch vector ~t, is prepared (using the quantum channel ρ), at the end of the RSP protocol, was quantified via the payoff function, The fidelity of RSP transport using the quantum channel is thence defined in [3] by averaging over all target states and minimizing over all choices of →:

The Fidelity function can then be explicitly calculated for arbitrary two qubit states. It was shown in [3] that if Alice and Bob share the quantum state given by ρ, the average fidelity F is the following:

The value of F for the given quantum state quantifies its usefulness for the given RSP task. The worst case fidelity for RSP, as defined by Eq.2, was shown to be equal to geometric discord for a wide class of quantum states, including the Bell-diagonal states that were defined earlier. Thus, by ordering bipartite quantum states using their geometric discord, one can meaningfully arrange them in order of their apparent usefulness for the RSP task, showing that quantum discord is a sufficient resource for ensuring effective remote preparation of quantum states. A non-vanishing value of F indicates that, given any choice of → that fixes the equatorial plane containing all target states, there is always a non-vanishing fraction for which the prepared quantum state at Bob's end is distinct from the maximally mixed state (that contains no information about the target state at all). However, this raises the important question of whether a vanishing value of F(and equivalently, geometric discord) implies that no meaningful RSP has taken place in other words, if zero discord resource states are at all useful for RSP in the sense outlined before. In this context, we make an important observation that there can still be target states in at least one equatorial plane de-fined by → that are remotely prepared effectively different from maximally mixed states. To capture this effect, we invoke following quantity to quantify usefulness of such a state for RSP:

Where, we see that, instead of examining the worst case scenario for the protocol, we look at the average payoff, further averaged over all equatorial planes that contain all target states. To interpret this new quantity, we note that if 풢 were to vanish, it would mean that there is not a single target state that can be prepared better than the maximally mixed state, thereby showing a complete failure of the RSP protocol. Hence, in this sense, the quantity 풢 represents a necessary condition for success of the RSP protocol. Now comes the important observation that the above fidelity quantity 풢 does not vanish for zero-discord Bell-digonal states (which are of the form E=diag [퐸, 0, 0]), showing that zero discord states may still be partially useful for remote preparation in that the optimised payoff function has support on the set of target states defined by at least some choice of the Bloch vector →. Now, to identify the correct resource underpinning this usefulness of zero discord Bell diagonal states, we make use of SCMUB, first introduced in [7] [6]. One can explicitly define one such measure of simultaneous correlations using triads of MUBs, referred to as the quantity C3, in the following way:

Where, the set of all triads of mutually unbiased bases on Alice's Hilbert space is denoted by Ʌ. The quantity 퐶 has been explicitly calculated for a wide class of states [6]. From these results, we obtain an analytical relationship between 퐶 and the fidelity quantity 풢 from which one can see that when 풢 = 0, 퐶 also vanishes, implying that the states in question are

71 product states. Therefore, we can conclude that for RSP using product states, and only product states, the fidelity function has no support on any plane defined by the direction →, that is to say, there are no target states that can be prepared better than maximally mixed states at Bob's end using the RSP protocol. Further, all non-product states (including zero- discord states) will have some fraction of states that can be prepared better than the bench- mark of maximally mixed states at Bob's end, indicating a usefulness of zero discord states not explored earlier.

III. SUMMARY: We have demonstrated that the measure of simultaneous correlations in complementary bases 퐶 is a necessary resource for successfully accomplishing the quantum task of remotely preparing a non-vanishing fraction of target states at spatially separate locations. This observation significantly goes beyond the work of Dakic et al [3], where it was argued that only non-vanishing quantum discord is necessary to ensure that there is a finite proportion of target states that are meaningfully prepared (in the sense of RSP) for any arbitrary choice of Bloch vector →. It is by considering the average effect and not the worst case scenario in the context of RSP that enables us to uncover the significance of correlations inherent in the quantum state beyond discord.

References: [1] A. Datta, A. Shaji, and C. M. Caves, Physical review letters 100, 050502 (2008). [2] V. Madhok and A. Datta, Physical Review A 83, 032323 (2011). [3] B. Daki_c, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, _C. Brukner, et al., Nature Physics 8, 666 (2012). [4] L. Henderson and V. Vedral, Journal of physics A: mathematical and general 34, 6899 (2001). [5] H. Ollivier and W. H. Zurek, Physical review letters 88, 017901 (2001). [6] Y. Guo and S. Wu, Scienti_c reports 4, 7179 (2014). [7] S. Wu, Z. Ma, Z. Chen, and S. Yu, Scienti_c reports 4, 4036 (2014).

Akshay Gaikwad Experimental demonstration of selective quantum process tomography on an NMR quantum information processor

1Akshay Gaikwad, 2Diksha Rehal, 1Amandeep Singh, 1Arvind, and 1Kavita Dorai

1Department of Physical Sciences, Indian Institute of Science Education & Research, Mohali 2Indian Institute of Science (IISc), Bangalore

Abstract: We present the first NMR implementation of a scheme for selective and efficient quantum process tomography without ancilla. We generalize this scheme such that it can be implemented efficiently using only a set of measurements involving product operators.

Summary: Quantum process tomography (QPT) is a way to characterize general quantum evolutions. The mathematical framework of such a characterization is based on the fact that any physically valid quantum dynamics is a completely positive (CP) map and can be expressed as an operator sum representation. If we choose a particular operator basis set, the map can in fact be represented via a process matrix χ. Hence, the task of the characterization of a quantum process is equivalent to the χ matrix estimation. This is the standard protocol

72 for QPT. In order to get a valid quantum map, the estimated χ matrix should be a unit trace, positive Hermitian operator. The complete characterization of the quantum process based on the standard QPT protocol is experimentally as well as computationally a daunting task, as it requires high cost state tomographs. Several attempts have been made in the past few years to simplify the QPT protocol, which involve prior knowledge about the commutation relations of the system Hamiltonian and the system environment interaction Hamiltonian, performing ancilla- assisted tomography, using techniques of direct characterization of quantum dynamics, and process tomography via adaptive measurements. Although these methods offer some advantages over standard QPT, they still are not very useful when only certain elements of the χ matrix need to be estimated. Hence much effort has recently focused on achieving a selective estimation of elements of the χ matrix via a technique called selective and efficient quantum process tomography (SEQPT) without ancilla. The SEQPT without ancilla method interprets the elements of the χ matrix as an average of the survival probabilities of a certain quantum map; while the method certainly has advantages over other existing schemes, it still requires a large number of state preparations and experimental settings to carry out complete process tomography.

Block diagram of MSEQPT protocol. Step 1: Preparation of basis operator Ei state. Step 2: Application of quantum channel. Step 3: Mapping of basis operator to measurements of individual spin magnetizations followed by expectation value measurements of Pauli z-operators on subsystems. Step 4: Detection on individual spins.

In this work, we propose a generalization of the SEQPT method without ancilla, which requires fewer experimental resources as compared to the SEQPT or the standard QPT protocols. We exploit the fact that the density operator proportional to identity does not produce any NMR signal and use the product operator formalism to achieve selective estimation of the quantum process matrix to a desired precision.

(a) Molecular structure of 13C labeled chloroform used as a two-qubit quantum system. The first and second qubits are encoded as the nuclear spin 1H and 13C, respectively. The values of the scalar coupling JCH (in Hz) and relaxation times T1 and T2 (in seconds) and chemical shifts νi are shown alongside. (b) Thermal equilibrium NMR spectrum after a π/2 detection pulse.

We call our scheme modified selective and efficient quantum process tomography (MSEQPT). Our scheme achieves a simplification of the QPT protocol as in this scheme the detection settings need not be changed each time while estimating different elements of χ matrix. Our scheme is efficient as it relies on calculating the expectation values of special Hermitian observables by locally measuring the expectation values of the basis operators in a predecided set of quantum states. We experimentally demonstrate our scheme by implementing it on a two-qubit NMR system, where we tomograph the “no operation,” the controlled-NOT, and the controlled-Hadamard gates.

Results:

Identity CNOT gate gate

Control Hadamard

The fidelity of identity operation, CNOT gate and C-hadamard gate turns out to be 0.98, 0.93 and 0.92 respectively.

References: [1] “Experimental demonstration of selective quantum process tomography on an NMR quantum information processor” by Akshay Gaikwad, Diksha Rehal, Amandeep Singh, Arvind, and Kavita Dorai. Phys. Rev. A 97, 022311 (2018) [2] “Selective and efficient quantum process tomography” by Ariel Bendersky, Fernando Pastawski, and Juan Pablo Paz. Phys. Rev. A 80, 032116 (2009)

[3] “Selective and Efficient Quantum Process Tomography without Ancilla” by Christian Tomás Schmiegelow, Ariel Bendersky, Miguel Antonio Larotonda, and Juan Pablo Paz. Phys. Rev. Lett. 107, 100502 (2011).

Experimental Entanglement Detection of Unknown Tripartite States on Spin Ensemble using NMR

Amandeep Singh, Harpreet Singh, Arvind, and Kavita Dorai

Department of Physical Sciences, Indian Institute of Science Education & Research Mohali, Sector 81 SAS Nagar, Manauli PO 140306 Punjab India. ([email protected])

Abstract: Entanglement detection of tripartite random states using four observables [1] have been implemented successfully using NMR quantum processor. Six SLOCC inequivalent [2] entangled classes were detected experimentally using protocol [1].

Summary: Introduction: For three qubit quantum system, there are six SLOCC inequivalent [2] classes i.e. GHZ, W, three biseparable and separable. One of the entanglement measures, to characterize the entanglement, is n-tangle. Three tangle characterize three qubit entanglement via interplay between concurrence and entanglement of formation and has shown to be an entanglement monotone. A non-vanishing three tangle is a signature, and hence can be explored for detection, of GHZ class of entangled states. This approach is particularly interesting and general as any three qubit pure state can acquire the canonical form:

|훹〉 = ɑ|000〉+ ɑ푒 |100〉 + ɑ|101〉 + ɑ|110〉 + ɑ|111〉 (1)

2 with ai ≥ 0, Σ (a i) = 1 and θ ϵ [0; π] in the computational bases. Three tangle for state 2 2 given in Eq. (1) comes out to be τ = 4a2 a4 . Three tangle for generic state (Eq. 1) can be measured experimentally by measuring the expectation value of the Pauli’s operator O = 2σx⊗ σx⊗ σx, acting on the three qubit Hilbert space, in the state |Ψ>. Equivalently, O can be rewritten as 2σ1xσ2xσ3x for compactness. σx, y, z are the Pauli’s matrices. One can easily verify 2 2 that <Ψ|O|Ψ> =( Ψ ) = 4τ. A nonzero tangle will imply that state under investigation is in GHZ class [2]. In order to further assess the classes of three qubit generic states, three more observables have been defined [1] as O1 = 2σ1xσ2xσ3z, O2 = 2σ1xσ2zσ3x and O3 = 2σ1zσ2xσ3x. Experimentally measuring the expectation values of the operators O, O1, O2 and O3 can reveal the entanglement class of the generic state under investigation as detailed in [1]. Further two quantities have been defined as P = Ψ Ψ and Q = Ψ + Ψ + Ψ. Based on the mathematical structure developed in [1], Table 1 summarizes the classification of six SLOCC inequivalent classes based on the expectation values of the observables O, O1, O2, O3 as well as the quantities P and Q.

Table 1. Decision table for the classification of three qubit pure entangled states based on the expectation values of operators O, O1, O2, O3 in state |Ψ>. Each class in the row is shown with the expected values of the observables.

Fig. 1. (a) Quantum circuit to achieve mapping of the state ρ to measure O, O1, O2, O3 followed by measurement of qubit# 3 in the computational basis. (b) NMR pulse sequence of the mapping quantum circuit. All the blank rectangles are π/2 spin selective hard pulses while black rectangles are π pulses. Phases are written above respective pulse. Bar over a phase represent negative phase. Delays τij = 1/8Jij .

Experimental Implementation: To implement the protocol experimentally, 13C labeled diethyl fluoromalonate dissolved in acetone-D6 sample was used. 1H, 19F and 13C spin-half nuclei were encoded as qubit 1, qubit 2 and qubit 3 respectively. Before proceeding with the actual experimental demonstration, the system was initialized in (pseudo) pure state i.e. |000> by spatial averaging method. NMR parameters like chemical shifts, T1 and T2 relaxation times, scalar couplings Jij as well as thermal equilibrium and pseudo pure state (PPS) NMR spectra are shown in Fig. (2). Let the observable, whose expectation value is to be measured in the state ρ =|Ψ><Ψ|, be † ʘ. Instead of measuring <ʘ>ρ, the state ρ is mapped to ρi using ρi = UiρUi followed by subsystem measurement of the expectation value of Pauli’s local σz. Ui can be worked out depending upon the observable to be measured. In the present study the observables of interest are O, O1, O2 and O3 as described above and Table 1. Quantum circuit to achieve required mapping is shown in Fig. 1 (a). This circuit is particularly designed for current study and mapping of the state ρ followed by the measurement of the third qubit’s σz in the mapped state. Depending upon the experimental settings, <σz> in the mapped states is indeed the expectation values of O, O1, O2 or O3 in the initial state ρ. Fig. 1 (b) depicts the NMR pulse sequence used to experimentally implement the discussed scheme.

Fig. 2. (a) Molecular structure of 13C-labeled Fig. 3. Bar plots of the expectation values of the diethyl fluoromalonate and NMR parameters. observables O, O1, O2 and O3 for states numbered NMR spectra of (b) thermal equilibrium state (c) from 1-27. Horizontal axes denote the state pseudo pure state. number while vertical axes represent the value of the respective observable. Black, orange and white bars represent theoretical (The.), direct (Dir.) experimentally measured and QST based expectation values respectively.

Three qubit entanglement detection protocol [1] have been tested on twenty-seven states to demonstrate the efficacy experimentally. Seven states were used from six different entangled classes i.e. GHZ (two states), W, three biseparable and a separable (State number 1-7). In addition, twenty random generic states were prepared and labeled as R1, R2, R3,...... ,R20 (State number 8-27 for plot in Fig. (3)). To prepare the random states the MATLAB R (2016a, The MathWorks, Mohali, India) random number generator was used. To experimentally prepare the random generic three qubit states our recent [3] scheme have been utilized. Results: All the representative states of six inequivalent entanglement classes have successfully been detected which is further supported by Negativity measurement (not shown here). Results of random states are more interesting. As an example R10 and R11 has negativity of approximately 0.3, which implies that states have genuine tripartite entanglement. But one can observe from Fig. (3) that R10 has a nonzero 3-tangle and indeed it is in GHZ class of states. On the other hand, R11 has a vanishing 3-tangle with non-vanishing expectation values of O1, O2 and O3 indicates the state to be in W class. For instance, all vanishing expectation values as well as a near zero negativity for R8 state imply it to be a separable state. States R17, R18 and R20 also belongs to separable class. Similarly, other results can be interpreted. Surprisingly, none of the randomly prepared state has fallen under biseparable class. This might had happened as we prepared only twenty random states. A possible reason could be that the chances of a randomly prepared generic state are too remote that a system will undergo an evolution only under one coupling which indeed is the requirement to create a biseparable class of state.

References: [1] S. Adhikari, C. Datta, A. Das, and P. Agrawal, “Distinguishing different classes of entanglement of three-qubit pure states in an experimentally implementable way” arXiv:1705.01377 (2017). [2] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and A. V. Thapliyal, “Exact and asymptotic measures of multipartite pure-state entanglement”, Phys. Rev. A, 63, 012307 (2000) [3] S. Dogra, K. Dorai, and Arvind, “Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor”, Phys. Rev. A, 91, 022312 (2015).

Effect of Thermal Noise on Optimal Cloning of Quantum states

Anirudh Reddy, Joseph Samuel, Supurna Sinha Raman Research Institute, Bangalore, India

Abstract: We study quantum cloning using atom-photon interactions in the presence of thermal noise in the system to see the effects on the fidelity of the cloning process. Summary: Copying information is a central aspect of information theory. While one can make an exact copy of classical information, it is not the same in the case of quantum information. In other words, we cannot make an exact replica of an arbitrary qubit, which is a direct result of the no-cloning theorem [1, 2]. This is because of linearity and unitarity of quantum mechanics. In 1996 when Buz̆ ek and Hillery [3] came up with the concept of imperfect cloning. They argued that the cloning procedure is dependent upon the input state, i.e., it copies the states |휙⟩ and |휓⟩ perfectly, while the copies of the superposition states are imperfect. They came up with the concept of a universal quantum-copying machine, which is independent of the input state. In other words, it copies all the states imperfectly, but the degree of imperfection is independent of the input state.

Here we use an atom-photon system [4, 5] for studying the cloning process. In this system, achievement of a perfect clone is obstructed by the presence of spontaneous emission. Even the study of the optimal cloning process is an ideal case. We study how the cloning process gets affected when the systems is not ideal. To achieve this we introduce thermal noise in the system at the atomic level and the photon level to see the effects and see how the fidelity gets affected. In the process we find that the traditionally used fidelity does not show the effects of temperature. Hence, we use a slightly different form of fidelity to analyze the effects of temperature.

References: [1] W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299(5886): 802– 803, Oct 1982. [2] D. Dieks. Communication by epr devices. Physics LettersA, 92(6): 271–272, 1982. [3] V. Bužek and M. Hillery. Quantum copying: Beyond the no-cloning theorem. Phys. Rev. A, 54:1844–1852, Sep 1996. [4] Christoph Simon, GregorWeihs, and Anton Zeilinger. Optimal quantum cloning via stimulated emission. Phys. Rev.Lett., 84:2993–2996, Mar 2000. [5] Antía Lamas-Linares, Christoph Simon, John C. Howell, and Dik Bouwmeester. Experimental quantum cloning of single photons. Science, 296(5568):712–714, 2002.

Anjali K NON-LOCALITY AND ENTANGLEMENT IN SYMMETRIC THREE QUBIT STATES

K. Anjali, H. S. Karthik, Akshata Shenoy, Sudha‡ and A. R. Usha Devi Department of Physics, Jnanabharathi, Bangalore University,Bangalore-560056 ‡ Department of Physics, Jnanasahyadri, Kuvempu University, Shankaraghatta – 577451, India

Abstract: We investigate non-local correlations in 3-qubitsymmetric states using Majorana geometric construction. We focus speciﬁcally on the violation of conditioned Clauser-Horne- Shimony-Holt (CHSH) inequalities by 3-qubit symmetric states and identify how non-locality manifests in terms of Majorana geometric representation.

Summary: There has been a growing demand for more general ways of characterizing nonlocality [1], pertaining not only to the quintessential bipartite case, but also the multipartite case. In this work, we focus on investigating non-locality of 3 qubit system, obeying exchange symmetry, based on the conditioned Clauser-Horne-Shimony-Holt (CHSH) inequalities [,2,3], given by

CHSH = 〈퐴퐵〉 + 〈퐴퐵〉 + 〈퐴퐵〉 − 〈퐴퐵〉 −2푝(푐) ≤ 0 (1) where

〈퐴퐵〉 = 푝(푐) 푎 푏 푝(푎, 푏|푐), 푖, 푗 = 1,2 (2) ,± Here, 푝(푎, 푏|푐) = 푝(푎, 푏, 푐)/푝(푐) denote the conditional probabilities extracted when two of the parties Alice and Bob measure the observables 퐴퐵, 푖, 푗 = 1,2, respectively -- conditioned on the outcomes c = +1 or c=-1of Charlie’s measurement -- to obtain dichotomic outcomes a, b = ±1. Violation of the conditioned CHSH inequalities (1) has been shown to be useful in the super activation of non-locality in quantum networks [2]. Moreover, robustness of non-locality of N-qubit permutation symmetric states under noises has also been investigated recently [4, 5]. As permutation symmetric N-qubit states can be geometrically represented in an elegant manner [6, 7] in terms of N points on the Bloch sphere S2, exploring on local features through the window of Majorana geometric parametrization would be interesting. In this work we report violation of conditioned CHSH inequality in symmetric 3 qubit systems corresponding to 2 and 3 distinct Majorana star constellations on the Bloch sphere. We identify that the maximum violation of the states belonging to the 3 distinct spinor class is bounded between the value 2+2√2 to 4√2,

79 whereas for the three qubit states of the 2 distinct spinor class the maximum violation lies between 4 to 2+2√2. Thus, it is possible to explicitly bring out the connection between nonlocality in the two different SLOCC classes of the three qubit states (illustrating examples of which are the GHZ and W states respectively). In other words, violation of the conditional CHSH inequality throws light on the nature of genuine entanglement in the three qubit states belonging to the 2 and 3 distinct spinor Majorana family. We also explore the sensitivity of violation to amplitude damping and phase damping noise.

Acknowledgements: KA acknowledges financial support by Rajiv Gandhi National Fellowship (RGNF), India. ARU is supported by a UGC-MRP grant (Ref. MRP-MAJOR- PHYS-2013-29318).

References: [1] D. Cavalcanti, M. L. Almeida, V. Scarani and A. Acin, “Quantum networks reveal quantum nonlocality,” Nat. Commun.2, 184 (2011). [2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys.86, 419– 478 (2014). [3] R, Chaves, A. Acin, L. Aolita, and D. Cavalcanti, “Detecting nonlocality of noisy multipartite states with the Clauser-Horne-Shimony-Holt inequality,” Phys. Rev. A89, 042106 (2014). [4] A. I.Zaquine, E.Diamanti, and D. Markham “Decoherence effects on the nonlocality of symmetric states,” Phys.Rev.A91,022101 (2015). [5] P. Divianszky, R.Trensenyi, E. Bene, and T.Vertesi, “Bounding the Persistency of the nonlocality of W states,” Phys.Rev.A93,042113 (2016). [6] E. Majorana, “Atomiorientati in campo magneticovariabile,” Nuovo Cimento9, 43–50 (1932). [7] A. R. UshaDevi, Sudhaand A. K. Rajagopal, “Majorana representation of symmetric multiqubitstates,” Quantum Inf. Proc.11, 685–710 (2012).

Manipulation of entanglement sudden death in an all-optical experimental setup

Ashutosh Singh, Subhajit Bhar, and Urbasi Sinha Raman Research Institute, Bangalore, India.

Abstract: In this poster, we will discuss the manipulation of entanglement by an all-optical implementation of the NOT operations that can hasten, delay, or entirely avert entanglement sudden death.

Summary: INTRODUCTION: Quantum entanglement is a non-classical correlation shared among quantum systems which could be non-local in some cases. It is a fundamental trait of quantum mechanics. Like classical correlations, entanglement also decays with time in the presence of noise in the ambient environment. The decay of entanglement depends on the initial state and the type and amount of noise (Amplitude damping, Phase damping, etc.) acting on the system. The non-maximally entangled states |휓 ±⟩ = |훼||푔푒⟩ ± |훽| exp(푖훿)|푒푔⟩ always undergo asymptotic decay of entanglement, whereas |휙 ±⟩ = |훼||푔푔⟩ ± |훽| exp(푖훿) |푒푒⟩ undergo asymptotic decay for |훼|>|훽| and a finite time end called entanglement sudden death (ESD) for |훼|<|훽| in the presence of an amplitude damping channel (ADC) [1]. Since entanglement is a resource in quantum information processing, manipulation that prolongs entanglement will help realize protocols that would otherwise suffer due to short entanglement times. The practical question we want to address here is; whether, given a two-qubit entangled state in the presence of amplitude damping channel which causes disentanglement in finite time, can we alter the time of disentanglement by a suitable operation during the process of decoherence? A theoretical proposal exists in the literature for such manipulation of ESD through a local unitary operation (NOT operation in computational basis:휎) performed on the individual qubits which swaps their population of ground and excited states [2]. Depending on the time of application of this NOT operation, it can avoid, delay, or hasten the ESD. Based on this proposal, we have extended the experimental set up for ESD [1] and proposed an all-optical experimental set up for manipulating ESD involving the NOT operation on one or both the qubits of a bipartite entangled state in a photonic system [3]. For details, please check reference [3].

RESULT: We give analytical expressions for probabilities 푝; 푝&푝 in terms of the parameters of the initial state (density matrix). The first of these is the setting (“time”) for ESD, the next setting for the NOT marking the border between hastening and delay; that is, if the NOT is applied after 푝 (and of course before 푝), it actually hastens, ESD happening before 푝, whereas application before delays ESD to stretch past 푝 to larger but still finite value less than one. The third, 푝, marks the border between delaying or completely avoiding ESD. Applying the NOT after 푝 delays to a larger 푝 value whereas applying before avoids ESD altogether. Consider a two qubit entangled state of the form 푢 0 0 푣 휌(0,0) = 0 0 0 0, (1) 0 0 0 0 푣 ∗ 0 0 푥 The condition for ESD is given by, || 푝′ = , (2) () Let us denote the effective end of entanglement due to combined evolution through two ADCs by 푝. The 푝 involves a multiplication of survival probabilities to give, ′ 1− 푝 =(1− 푝)(1 − 푝) with 푝 = |푣|/푥 (3) depending only on the initial state parameters in (1).

For the manipulation of ESD using NOT operation on both the qubits, we get || 푝 = , 푝 = (4) () ||

For the manipulation of ESD using NOT operation on only one of the qubits, we get || || 푝 = , 푝 = (5) || ||

For 푢 = 0.2and |푣| = 0.15; 푝 = 0.1875, 푝 = 0.0274 (푝 does not exist), and 푝 does not exist in the physical domain for single (double) NOT operation. Therefore, NOT operation applied on only one (both) of the qubits delays as well avoids (only delays) the ESD. The corresponding plot of 푝 vs. 푝 is shown by solid (dashed) blue curve in figure (1).

CONCLUSIONS: We have proposed an all-optical experimental setup for the demonstration of hastening, delay, and avoidance of ESD in the presence of ADC in a photonic system. The simulation results of the manipulation of ESD considering a photonic system, when NOT operations are applied on one or both the qubits, are completely consistent with the theoretical predictions of reference [2] for the two -level atomic system where spontaneous emission is the ADC. We give analytical expressions for 푝, 푝&푝 which depend on the parameters of the density matrix of the system.

Our proposal also has an advantage over decoherence suppression using weak measurement and quantum measurement reversal, and delayed choice decoherence suppression. There, as the strength of weak interaction increases, the success probability of decoherence suppression decreases. In our scheme, however, we can manipulate the ESD, in principle, with unit success probability as long as we perform the NOT operation at the appropriate wave plate angle which is analogous to time in the atomic system. Delay and avoidance of ESD, in particular, will find application in the practical realization of quantum information and computation protocols which might otherwise suffer a short lifetime of entanglement. Also, it will have implications towards such control over other physical systems. The advantage of the manipulation of ESD in a photonic system is that one has complete control over the

82 damping parameters, unlike in most atomic systems. An experimental realization of our proposal will be important for practical noise engineering in quantum information processing, and is under way.

References: [1] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, Science 316, 555 (2007). [2] A.R.P. Rau, M. Ali, and G. Alber, EPL 82, 40002 (2008). [3] Ashutosh Singh, Siva Pradyumna, A. R. P. Rau, and Urbasi Sinha, J. Opt. Soc. Am. B 34, 681-690 (2017).

Swapping Intra-photon entanglement to Inter-photon entanglment using linear optical devices Asmita Kumari, Abhishek Ghosh, Mohit Lal Bera and A. K. Pan

National Institute Technology Patna, Ashok Rajhpath, Patna, Bihar 800005, India

Abstract: We propose a curious protocol for swapping the intra-photon entanglement between path and polarization degrees of freedom of a single photon to intra-photon entanglement between two spatially separated photons. Which have never interacted. We have also demonstrated an interesting quantum state transfer protocol, symmetric between Alice and Bob. Importantly, the Bell-basis discrimination is not required in both the swapping and state transfer protocols.

Summary: Introduction: In 1993, Bennett and colleagues [1] put forwarded a path breaking protocol for transporting an unknown quantum state from one location to a spatially separated one - a protocol now widely known as quantum teleportation. By exploiting the notion of quantum teleportation a fascinating consequence emerges known as entanglement swapping [2, 3]. In a swapping protocol, the entanglement can be generated between two photons which have never interacted. If photon A entangled with photon B and C entangled with photon D, then the entanglement can be created between A and D, although they never interacted in the past. However, the photons B and C need to be interacted with each other. Both the teleportation and entanglement swapping protocols require the Bell basis discrimination which is practically a difficult task to achieve using linear optical instruments. Results: Our experimental setup consists of three suitable MZIs where MZ and MZ belong to Alice and Bob respectively, and the third interferometer MZ is shared by both as shown in the Figure 1. The entire setup consists of six 50 : 50 beam splitters, four polarizing beam splitters, two polarization rotators, eight detectors and two mirrors are denoted by BS (i = 1, 2...6), 푃BS (j = 1, 2, 3, 4), PR (k = 1, 2), D (l = 1, 2..8) and M (m = 1, 2) respectively.

Figure 1: (color online) Implementing swapping setup on the state transfer protocol The task of this protocol is to generate a polarization-polarization or path-path entangled state between the photons in ‘1’ or ‘3’ while ensuring that they never interact. Depending on a suitable joint path measurement chosen by Alice and Bob on the joint state of the photons ‘1’and ‘3’ after the BS, the following polarization-polarization intra-photon entangled state | Ψ > = ac|H>|H>− bd|V>|V> can be generated. Hence, using our setup we have generated a polarization-polarization entanglement between the photons ‘1’ and ‘3’ even when they have never interacted with each other. It is important to note that, both the photons contain an intra-photon path-polarization entanglement that is swapped to the inter- photon entanglement between them. For state transfer protocol the measurements at Bob’s end produce the following states at Alice’s end are given by

| Ψ > = |ψ>(ac|H> − bd|V>) + |ψ>(ac|H> + bd|V>) | Ψ > = |ψ>(ac|H> + bd|V>) + |ψ>(ac|H> − bd|V>) | Ψ > = |ψ>(ac|H> + bd|V>) + |ψ>(ac|H> − bd|V>) | Ψ > = |ψ>(ac|H> − bd|V>) + |ψ>(ac|H> + bd|V>) According to the Bob’s instruction Alice perform some gate operations to obtain her desire state state | χ′ > = c|H> + d|V> . Hence, we demonstrated a state transfer protocol from Bob to Alice without any direct interaction between photons ‘1’ and ‘3’ in two interferometers MZ1 and MZ3.

References: [1] C. H. Bennett and G. Brassard, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, (Bangalore, India, 1984), p. 175. [2] M. Zukowski et al., Phys. Rev. Lett. 71, 4287 (1993). [3] J. W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998).

How to Measure the Quantum Measure

Álvaro Mozota Frauca1, Rafael Dolnick Sorkin1,2 1Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada 2Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA (Int J Theor Phys (2017) 56:232–258. DOI 10.1007/s10773-016-3181-x)

Abstract: The histories-based framework of Quantum Measure Theory assigns a generalized probability or measure μ(E) to every (suitably regular) set E of histories. Even though μ(E) cannot in general be interpreted as the expectation value of a self adjoint operator (or POVM), we describe an arrangement which makes it possible to determine μ(E) experimentally for any desired E. Taking, for simplicity, the system in question to be a particle passing through a series of Stern-Gerlach devices or beam-splitters, we show how to couple a set of ancillas to it, and then to perform on them a suitable unitary transformation followed by a final measurement, such that the probability of a final outcome of “yes” is related to μ(E) by a known factor of proportionality. Finally, we discuss in what sense a positive outcome of the final measurement should count as a minimally disturbing verification that the microscopic event E actually happened.

B Sharmila Entangled quantum states: Nonclassical effects and entanglement indicators

Pradip Laha, B Sharmila, S Lakshmibala and V Balakrishnan Department of Physics, IIT Madras, Chennai 600036, India

Abstract: In generic models of bipartite and tripartite systems of field-atom interactions, we estimate nonclassical effects, and investigate entanglement dynamics by proposing an entanglement indicator which is directly calculable from optical tomograms.

− −푖휃 + 푖휃 푋휃= (ɑ 푒 + ɑ 푒 )/√2 where θ(0<θ<π), is the phase of the single mode in the homodyne detection setup. It is evident that for θ=0 and π/2, respectively, we have the two field quadrature operators analogous to position and momentum, respectively. The optical tomogram W(푋, θ) corresponding to a density matrix ρ is given by w (X θ ,θ)=⟨ X θ ,θ|ρ|X θ ,θ ⟩ . These definitions can be extended in a straightforward manner to include multipartite systems as well. We first demonstrate the use of this approach for output states of a beamsplitter. For this purpose, experimentally relevant optical states such as the cat states are used as input states. We have quantified both quadrature squeezing and entropic squeezing directly from the optical tomograms also, for bipartite systems evolving in time with inherent nonlinearity present in the atom-field system. Further, the rich entanglement dynamics of tripartite systems such as a three-level atom interacting with radiation fields is also examined in a general setting with intensity-dependent field-atom couplings. The bipartite system examined is a multilevel atom interacting with a non-linear medium [7]. We investigate the roles played by classical and quantum correlations in determining the extent of entanglement at various instants during dynamical evolution of this system. (A similar system in a somewhat different setting is the double-well Bose-Einstein condensate). For our purpose, we compare standard measures of entanglement such as subsystem von Neumann entropy (SVNE) and subsystem linear entropy (SLE) with two tomographic entropy-based indicators that we have developed. We have shown that in the two systems considered, one of these indicators, namely, tomographic entanglement indicator (TEI) is in fair qualitative agreement with SVNE and SLE [8]. Inspired by [9], we have improved the TEI to obtain the other indicator which we refer to as the ‘dominant tomographic entanglement indicator’ (DTEI). We emphasize that while both these indicators primarily capture the classical correlations, for certain regions of nonlinearity and interaction strength parameters, and for certain initial states, they compare very favourably with SVNE and SLE. Lessons learnt from our preliminary tomographic analysis (in the case of the beamsplitter) of a wide class of states including ‘Janus-faced’ partner states [10] are used in adopting this approach to a tripartite quantum system [11]. This is a Λ atom interacting with two radiation fields, with nonlinearities in the two fields and an intensity-dependent coupling of the form √1+ 휅푎푎. Here, the intensity parameter 휅 varies from 0 to 1. This form of the coupling [12] is a straightforward generalisation of the two-photon realisation of the SU (1, 1) algebra in Jaynes-Cummings models with intensity dependence. We investigate the roles played by field nonlinearity and intensity-dependent coupling in determining the extent of quadrature and entropic squeezing of relevant subsystem states, during temporal evolution. The extent of entanglement and the time duration over which the entanglement collapses to a constant non- zero value (in contrast to sudden death) are sensitive to the extent of nonlinearity, the precise nature of the initial unentangled state and the extent of decoherence.

[6] S. D. Nicola, R. Fedele, M. A. Man’ko and V. I. Man’ko, J. Phys.: Conf. Ser. 70012007 (2007). [7] G. S. Agarwal and R. R. Puri, Phys. Rev. A 39 2969 (1989). [8] B. Sharmila, K. Saumitran, S. Lakshmibala, and V. Balakrishnan, J. Phys. B: At. Mol. Opt. Phys. 50 045501 (2017). [9] S. Ghose and B. C. Sanders, Phys. Rev. A 70 062315 (2004). [10] P. Shanta et. al., Phys. Rev. Lett. 72 1447 (1994). [11] Pradip Laha et. al., Int. J. Theor. Phys. 55 4044 (2016); arXiv:1705.08190 (2017). [12] S. Sivakumar, J. Phys. A: Math. Gen. 35, 6755 (2002).

Bikash K. Behera Experimental Demonstration of Quantum Tunneling in IBM Quantum Computer

Narendra N. Hegade1, Bikash K. Behera2 and Prasanta K. Panigrahi2

1 Department of Physics, National Institute of Technology Silchar, Silchar 788010, India 2 Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India (*These authors have equal contribution to the work.)

Abstract: Here we simulate quantum tunneling through a potential barrier by using IBM quantum computer. We clearly visualize the tunneling process by taking a two-qubit system and verify our experimental results.

Summary: Quantum simulation is one of the problems that a quantum computer could perform more efficiently than a classical computer as it provides significant improvement in computational resources [1, 2]. It has been applied in a wide range of areas of physics like quantum many-body theory [3], quantum phase transitions [4] and molecular physics [5, 6] etc. Algorithms have been used in simulating many quantum field theory problems where Hamiltonian of the system [7-11] splits into kinetic and potential energy operators which are then simulated using Trotter's formula [12]. Experimental realizations of quantum simulations have already been made in systems like NMR [13, 14], ion-trap [15-18], atomic [19] and photonic [20] quantum computers. The current status of this field can be found out from this review paper [21].

Quantum tunneling acts as one of the exciting phenomena and the unique fundamental phenomena in quantum mechanics. It has been observed in superconducting Cooper pairs [22]. It has also been utilized in modern technologies [23]. Important science puzzles like lattice quantum chromodynamics can be solved using this tunneling simulation approach.

This type of simulation has remained untested in a quantum computer due to the requirement of large number of ancillary qubits and quantum gates. Recently, an algorithm proposed by Sornborger [24] illustrates the simulation by using no ancillary qubits and a small number of quantum gates which motivates the possibility of simulating in today's quantum computer consisting of few qubits. Feng et al. [25] have demonstrated the tunneling effect using NMR. Ostrowski [26] has also explicated this process and calculated the transmission and reflection coefficients for the Gaussian wave packet scattered on a rectangular potential. Here, in the 87 present work, we illustrate the simulation using IBM's 5-qubit quantum computer. Using only two qubits and a set of Hadamard and controlled phase gates, we were able to simulate the tunneling process in a double well potential for a single particle.

References: [1] Lloyd, S. Universal Quantum Simulators. Science 273, 5278 (1996). [2] Zalka, C. Simulating quantum systems on a quantum computer. Proc. R. Soc. Lond. A 454, 313-322 (1998). [3] Peng, X. H., Zhang, J. F., Du, J. F. & Suter, D. Quantum simulation of a system with competing two- and three-body interactions. Phys. Rev. Lett. 103, 140501 (2009). [4] Edwards, E. E. et al. Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins. Phys. Rev. B 82, 060412 (2010). [5] Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 5741 (2005). [6] Du, J. F. et al. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. Phys. Rev. Lett. 104, 030502 (2010). [7] Buchler, H. P. et al. Atomic quantum simulator for lattice gauge theories and ring exchange models. Phys. Rev. Lett. 95, 040402 (2005). [8] Cirac, J. I. et al. Cold atom simulation of interacting relativistic quantum field theories. Phys. Rev. Lett. 105, 190403 (2010). [9] Maeda, K. et al. Simulating dense QCD matter with ultracold atomic boson-fermion mixtures. Phys. Rev. Lett. 103, 085301 (2009). [10] Weimer, H. et al. A Rydberg quantum simulator. Nat. Phys. 6, 382 (2010). [11] Casanova, J. et al. Quantum simulation of quantum field theories in trapped ions. Phys. Rev. Lett. 107, 260501 (2011). [12] Jordan, S. P. et al. Quantum algorithms for quantum field theories. Science 336, 1130- 1133 (2012). [13] Somaroo, S. et al. Quantum simulations on a quantum computer. Phys. Rev. Lett. 82, 5381-5383 (1999). [14] Brown, K. R. et al. Limitations of quantum simulation examined by a pairing Hamiltonian using nuclear magnetic resonance. Phys. Rev. Lett. 97, 050504 (2006). [15] Gerritsma, R. et al. Quantum simulation of the Dirac equation. Nature 463, 68-71 (2010). [16] Gerritsma, R. et al. Quantum simulation of the Klein paradox with trapped ions. Phys. Rev. Lett. 106, 060503 (2011). [17] Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57-61 (2011). [18] Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106-111 (2010). [19] Kinoshita, T. et al. Observation of a one-dimensional Tonks-Girardeau gas. Phys. Rev. B 82, 060412 (2010). [20] Ma, X. S. et al. Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems. Nat. Phys. 7, 399-405 (2011). [21] Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. 8, 264-266 (2012). [22] Josephson, B. D. Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251 (1962). [23] Binnig, G. & Rohrer, H. Scanning tunneling microscopy. IBM J. Res. Dev. 44, 279-293 (2000).

[24] Sornborger, A. T. Quantum simulation of tunneling in small systems. Sci. Rep. 2, 597 (2012). [25] Feng, G.-R., Lu, Y., Hao, L., Zhang, F.-H. & Long, G.-L. Experimental simulation of quantum tunneling in small systems. Sci. Rep. 3, 2232 (2013). [26] Ostrowski, M. Quantum simulation of the tunnel effect. Bull. Pol. Acad. Tech. Sci. 63, 0042 (2015).

Chandan Datta An authentication protocol based on polygamous nature of quantum steering

Debasis Mondal1, Chandan Datta 2,3, Jaskaran Singh 4, and Dagomir Kaszlikowski 1,5

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543. 2 Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, Odisha, India. 3 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India. 4 Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81 SAS Nagar, Manauli PO 140306 Punjab India. 5 Department of Physics, National University of Singapore, 2 Science Drive 3, 117542 Singapore, Singapore.

Abstract: In this paper, we exploit the asymmetric nature of quantum steering and show that there exist three qubit states which are correlated only via a polygamous relationship.

Summary: Quantum mechanical correlations offer many surprises whenever one digs into the theory to understand its nature and differences from the classical world. Some examples include correlations arising from entanglement [1], Bell non-locality [2-4], contextuality [5, 6], coherence [7, 8] and steering [9]. Such correlations have the unique and surprising property of being monogamous [10-16]. Correlations between certain parties are said to be monogamous if they diminish when shared among more additional parties. A simple example is illustrated by Bell-CHSH inequality [14]: Two parties Alice and Bob share non-local correlations and are able to violate the Bell inequality. If the state of Alice is also entangled with a third party Charlie, the non-local correlations between Alice and Bob diminish as the correlations between Alice and Charlie increase. It is therefore implied that the Bell-CHSH correlations are monogamous. Monogamy of correlations has been extensively studied and has found widespread applications in information theoretic tasks like key distribution [17, 18]. Quantum steering [9] was first introduced for bipartite parties. In this scenario, Bob prepares an entangled state of joint systems A and B. He keeps the system B with himself and transmits the system A to Alice. Alice does not trust Bob, but believes that the system sent to her is quantum. Bob's job is to convince Alice that the system he sent indeed belongs to an entangled state and it is possible for him to steer her state. While it is well known that steering is monogamous [19, 20], a major aspect of it has not been addressed yet. Steering at

89 its core is asymmetric and although it's monogamous from one side, the nature of correlations from the other side is yet to be studied. Consider a scenario, where three parties Alice, Bob and Charlie share an entangled state ρabc. From monogamy of steerability [19, 20] if Alice can steer Bob, she cannot steer Charlie and vice-versa. The Idea originates from the resource theory of quantum correlations like entanglement [1, 21]. However, the question we address here is to check whether there exist states for which a particular party (say Alice) cannot be steered independently either by Bob or Charlie but rather only if they steer together. The scenario is exact opposite of what is generally considered to show monogamous nature of quantum steering. We consider a tripartite state ρabc distributed between Alice (A), Bob (B) and Charlie (C). Alice asks Bob and Charlie to perform projective measurements on their respective systems (B) and (C) on stated bases. We consider Bob and Charlie to perform projective measurements in Pauli eigenbases (or in general on a set of mutually unbiased bases) and communicate the results to Alice. Upon receiving the results, Alice measures coherences on her conditional states with respect to her Pauli eigenbases (or a mutually unbiased bases) with the choice of basis being based on the measurement results from Bob and Charlie. Using this we have constructed six inequalities. It is seen that Bob, together with Charlie can steer the state of Alice if any of the six inequalities are violated. We denote this set of inequalities as first set. We consider another scenario where Alice ignores the results sent by one party while acknowledging the other. We construct a set of four steering inequalities in this two-qubit scenario, where Alice ignores the results of Charlie. Similar steering inequalities can be constructed when Alice ignores the results of Bob. We denote these inequalities as second set. It is our aim to look for a set of states which violate at least one of the first set but not the second set of inequalities. This would ensure that the state of Alice can be steered by Bob and Charlie together but not individually. In this paper we provide a detailed analysis of quantum steering in such a scenario. We identify a set of states for which Alice can share a polygamous relationship with Bob and Charlie and also lay down the foundation for identifying the complete set of such states. We show a possible advantage offered by a polygamous relationship much like its monogamous counterpart in the security of quantum key distribution (QKD) protocols. To that end we propose a new key distribution protocol, termed as quantum key authentication (QKA), which utilizes the GHZ state. The protocol is different than the standard QKD protocols and is made semi-device independent by the aforementioned polygamous relationship. The security of the protocol remains the same as for entanglement based QKD schemes [22-24]. QKA offers various advantages over QKD and is manifested in the example we provide where it is used in conjunction with quantum private comparison (QPC) protocols [25-27].

References: [1]R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [2]A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [3]J. S. Bell, Physics 1, 195 (1964). [4]J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). [5]S. Kochen and E. P. Specker, Journal of Mathematics and Mechanics 17, 59 (1967). [6]A. Cabello, S. Severini, and A. Winter, Phys. Rev. Lett. 112, 040401 (2014). [7] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014). [8] D. Girolami, Phys. Rev. Lett. 113, 170401 (2014). [9] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007). [10] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004).

[11] D. Yang, Physics Letters A 360, 249 (2006), ISSN 0375-9601. [12] M. P. Seevinck, Quantum Information Processing 9, 273 (2010), ISSN 1573-1332. [13] P. Kurzynski, T. Paterek, R. Ramanathan, W. Laskowski, and D. Kaszlikowski, Phys. Rev. Lett. 106, 180402 (2011). [14] M. Pawlowski and C. Brukner, Phys. Rev. Lett. 102, 030403 (2009). [15] R. Ramanathan, A. Soeda, P. Kurzynski, and D. Kaszlikowski, Phys. Rev. Lett. 109, 050404 (2012). [16] D. Saha and R. Ramanathan, Phys. Rev. A 95, 030104 (2017). [17] M. Pawlowski, Phys. Rev. A 82, 032313 (2010). [18] J. Singh, K. Bharti, and Arvind, Phys. Rev. A 95, 062333 (2017). [19] M. D. Reid, Phys. Rev. A 88, 062108 (2013). [20] A. Milne, S. Jevtic, D. Jennings, H. Wiseman, and T. Rudolph, New Journal of Physics 16, 083017 (2014). [21] M. B. Plenio and S. S. Virmani, An Introduction to Entanglement Theory (Springer International Publishing, Cham, 2014), pp. 173{209, ISBN 978-3-319-04063-9. [22] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [23] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009). [24] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002). [25] A. C. Yao, in Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, USA, 1982), SFCS '82, pp. 160-164. [26] H.-Y. Tseng, J. Lin, and T. Hwang, Quantum Information Processing 11, 373 (2012), ISSN 1573 1332. [27] X.-B. Chen, G. Xu, X.-X. Niu, Q.-Y. Wen, and Y.-X. Yang, Optics Communications 283, 1561 (2010), ISSN 0030-4018.

Debarshi Das Steering a single system sequentially by multiple observers

DEBARSHI DAS1, SOURADEEP SASMAL1, SHILADITYA MAL2,3, A. S. MAJUMDAR2

1Centre for Astroparticle Physics and Space Science (CAPSS), Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700 091, India. 2S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, India; 3Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India. [arXiv:1712:10227 (2017)]

Abstract: In this work we find the limit on the number of observers (measuring sequentially and independently on one particle) who can demonstrate EPR steering of a spatially separated single observer's particle.

Summary of the work: The subject of quantum correlations first received attention due to the seminal paper by Einstein, Podolsky and Rosen (EPR) [1], leading subsequently to the exploration by Bell that a local realist description, which reflects classical mechanics, is incompatible with quantum mechanics (QM) [2]. Schrodinger, unlike EPR, did not believe in the incompleteness of QM, but he was surprised by the fact that an observer can steer a system which is not in her possession [3]. Later Wiseman et al. [4, 5] have shown that

91 steering is the lack of a local hidden state (LHS) model through which an observer can simulate classically another remote party's state.

A key property of all quantum correlations is that they are monogamous in nature. The restriction on sharing of quantum correlations between several number of observers is quantitatively expressed through monogamy relations for entanglement, Bell-nonlocality, and EPR steering. Unlike entanglement and Bell-nonlocality, EPR steering is asymmetric with respect to the observers resulting in extra sophistication in the monogamy relations.

Monogamy relation for the analog of the Clauser-Horne-Shimony-Holt (CHSH) inequality for steering [6] implies that for three parties sharing a quantum state, any two of them cannot steer other parties state simultaneously [7]. On the other hand, in the context of n-settings linear steering inequality [8], it is shown that for spatially separated multiple observers sharing a quantum state, at most (n-1) parties can steer the nth party's system [9].

Quantum correlations also satisfy a fundamental physical postulate, i. e., the no-signalling condition (the probability of obtaining one party’s outcome does not depend on spatially separated other party's setting). Recently, it has been shown in [10] that monogamy relations no longer hold for the correlations obtained from measurements performed by different parties if the no-signalling condition is relaxed. Specifically, the scenario is that one observer (say, Alice) share half of an entangled pair whereas several observers (say, several Bobs) can share and measure on another half of that pair sequentially. It has been shown through numerical evidences that at most two Bobs can demonstrate violation of the CHSH inequality [11] with single Alice when the measurements of each of the several observers at one side are unbiased with respect to the previous observers. This result is proved analytically in [12] and confirmed by experiment [13,14].

In the present work we investigate how many Bobs can steer single Alice, when Alice shares half of a bipartite entangled state, and multiple Bobs share and measure sequentially on another half. In particular, we have considered two spatially separated particles in maximally entangled state. Alice measures on the first particle and multiple Bobs (Bob1, Bob2, Bob3, ..., Bobn) measure on the second particle sequentially. After doing measurements on his particle Bob1 delivers the particle to Bob2, Bob2 also passes his particle to Bob3 after doing measurement and so on. Here, it is important to note that each Bob measures independently of the previous Bobs on the particle of his possession and we have considered unbiased input scenario, i. e., all possible measurement settings of each Bob are equally probable. Here no- signalling condition is not satisfied between each pair of observers (between different Bobs).

As we want to explore how many Bobs can have measurement statistics violating different steering inequalities with a single Alice, Bob1 cannot measure sharply. This would destroy the entanglement between the particles shared between Alice and Bob1, and there would be no possibility of violation of the steering inequality for the Bob2. Hence, in order to address the aforementioned problem with n Bobs, measurements of (n-1) Bobs should be weak. In the present work we have recast weak measurement formalism in terms of unsharp measurement [which is a particular form of positive operator valued measurement (POVM)] as the optimal pointer state condition derived in case of weak measurement is satisfied in unsharp measurement formalism [12].

To address the above issue we consider the analog of the CHSH inequality for steering, which is a necessary and sufficient condition for steering in the experimental scenario where

92 each party measures two dichotomic observables with mutually unbiased measurements for the party whose system is being steered [6].

In this case we find that at most two Bobs can steer Alice's system going beyond the monogamy restriction. We also consider linear steering inequalities with two as well as three settings which were proposed in [8].

In the scenario of a single Alice and multiple Bobs [10, 12], we find that at most two Bobs can steer Alice via violation of the two settings linear steering inequality. On the other hand, at most three Bobs can steer Alice via violation of the three settings linear steering inequality going beyond the monogamy restriction. Based on the above result, we conjecture that when steering is probed through n-settings linear steering inequality, at most n Bobs can steer Alice's system.

References: [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777-780 (1935). [2] J. S. Bell, Physics 1, 195 (1964). [3] E. Schrodinger, Proc. Cambridge Philos. Soc. 31, 553 (1935); 32, 446 (1936). [4] H. M. Wiseman, S. J. Jones and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007). [5] S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007). [6] E. G. Cavalcanti, C. J. Foster, M. Fuwa, H. M. Wiseman, J. Opt. Soc. Am. B 32, A74 (2015). [7] S. Mal, D. Das, S. Sasmal, A. S. Majumdar, arXiv:1711.00872 (2017). [8] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Phys. Rev. A 80, 032112 (2009). [9] M. D. Reid, Phys. Rev. A 88, 062108 (2013). [10] R. Silva, N. Gisin, Y. Guryanova, and Sandu Popescu, Phys. Rev. Lett. 114, 250401 (2015). [11] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). [12] S. Mal, A. S. Majumdar, D. Home, Mathematics 4 48 (2016). [13] M. Hu, C. Li, Z. Zhou, G. Guo, X. Hu, Y. Zhang, arXiv: 1609.01863. [14] M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, P. Villoresi, arXiv: 1611.02430.

Non-Universality of Quantum Dynamics Computed from Time-Correlation Functions

Devashish Pandey†, Guillem Albareda*, Xavier Oriols† * Theory Department, Max Planck Institute for the Structure and Dynamics of Matter Luruper Chaussee 149, 22761 Hamburg, Germany † Department of Electronic Engineering, Universitat Autónoma de Barcelona, 08193- Bellaterra, Barcelona, Spain

Abstract: Contrarily to classical mechanics, we show that quantum time-correlation functions are neither independent of the measurement process nor an intrinsic value of the ensemble dynamics.

Summary: Introduction: In classical mechanics, the outcome of a measurement is expected to be a pre- existing value of the system. This is because the perturbation induced by the classical measurement can be conceptually neglected, i.e., a measurement at time t1 does not affect a second measurement at a later time t2. Thus, the dynamics of a classical system defined in terms of a two-time correlation function of a single experiment is universal, i.e. it is independent of the measurement process and hence it reveals the intrinsic dynamics of the unmeasured system, viz., when nobody is looking. In quantum mechanics, however, the measurement process unavoidably perturbs the system and there is no reason to expect that the two-time correlation function of a single experiment will provide the dynamics of the system when nobody is looking. In fact, there are quantum theories (like Copenhagen one) where the concept of pre-existing values of the system (when nobody is looking) are ill-defined. Other quantum theories (known as quantum theories without observers) define pre-existing values at the ontological level, but they are contextual [1]. A quantum measurement process implies a stochastic outcome. Thus, quantum observables are obtained from an ensemble of identical experiments. For example, for a measurement of the position 푥(푡) at time 푡, the definition of its expectation value ⟨푥(푡)⟩ is universal in the sense that either a strong or a weak measurement provide the same ensemble average [2,3]. In this sense, one concludes that ⟨푥(푡)⟩ provides an intrinsic property of the system. In this conference, we will discuss whether or not quantum dynamics observables computed from two-time correlations for an ensemble of identical experiments are Universal and provide intrinsic dynamical information of the quantum system. Two-time Heisenberg-operator correlation function from weak measurements: We show that the standard expression of the two-time correlation function in terms of Heisenberg operators correspond, in fact, to a two-times consecutive weak measurement modeled by a Gaussian POVM with a spatial dispersion (휎) much larger than the characteristic one of the system (휎) [4,5]. We show that even with an apparatus modeled by a POVM with an almost infinite spatial dispersion 휎 ≫ 휎, there always exist some values of the weak measurement outcome (labelled here by the pointer position푦) that induce a strong perturbation on the system such that the expectation value of the two-time correlation function does not provide information of the intrinsic dynamics of the system.

In fact, an infinite number of two-time correlation (measurement-dependent) functions exist giving an infinite number of (measured) dynamics of the system. Therefore, unlike one-time quantum expectation values, two-time correlation functions cannot be Universal. The failure of acknowledging these limitation leads to misunderstandings in the literature [6, 7] when comparing measured and unmeasured dynamics.

Numerical results: For a one-dimensional harmonic oscillator (HO) defined by the degree of freedom 푥, we evaluate the autocorrelation function ⟨푥(푡)푥(푡)⟩ at two different times, 푡 and 푡. We follow a standard weak measurement protocol effectively described by a Gaussian Kraus operator, 퐺(푥, 푦), with the condition 휎 ≫ 휎. We show that the measurement process at time 푡 induces a spatial translation of the ground state by the amount 훼 = 푦휎⁄(휎) , and hence the measured state corresponds to a coherent state |훼⟩ (see in Figs. 1 and 2). Thus, it is the perturbation of the system due to the measurement that leads to the oscillatory behavior of the coherent state that accounts for the well-known result ⟨푥(푡)푥(푡 + 푇⁄2)⟩ = −⟨푥⟩, where T is the natural period of the HO. Due to its ontological nature, Bohmian mechanics allows the definition of unmeasured time-correlation functions [1]. Unmeasured, immovable, trajectories give rise to the result ⟨푥(푡)푥(푡 + 푇⁄2)⟩ =⟨푥⟩. Nevertheless, when

94 the system is measured by 퐺푥,푦(푡) with the condition 휎 ≫휎, the trajectories do provide the oscillatory result that fits the orthodox result (see Fig.1 and Fig.3). Any attempt to discuss this dynamical information without considering the effect of the measurement apparatus on the system evolution can easily induce misinterpretations [5,6].

Fig.1: Oscillatory Bohmian trajectory associated to the ground state of the HO being measured at time t1=0 through a Gaussian Kraus operator (described in Fig. 2). Each trajectory corresponds to a different value of the pointer measurement 푦 at time t1=0 which is related to the value 훼=푦휎⁄(휎) corresponding to different position displacements of the coherent state. In particular, 훼 = 0 (푦 = 0) corresponds to one immovable Bohmian trajectory associated to the unmeasured (non-perturbed) ground state giving ⟨푥(푡)푥(푡+푇⁄ 2)⟩ =⟨푥⟩ where T is the period of the HO, while we get ⟨푥(푡)푥(푡 + 푇⁄2)⟩ = −⟨푥⟩ for the rest of measured (perturbed) trajectories..

Fig.2: Logarithmic plot for the x- dependence ofground state of the HO (dashed line) and of the Gaussian Kraus operator G(x,y) (solid line) for different values of yrelated to different values of 훼=푦휎⁄(휎) . Under the condition 휎 ≫휎, 퐺(푥,푦) = exp (−(푥−푦) /2휎) can be approximated to exp (푥푦/휎), which can be identified as the term exp (푥훼/휎) of the coherent state. The green line corresponds to the term 푒푥푝(−푥 /(2휎)) ≈ 0.

Fig.3: Mean position as a function of time t2 for a two-time measured ground state of the HO. A first weak measurement at time t1=0 generates a perturbation on the system quantified by 훼=푦휎⁄(휎) =4 which affect the posterior mean position measurement at time t2. Black solid line corresponds to orthodox computation and red dashed line to a computation from an ensemble of Bohmian trajectories. Both theories give the same results except for irrelevant differences (due to numerical computations).

Conclusions: The dynamics extracted from the computations of the ensemble value of two- time Heisenberg-operators are discussed in terms of two consecutive weak measurements. It is shown that such ensemble definition of the dynamics is neither Universal nor provides information of the unmeasured dynamics. Important misunderstanding appears in the scientific community when trying to compare unmeasured (dashed line in fig.1) with measured (solid lines in fig.1) dynamics. The error comes on the account of “comparing apples and pears”. This discussion shed some light on the concept of the unmeasured dynamics of a quantum system which is widely used in many fundamental and practical quantum research fields.

References:

[1] X.Oriols and J. Mompart, “Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology” (Pan Stanford, Publishing, Singapore 2012) [2] F. L. Traversa, G. Albareda, M. Di Ventra, and X. Oriols "Robust weak-measurement protocol for Bohmian velocities" Phys. Rev. A 87, 052124 (2013). [3] D. Marian, N. Zanghi, X. Oriols "Weak Values from Displacement Currents in Multiterminal Electron Devices " Phys. Rev. Lett. 116, 110404 (2016). [4] G. Albareda, D. Pandey, and X. Oriols, “Understanding the Fundamental Connection between Time-Correlation Functions and the Underlying Quantum Dynamics”. in preparation [5] D. Pandey, G. Albareda and X. Oriols, “Non-Universality of Quantum Dynamics Computed from Time-Correlation Functions ” in preparation [6] Feligioni, L., Panella, O., Srivastava, Y. et al. “Two-Time Correlation Functions: Stochastic and Conventional Quantum Mechanics”. Eur. Phys. J. B, 48, 233 (2005). [7] A. Neumaier, “Bohmian contradicts quantum mechanics” in arXiv:quant-ph/0001011.

Gautam Sharma Quantum uncertainty relation based on the mean deviation

GAUTAM SHARMA, CHIRANJIB MUKHOPADHYAY, SK SAZIM, ARUN KUMAR PATI

Quantum Information and Computation Group, Harish-Chandra Research Institute, Homi Bhabha National Institute, Allahabad 211019, India (https://arxiv.org/abs/1801.00994)

Abstract: We present uncertainty relation, based on the mean deviation. We illustrate its robustness even when standard deviation blows up. We show its usefulness in detecting steering and entanglement detection.

Summary: MEAN DEVIATION BASED UNCERTAINTY RELATIONS: We define the mean deviation based uncertainty for an observable with respect to a quantum state in the following way –

Definition: (MD based uncertainty) – For any physical observable 퐴 = ∑ 푎|푎⟩⟨푎|, the mean deviation based uncertainty of 퐴 with respect to the state |휓 ⟩ is defined as ∆퐴 = |푎 − 〈퐴〉||⟨휓||푎⟩| = ⟨휓|퐴′|휓⟩ Let us define 퐴′= ∑ |푎 − 〈푎〉| |푎⟩⟨푎|, and hence 퐴′ = ∑ |푎 − 〈퐴〉| |푎⟩⟨푎|. Similarly for the operator 퐵 = ∑ 푏 |푏⟩⟨푏|, we define 퐵′ and write down the uncertainty of Bwith respect to the stateas ∆퐵 = |푏 − 〈퐵〉||⟨휓||푏⟩| = ⟨휓|퐵′|휓⟩.

Now, let consider the product of ∆퐴 and∆퐵 in the state |휓⟩ using (1) and (2) ∆퐴∆퐵 = ⟨휓퐴 휓⟩⟨휓퐵 휓⟩ ≥|⟨휓|퐴′퐵′|휓⟩| ,

⟹∆ 퐴∆ 퐵 ≥ |⟨휓|[ 퐴, 퐵]|휓 ⟩|. We also present sum uncertainty relations based on mean deviations. Theorem: (MD based uncertainty relation for incompatible observables) - For observables A and B, system state |Ψ ⟩and any state |Ψ ⟩orthogonal to the system state, the following uncertainty relation holds- ∆퐴 +∆퐵 ≥±푖〈[퐴 , 퐵 ]〉 + |⟨Ψ|퐴 ± 푖퐵 |Ψ ⟩| . (4)

1.a Intelligent states In this section we try to look for the intelligent states for the mean deviation based uncertainty relations. Although we could not find the most intelligent state. We found that the Laplace states are as intelligent as the Gaussian states for mean deviation based uncertainty relations 1.b State independent MD based uncertainty relation We prove the following state independent uncertainty relation. Theorem: (state independent MD based uncertainty relation) - The following MD based sum uncertainty relation holds for multiple observables {푂} and any 훼 ∈ ℝ 훼 ∑ ∆ (푂) ≥ 퐶 − ∑ lnmax ∑ 푒 . (10)

2. NEW UNCERTAINTY RELATION: EXAMPLES We illustrate two different scenarios where, in some regimes, the SD based uncertainty relations are inapplicable, but the new uncertainty relations hold true. The two scenarios are when the probability distribution functions are :- 2.a F-distribution 2.b Pareto Distribution

3. SOME APPLICATIONS OF THE MEAN-DEVIATION BASED UNCERTAINTY RELATIONS

In the present work, we seek to provide an answer in two such situations. First, we consider the problem of detecting EPR steering. Finally, we analyze the utility of mean deviation based uncertainty relations in entanglement detection.

3.a Detection of EPR violation in lossy scenario through MD uncertainty relation

In the lossy scenario one can detect steering using mean deviation based uncertainty relations for efficiency of 0.33 where as it was possible with standard deviation based uncertainty for efficiency of 0.579.

3.b Entanglement detection

We find that entanglement detection with mean deviation based uncertainty relations is as good as with standard deviation based uncertainty relations.

References: [1] E. G. Cavalcanti, P. D. Drummond, H. A. Bachor, and M. D. Reid, Opt. Express 17, 18693 (2009). [2] O. G¨uhne, Phys. Rev. Lett. 92, 117903 (2004). [3] A. K. Pati, Physics Letters A 262, 296 (1999), quantph/9901033.

[4] F. Grosshans, “Heisenberg’s uncertainty principle for mean deviation?” Physics Stack Exchange, https://physics.stackexchange.com/q/251858

‘Loophole’ free test of Macro-realism in optical system

Kaushik Joarder1, Debasis Saha2, Dipankar Home3, Urbasi Sinha1

1Raman Research Institute, Bangalore, India 2University of Gdansk, Poland 3Center for Astroparticle Physics and Space Science, Bose Institute, Kolkata

Abstract: We are going to demonstrate an optical setup to test macro-realism. The experiment will be the first to test all three conditions of macro-realism in the same setup, and in a ‘loophole-free’ way.

Summary: Macro realism or macroscopic realism is a viewpoint where a macroscopic system with two or more macroscopically distinct states is believed to be in one or the other of these states, at all times. To test if nature really support this viewpoint or rather follow quantum mechanical description of superposition, Leggett and Garg formalized an inequality (LGI) (1) which is similar in spirit to Bell’s inequality verifying Local-realism. But, LGI is only necessary condition for macro realism, not sufficient. On the other hand ‘no signaling in time (NSIT)’ condition was found both necessary and sufficient (2, 3). There are also other form of inequalities which can serve the purpose of verifying macro-realism, and lie in between NSIT and LGI. For example, Wigner form of LGI (WLGI) (4) is one of the candidates. In our lab we are trying to develop an experimental setup to test macro-realism in photonic system. We will perform all three types of test (LGI, WLGI, and NSIT) in our setup; to execute comparative study of these inequalities/equalities. Till date many experiments have been successfully implemented on various systems, but almost all experiments concerning macro-realism have essentially tested LGI, with only one experiment so far testing NSIT using superconducting flux qubit (5). Also most of them suffer from so called ‘loopholes’ in their experiment. One of our main goals is to develop a ‘loophole-free’ test of macro-realism.

o LGI inequality was formed considering macro realism in conjunction with another condition called Non-invasive Measurement (NIM). Till date, most of the LGI experiments did not implement ideal NIM. The closest to an ideal NIM is a type of measurement called Ideal Negative Result Measurement (INRM) which was introduced in the original LGI paper. In the INRM, the detector is coupled to only one of the two possible states at some time ‘t’. So, non-detection in that detector implies that the system is in the other state at time ‘t’. We are going to implement INRM in our photonic system.

o Another loophole in any LGI test comes from the fact that no detector is 100% efficient. This is similar to ‘fair sampling loophole’ in Bell test, where it is assumed that photons that haven’t been detected have similar probability distribution to the ones that got detected. We have considered different ontic state models to address this loophole, and were able to calculate minimum detector efficiency requirement for a conclusive LGI/WLGI/NSIT violation. We also found that while performing INRM, using perfect blocker instead of detector will reduce the efficiency requirements.

o For the single photon source we are using spontaneous parametric down conversion (SPDC) in a BBO crystal with type 1, degenerate, non-collinear arrangement, where the signal photons (810 nm) go to the experimental setup and the idler photons are used for heralding.

The experiment underway will be the first to test and compare all three conditions of macro- realism (LGI, NSIT, and WLGI) in the same setup, and that too in a ‘loophole-free’ way. All experimental results will be presented in the poster.

References: [1] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985). [2] J. Koﬂer and C. Brukner, Phys. Rev. A 87, 052115 (2013). [3] L. Clemente and J. Koﬂer, Phys. Rev. A91, 062103 (2015); Phys. Rev. Lett.116, 150401 (2016). [4] D. Saha, S. Mal, P. K. Panigrahi and D. Home, Phys. Rev. A91, 032117 (2015). [5] G. Knee et al. Nature Communications 7, 13253 (2016)

Our experience with the IBM quantum experience: The story of successful achievements and failures due to the limitations of the IBM quantum computers

Mitali Sisodia**, Abhishek Shukla* and Anirban Pathak** **Jaypee Institute of Information Technology, A-10, Sector 62, Noida, UP 201307, India E-mail address:[email protected] * University of Science and Technology of China, Hefei-230026, China, E-mail address: [email protected]

Abstract: We briefly review our experimental works on quantum computation and communication [1- 3] using the IBM quantum processors- IBMQX2 and IBMQX4 through cloud computing, which are the superconductivity-based quantum computer placed in the cloud by the IBM corporation [4].

Summary: First quantum computing platform is delivered by the IBM Corporation at T. J. Watson Research Center in New York in 2016 [4]. It is a superconductivity-(SQUID)-based

99 quantum computer, which is easily available over the internet to run and simulate the algorithms and experiments. The architecture of their quantum processor is based on transmon qubits. A transmon qubit is a type of superconducting charge qubit, which diminishes the sensitivity of charge noise. In IBM quantum computer, we are allowed to perform many single qubit and two qubit (CNOT) operations. Note that the CNOT gates are not allowed between all possible combinations of two qubits in the accessible five qubit quantum computer. Due to the absence of such couplings, we are restricted to apply CNOT gate in IBM quantum experience only between certain qubits that too in a directional manner. Here, by the directional manner we mean that in IBMQX2 a CNOT with control on qubit q[0] to target qubit q[1] is allowed but not vice versa. This is so as the IBM quantum experience uses the cross-resonance interaction as the basis for the CNOT-gate. Thus, this interaction is stronger when choosing the qubit with higher frequency to be the control qubit, and the lower frequency qubit to be the target, so the frequencies of the qubits determines the direction of the allowed CNOT gate [4].

This available quantum computer can be exploited to perform several tasks. We have also performed some experiments [1, 2] using the IBM quantum processors-IBMQX2 and IBMQX4. Specifically, we discuss and experimentally demonstrate a scheme for nondestructive Bell state discrimination [5, 6] on the IBM quantum computer and compare the results of this SQUID-based implementation [1] with the earlier non-destructive Bell state discrimination using an NMR based 3-qubit quantum computer [7].

The comparison would establish that to obtain scalability, IBM quantum computers (more precisely the quantum gates used in realizing them) have to reduce error rate. This point obtained via quantum state tomography is further established by performing the quantum process tomography of the Clifford group gates used in the IBM quantum experience. Apart from the gate fidelity, the restrictions imposed by the architecture of the quantum computers also often increase the gate count and thus reduce the fidelity with which the desired state is prepared. In our efforts to realize some quantum games, the fidelity is found to reduce below the classical limit [8]. However, in many occasions, IBM quantum computers are found to provide excellent results. Here, to establish the benefits of IBM QE we also discuss one such successful case in the context of quantum communication [2]. Specifically, we discuss a proof-of-principle experiment performed using IBMQX2 to realize an optimal scheme for teleportation of an n- qubit quantum state [2]. In these works [1,2], quantum state tomography is employed to extract information about the output states (which requires 8192 experimental runs). Once obtained density matrices can be used to compute fidelity of the states prepared in the experiments with that of the theoretical results, which can be used to check the accuracy of the IBM quantum computer. In our study [1],we realized and observed that the fidelity decreases as the number of gates in a quantum circuit increase. Thus, optimization of quantum circuit before implementing on IBM quantum computer is relevant.

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References: [1] Sisodia M, Shukla A, and Pathak A. "Experimental realization of nondestructive discrimination of Bell states using a five-qubit quantum computer." Physics Letters A 381, no. 46 (2017): 3860-3874. [2] Sisodia M, Shukla A, Thapliyal K, and Pathak A. "Design and experimental realization of an optimal scheme for teleportation of an n-qubit quantum state." Quantum Information Processing 16, no. 12 (2017): 292. [3] Sisodia M, Shukla A, and Pathak A. "Characterizing quantum operations on IBM quantum computer using quantum process tomography." Under preparation. [4] IBM quantum computing platform, http://research.ibm.com/ibm-q/qx/, 2016. [5] Panigrahi PK, Gupta M, Pathak A, Srikanth R. Circuits for distributing quantum measurement. InAIP Conference Proceedings 864, no.1 (2006 Nov 15): 197-207. [6] Gupta M, Pathak A, Srikanth R, Panigrahi PK. "General circuits for indirecting and distributing measurement in quantum computation." International Journal of Quantum Information 5, no. 04 (2007): 627-640. [7] Samal JR, Gupta M, Panigrahi PK, Kumar A. "Non-destructive discrimination of Bell states by NMR using a single ancilla qubit." Journal of Physics B: Atomic, Molecular and Optical Physics 43, no. 9 (2010): 095508. [8] Das D, Shukla A, and Pathak A. "A quantum magic-square game implemented on IBM Quantum computer."Under preparation.

Tight upper bound for the maximal quantum value of the Mermin operators

Mohd Asad Siddiqui1,*and Sk Sazim 2

1Department of Physical Sciences, IISER Mohali, 140306, India. 2Quantum Information & Computation group, HRI Allahabad, 211019, India. ( E-mail: [email protected], # arXiv:1710.10802)

Abstract: The violation of the Mermin inequality (MI) for multipartite quantum states guarantees the existence of nonlocality between either few or all parties. The detection of optimal MI violation is fundamentally important, but current methods only involve numerical optimizations, thus hard to find even for three qubit states. In this work, we provide a simple and elegant analytical method to achieve the upper bound of Mermin operator for arbitrary three qubit states. Also the necessary and sufficient conditions for the tightness of the bound for some class of states has been stated.

Summary: Introduction: Although quantum nonlocality is a puzzling aspect of microscopic world, it is an important resource for quantum information processing tasks. It has been shown previously that the quantum nonlocality between two parties can be revealed via various inequalities – popularly known as Bell-CHSH inequalities. However the scenarios in multipartite cases are yet to be discovered as they are much rich and complex in nature. For example, in the case of tripartite systems, two class of nonlocality exists, genuine tripartite nonlocality and the bipartite nonlocality.

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The genuine tripartite nonlocality can be unveiled by famous Svetlichny inequality (SI) whereas Mermin inequality (MI) reveals any form of non-locality present in the tripartite quantum states. The tight upper bound for these inequalities is fundamentally important. Many attempts are made to achieve the bounds using the numerical simulations. The first analytical solution was given by Horodecki et-al for Bell-CHSH inequality in [1]. Recently the attempt towards the generalization towards the Horodecki formula is made for three qubit states in [2], however the obtained bound are far from general and not always tight. In this work, we focus to find the tight quantum upper bound of Mermin operator, analytically, for arbitrary three qubit states. We find the maximum value of Mermin operator in terms of the singular values of tripartite correlation matrix, which not only works for pure states but also for mixed states. The Mermin operator and its quantum upper bound: The Mermin operator M is given as ℳ = 푎. 휎⃗ ⊗ 푏. 휎⃗ ⊗ 푐. 휎⃗ + 푎. 휎⃗ ⊗ 푏. 휎⃗ ⊗ 푐̂. 휎⃗ + 푎. 휎⃗ ⊗ 푏. 휎⃗ ⊗ 푐̂. 휎⃗ − 푎′. 휎⃗ ⊗ 푏′. 휎⃗ ⊗ 푐. 휎⃗.

where 푎, 푏푖 and 푐 ∈ 푅 and 휎⃗ =(휎, 휎, 휎), are vector of Pauli spin matrices. For three qubits the MI reads 〈ℳ〉 ≤2. The general three qubit state 휌 in the Hilbert space 퐻 = 퐶 ⊗ 퐶2 ⊗ 퐶2can be expressed as 1 휌 = Λ 휎 ⊗ 휎 ⊗ 휎 8 ,, where 훬휇휈훾 = 푇푟 [(휎휇 ⊗ 휎휈 ⊗ 휎훾)휌]. The coefficients 훬000 =1 is the normalization condition;푞 = {훬푖00; 푖 = 1,2,3}, 푟 ={훬0푗0; 푗 = 1,2,3}, 푠 ={훬00푘; 푘 = 1,2,3} are the Bloch vectors forthree parties respectively; 훩 =[훬푖푗0], 훷 =[훬푖0푘], 훺 =[훬0푗푘] are the two party correlationmatrices and 푇 = [훬푖푗푘] is the tripartite correlation matrix. The expectation value of the Merminoperator, ℳ for state 휌 is given by 〈ℳ〉 = 푇푟[ℳ휌].

Theorem: The maximum quantum value of Mermin operator for arbitrary three qubit stateρ is given by 푄:= max|〈푀휌〉|≤22휆 , where 휆 is the largest singular value of T .

The above theorem is proved following the lemma of [3], stated below Lemma: For any vectors 푥 ∈ 푅and 푦 ∈ 푅 and a 푚 × 푛 rectangular matrix B, we have|푥 ||퐵|≤ 휆푚푎푥|푥||푦|, where 휆푚푎푥 is a largest singular value of B. The equality holds when 푥and 푦 are the corresponding singular vectors of B with respect to 휆max, where T stands fortransposition.

Proof. The expression of Mermin operator can be recast as ℳ = (푎푏푐 + 푎푏 푐 + 푎 푏푐 − 푎′푏′푐′) 휎 ⊗ 휎 ⊗ 휎 ,, The expectation of Mermin operator is thus given by 〈ℳ〉 = (푎푏푐 + 푎푏 푐 + 푎 푏푐 − 푎′푏′푐′) × Tr [(휎 ⊗ 휎 ⊗ 휎)] ,, = [푎푏푐 + 푎푏 푐 + 푎 푏푐 − 푎 푏 푐 ] Λ ,, = 푎푇푏 ⊗ 푐̂ + 푏′⊗ 푐̂ + 푎′푇(푏 ⊗ 푐̂ + 푏′⊗ 푐̂′).

Let 휃 and 휃 are the angles between 푏and 푏′, and 푐̂and 푐̂′ respectively, then

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|푏 ⊗ 푐̂ + 푏 ⊗ 푐̂| =2+2〈푏, 푏′〉〈푐̂, 푐̂′〉 =2+2cos 휃 cos 휃, |푏 ⊗ 푐̂ − 푏 ⊗ 푐̂′| =2−2〈푏, 푏′〉〈푐̂, 푐̂′〉 =2−2cos 휃 cos 휃,

The above inequality is achieved using 푥 cos 휃 + 푦 sin 휃 ≤ (푥 + 푦), equality holds for tan 휃 = 푦/푥. Tightness of the bound: The tightness conditions is said to achieved when the above expression of upper bound saturates. To saturate the upper bound in the Theorem, one can choose 휃푏푐 = 휋/2. Also, the Lemma tells us that the first inequality in Eq. (1) saturates if the degeneracy of 휆maxismore than one, and corresponding to 휆max, there exist two nine- dimensional singular vectors of the form 푏 ⊗ 푐̂′+ 푏′⊗ 푐̂′and푏 ⊗ĉ− 푏′⊗ 푐̂′, respectively.

In our work (arXiv:1710.10802) we have also illustrated the upper bounds of the Mermin operator for various class of states and also highlighted their corresponding saturation conditions.

Conclusions and Discussion: In this work, we proffer a quantitative analysis of MI for three qubits by computing the maximum quantum value of the Mermin operator analytically. The obtained maximal value is shown to be tight for the state which produces the correlation matrix with more than one degenerate largest singular values. Also the maximal quantum bound for several noisy states is shown to be tight. The proposed method not only works for pure states but also for mixed states. As MI is very useful in detecting nonlocality of multipartite states, our method will open a new pathway in this direction. We have also discussed the possible relation between multipartite entanglement and the maximum quantum value of Mermin operator for mixed states which might help in experimental characterization of multipartite entanglement. Our analysis might also be applicable to other facet inequalities.

References: [1] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995). [2] S. Adhikari and A.S. Majumdar, arXiv:1602.02619. [3] M. Li, S. Shen, N. Jing, S.-M. Fei, and X. Li-Jost, Phys. Rev. A 96, 042323 (2017).

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Non-classicality with q-calculus

Mushtaq B. Shah1, Prince A. Ganai1 1Department of Physics, National Institute of Technology, Srinagar, Kashmir-190006, India

Abstract:The geometry of the q-deformed quantum mechanical phase space has led to an interesting results in physics. We will study q-deformed harmonic oscillator in the light of phase space variables. In this paper, we shall investigate the non classicalbehavior of a q- deformed coherent states of a harmonic oscillator, using Mandel parameter. It is found that these states (q-deformed Klauder states) show an appreciable degree of non classicality for a particular value of the q-parameter.

Summary: Introduction: The algebra of q-deformation has been an object of vast interest in mathematical physics over the past few years, and huge efforts have been invested for its exploration, understanding and development. It finds its premium applications in quantum optics, condensed matter, computation etc. Over few years, people have developed a keen interest in the q-deformed algebras. The reason for this is that the q-deformed algebra being the deformed versions of the standard Lie algebras are recovered when the deformation parameter goes to unity. As very rich and sophisticated set of symmetries is embedded in the deformed algebras than that of the standard Lie algebras, therefore, it is very useful as well as important to use q-deformed algebras as an appropriate tool to describe the symmetries of the physical systems which cannot be studied with the Lie algebras.

Non-classicality Vs Coherence: The Mandel parameter: In this section, we will discuss the non-classical attribute of coherent states in non-commutative spaces as implied by q- deformed algebra. We will explicitly derive the Mandel parameter for such a state, which encodes the scheme for studying non-classical behaviour and eventually elucidate its dependence on q-parameter. Specifically, the cases of time dependent and independent states will be dealt thoroughly. First, let’s briefly have an overview of this deformed oscillator algebra and Klauder state.

Under certain conventions, the one dimensional q-deformed osillator algebra reads [1]

퐴퐴 − 푞퐴퐴 = 1, |푞| ≤1 (2.1)

Where, 퐴 and 퐴 are annihilation and creation operators respectively. The associated Fock space state representation for these states is given by,

() |푛⟩ = |0⟩; [푛] = (2.2) []!

Where, mathematical operation of 퐴 and퐴 shows their lowering and raising property respectively asshown below

퐴 |푛⟩ = [푛 + 1]|푛 + 1⟩ (2.3) and

퐴|푛⟩ = [푛]|푛 − 1⟩ (2.4)

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A linear combination of the states |푛⟩

∞ |휓⟩ = ∑ 퐶 |푛⟩ (2.5) is a state spanning a deformed Hilbert space 퐻 preserving orthonormality such that

∞ ⟨휓|휓⟩ = ∑ |퐶| < ∞ (2.6)

Using this formalism, the construction of a special class of coherent states called Klauder states corresponding to deformed algebra becomes handy [2]- [3]. Thus a normalised q- deformed Klauderstate is given by

( ) ∞ |퐽, 훾⟩ = ∑ |푛⟩ (2.7) () []!

Where, the normalization condition ⟨퐽, 훾|퐽, 훾⟩ =1 yields

∞ 퐸(퐽) = ∑ (2.8) []!

A very powerful tool for investigating the behaviour of a coherent state is Mandel parameter that quantifies this behaviour. A well established form for it is given by

푄 = (2.9)

The essence of this parameter is that a negative value indicates a non-classical behaviour while zero means a coherent state. Regarding a positive value, nothing can be said with certainty. For Fock space, its value is -1. We now calculate the Mandel parameter for a time dependent q-deformed Klaudercoherent state. In reference to above eqn. (8), the parameters come out to be

1 퐽 퐴퐴 = 푒[]푚퐴퐴푛푒[] 퐸(퐽) [푚] ! [푛] ! ,

[] [] = ∑, 푒 [푛]훿푒 () []![]!

∞ = ∑ (2.10) () []!

Changing n to (n+1), the above equation becomes

∞ 퐴 퐴 = ∑ = 퐽 (2.11) () []!

This yields,

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⟨퐴퐴⟩ = 퐽 (2.12)

Also

⟨(퐴퐴)⟩=⟨퐴 (1+ 푞퐴퐴)퐴 ⟩

= ⟨퐴퐴⟩+ 푞⟨퐴 퐴⟩ (2.13)

Where ⟨퐴 퐴⟩= 퐽Thus eqn. 2.13 becomes ⟨(퐴퐴)⟩= 퐽 + 푞퐽 (2.14)

Using (2.11); (2.12); (2.14) in (2.9), we have 푄 = (푞 − 1) 퐽 (2.15)

Conclusion: We studied the q-deformed quantum mechanical phase space of the harmonic oscillator through phase space variables. We also studied the non classical behaviour of a q- deformed coherent states of a harmonic oscillator, using Mandel parameter. It is shown that these q-deformed Klauder states show flavour of non classicality for a particular value of the q-parameter.

References: [1] Sanjib. Dey, Andreas. Fring, Laure. Gouba and Paulo. G. Castro, Time-dependent q-deformed coherent states for generalized uncertainty relations Phys. Rev. D 87, 084033 (2013) [2] J. Klauder Quantization without quantization Annals Phys.237, 147-160(1992) [3] J.-P. Antoine, J.-P Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, Temporally stable coherent states for infinite well and Poschl-Teller potential J.Math.Phys.,42, 2349-2387(2001)

Muthuganesan R Robustness of measurement induced nonlocality to sudden death

R. Muthuganesan and R. Sankaranarayanan

Department of Physics, National Institute of Technology, Tiruchirappalli-15, Tamil Nadu, India

Abstract: Measurement induced nonlocality (MIN) captures global nonlocal effect of bipartite quantum state due to locally invariant projective measurements. Here, we analytically investigate the dynamical behavior of Hilbert – Schmidt norm and fidelity based MINs under decoherence and compared with entanglement. It is shown that MINs are more robust to sudden death. Summary: Introduction: Entanglement, a special kind of quantum correlation useful resource for various information processing, is quantified by the concurrence [1]. One notable measure, which goes beyond entanglement, is measurement induced nonlocality proposed by S. Fu and S. Luo [2]. Note that quantum systems are unavoidably subjected to their surrounding environments that will damage the quantum correlation. Hence, it is extremely intriguing and important to probe the dynamical behaviour of quantum correlation when a system interacts with noisy environment. Motivated by this, here we will focus on this theme

106 as well, explicitly, exploration of dynamical behaviours of quantum correlation under noisy channel. Measurement induced nonlocality: It is defined as the maximal square of Hilbert–Schmidt distance between pre- and post-measurement state: [2] 푚푎푥 푁(휌) = ‖휌 − Π (휌) Π

Where the maximum is taken over all local projective measurements. HereΠ(휌) = ∑(Π ⨂핀 )휌 (Π ⨂핀 ), with Π = {|푘〉〈푘|} being the projective measurements on the subsystem퐴, which do not change 휌 locallyi.e., ∑ Π 휌 Π = 휌 . This quantity is easy to compute and experimentally realizable. However, it is not a bonafide measure of quantum correlation due to local ancilla problem. In order to resolve this issue, defining MIN in terms of fidelity induced metric as [3] 푚푎푥 푁 (휌) = 퐶휌,Π(휌) , ℱ Π where 퐶(휌, 휎) = 1−ℱ(휌, 휎) is a sine metric andℱ(휌, 휎) is quantum fidelity measure of closeness between two arbitrary states 휌 and 휎. Here we will consider the system – environment interaction through the operator sum representation formalism. Given an initial state for two qubits 휌(0), its evolution can be written as 휌(푡) = ∑, 퐸,휌(0) 퐸,with Kraus operators 퐸, = 퐸 ⊗ 퐸 and satisfying ∑, 퐸, 퐸, = 핀. Depolarizing Channel: This channel is a type of quantum noise which transforms a single qubit into a maximally mixed state 휌 = 핀/2 with probability훾.This channel is represented by the Kraus operators: [4]

퐸 = 1−3훾/4핀, 퐸 = √훾휎, 퐸 = √훾휎 and퐸 = √훾휎 where휎 are Pauli spin matrices 훾 =1− 푒 with 훾′ being dampingconstant. For the initial state 휌(0) =|휓 ⟩⟨휓 |and|휓 ⟩ = √훼|00 ⟩ + √1− 훼|11 ⟩, with 훼 ∈ [0,1] the matrix elements of evolved state are 휌(푡) = 휌(0)(1− 훾) + 훾 /4, 휌(푡) = 휌(푡) = 훾(1− 훾/2)/2, 휌(푡) =1− 휌(푡) − 휌(푡) and 휌(푡) = 휌(푡) = 휌(0)(1− 훾).The concurrence and MINs for evolved state are퐶휌(푡) =2푚푎푥{0, 휌(푡) − 휌(푡)}, 푁ℱ휌(푡) = 2 휌(푡) (∑ 휌(푡) + 2 휌(푡) ) and 푁휌(푡) = 2 휌(푡) . We find that both the forms of MIN vanish identically for the initial product states 훼 = 0,1and for all other states they vanish in the asymptotic limit, 훾 =1. However, we find that the concurrence vanishes훾 ≥

훾 =1+ 4훼(1− 훼) − 1+4훼(1− 훼). Thus, the depolarizing channel induces zero entanglement for훾 ≥ 훾, which is known as entanglement sudden death. The critical value 훾 vanishes for훼 = 0,1 (product states) with the maximum of 2− √2 = 0.586 for 훼 = 1/2 (maximally entangled state) as shown in Fig. (1).

Generalized amplitude damping channel (GAD): Kraus operators for GAD channels are[4] 0 훾 0 0 퐸 = 푝 푑푖푎푔1, 1− 훾,퐸 = 푝 √ ,퐸 = 훿푑푖푎푔1− 훾,1, and퐸 = 훿 0 0 √훾 0

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Figure 1: The dynamics of Hilbert- Schmidt norm and fidelity based MIN and concurrence in depolarizing (left) and GAD channels(푝 = 2/3) (right) for the initial state |휓 ⟩ = (|00 ⟩ +|11 ⟩). √ where 훿 = (1−푝),훾=1−푒, Γ is decay rate and 푝 defines the final probability distribution of stationary state. The non-zero matrix elements of evolved state are 휌(푡) = 휌(0){(1−훾)[2(1−푝) −훾(1 − 2푝)]} +훾 푝 , 휌(푡) =휌(푡) =훾[휌(0)(1−훾)(1− 2푝) +훾(1−훾푝)], 휌(푡) =1 − 휌(푡) −휌(푡) and 휌(푡) =휌(푡) =휌(0)(1−훾).It is clear that the evolution of state in this channel depends on 훼and 푝.We find that entanglement of the evolved state is zero for훾≥훾(훼,푝). Toexamine the dynamics, we consider the initial state훼 = 1/2. Setting 푝 = 2/3we have훾 = 0.6. In other words, evolution of the maximally entangled state under this channel also exhibits entanglement sudden death. It is also clear from our results that, dynamics of both the MINs are qualitatively same. In particular, as time increases the MIN and F-MIN decrease showing that quantum correlation vanishes asymptotically. In this channel also the non-zero MINs in the region of zero concurrence show the existence of quantum correlation without entanglement. Conclusion: The dynamics of MIN and F – MIN are studied under various noisy channels such as depolarizing and generalized amplitude damping. Our findings suggest that MIN and fidelity based MIN are more robust than entanglement against decoherence.

References: [1] S. Hill and W.K. Wootters, Phys. Rev. Lett.78, 5022 (1997). [2] S. Luo and S. Fu. Phys. Rev. Lett. 106 120401(2011). [3] R. Muthuganesan and R. SankaranarayananPhys. Lett. A381, 3855 (2017). [4] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge (2010).

Higher order nonclassicality: Where can we find it?

Nasir Alam, Kathakali Mandal, Kishore Thapliyal and Anirban Pathak

Abstract: In view of our recent works, it is shown that useful higher order nonclassical states can be found in various physical systems, like BEC, finite-dimensional coherent state, cat states, etc.

Summary:

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Introduction: Nonclassical states defined by the negative values of Glauber-Sudarshan P - function [1-2] have recently found applications in various situations, like gravitational wave detection, single photon source construction, etc. In the recent past, it is also recognized that the weak nonclassicalities not-detected by the lower order counterparts can be detected by the higher order nonclassicality criteria. Motivated by this fact and the recent success in experimental realization of higher order nonclassical states, we have systematically studied the possibilities of observing higher order nonclassicality in various physical systems (see [3-6] and references there in) which includes optomechanical and optomechanical like systems, atom-molecule BEC, finite dimensional coherent state, etc. In what follows, we briefly review some of the results of our group and illustrate a few results using examples of an optomechanical system [3], atom-molecule BEC [4], a finite dimensional coherent state [5] and a cat state [6-7].

Figure 1: (a) HOE in optomechanical system using HZ-1 where smooth, dashed and dashed- dotted line correspond to the l = 1,l = 2 and l = 3, respectively.(b) HOA in superposition of even odd coherent state. (c) HOS in atom molecule BEC where smooth and dotted lines correspond to the A,andA,, respectively. (d) HOSPS in finite dimensional coherent state.

Results: Our systematic study of higher order nonclassicality have already shown the existence of higher order entanglement (HOE) in optomechanical system, higher order antibunching (HOA) in superposition of even odd coherent state, higher order subpoissonian photon statics (HOSPOS) in finite dimensional coherent state and higher order squeezing (HOS) in atom-molecule BEC. The observations reported in those works, clearly established

109 that higher order nonclassicality is not a rare phenomenon and it can be seen in various physical system. This fact is illustrated here in Fig. 1, where we show variation of HOE with dimensionless interaction time (푘푡), HOA with phase angle (휑), HOS with dimensionless interaction time (Ω푡), HOSPS with complex parameter (훽). Acknowledgment: A.P. and N.A. thank the Department of Science and Technology (DST), India, for support provided through the DST project No. EMR/2015/000393.

References: [1] J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev.131, 2766 (1963). [2] C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Rev. Lett.10, 277 (1963). [3] N. Alam, K. Thaplyal , A. Pathak, B. Sen, A. Verma, S. Mandal, Normal order and higher order nonclassicality in a Fabry–Perot cavity with one movable mirror and a Bose-condensed optomechanical-like system: squeezing, antibunching and entanglement, arXiv:1708.03967 [4] S. K.Giri, B. Sen, C. H. R.Ooi, A.Pathak, Single-mode and intermodal higher-order nonclassicalities in two-mode Bose-Einstein condensates, Phys. Rev. A 89, 033628 (2014) [5] N. Alam, A. Pathak and A.Verma, Higher-order nonclassicalities in truncated Hilbert space: A comparative study, arXiv:1712.10135 [6] Yurke, Bernard, and D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett.57, 13 (1986). [7] G. Ren, J. Du, and H. Yu, Non-classical state via superposition of two opposite coherent states. Int. J Theor. Phys. 1 11 (2017)

Universal quantum uncertainty relations between non ergodicity and loss of information

Natasha Awasthia,1, Samyadeb Bhattacharyab, Aditi Sen (De)b, and Ujjwal Senb aCollege of Basic Sciences and Humanities, G.B. Pant University Of Agriculture and Technology, Pantnagar, Uttarakhand - 263153, India bHarish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, India

Abstract: We establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics (arXiv:1707.08963). The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices. The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that non-ergodicity plays an important role in

110 preserving information of quantum states undergoing lossy evolution. We also consider a model of a central qubit interacting with a fermionic thermal bath and derive its reduced dynamics, to subsequently investigate the information loss and non-ergodicity in such dynamics. We comment on the “minimal” situations that saturate the uncertainty relations.

Summary: Introduction: It is impossible in practical situations to isolate a quantum system from its environment. Such open quantum systems are subject to dissipation and decoherence: their evolutions are no longer governed by unitary dynamics. The state of the corresponding physical system now evolves under completely positive trace preserving maps (CPTP). Since the joint entity of the system and the environment is closed, it evolves unitarily. But if we trace out the environment to observe the evolution of the system alone, we find that the description is CPTP, with the initial system- environment state being assumed to be product [1]. Since the environment is typically a large system with many degrees of freedom, it is generally difficult to calculate the joint system-environment dynamics. In this work, we derive the exact reduced dynamics of a quantum system interacting with a fermionic bath, after solving the Schrodinger equation for the joint system- environment state and then tracing out the environment part. We then investigate the evolution of certain information- theoretic measures for this quantum system and analyse their connections with physical processes typically linked to system-environment interactions.

Methodology and the main results: Exact theories for dynamics of open quantum system requires the solution of Schrödinger equation for the total system and bath state, by the method given in [2, 3]. For this purpose we consider a model consists of single qubit interacting centrally with many spins. We assume the bath spins to be mutually non- interacting. To do the exact calculation we solve the Schrödinger dynamics of the total state and derive the reduced dynamics by tracing out the bath degree of freedom after performing. Here we have taken initial bath to be in thermal state at an arbitrary temperature. We then investigate the flow of information between the system and environment and draw novel connection of that with physical processes related to system-environment interaction, realized over long time range.

References: [1] H. P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, Great Clarendon Street, 2002).. [2] S. Bhattacharya, A. Misra, C. Mukhopadhyay, and A. K. Pati, Phys. Rev. A 95, 012122 (2017). [3] C. Mukhopadhyay, S. Bhattacharya, A. Misra, and A. K. Pati, arXiv:1704.08291 (2017).

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Analyzing nested Mach-Zehnder interferometer with weak path marking on internal degrees of freedom

Rajendra Singh Bhati1 and Arvind1

1Department of Physical Sciences, Indian Institute of Science Education & Research, Mohali

Abstract: We analyze Vaidman's nested Mach-Zehnder interferometer using weak path marking on quantum particles' internal degrees of freedom with predictions of two-state vector formalism remaining unaltered. Our results present a challenge to Vaidman's conclusions. Summary: Vaidman argues that talking about the past of a quantum particle is meaningful and possible in the two-state vector formalism (TSVF) of quantum mechanics [1]. In a very controversial and highly debated work, Vaidman and co-authors concluded that quantum particles do not always take continuous trajectories [2]. Their claim is that the results of their experiment reject the 'common sense' approach to the past of a quantum particle but fit very well in the TSVF. The criticism of this work has also been intensive. With various approaches people have argued that in fact the 'common sense' approach does agree with the experimental results [3, 4, 5].

In this work we present an analysis of nested Mach-Zehnder interferometer with a different technique of weak path marking on particles' state. Instead of weak marking on spatial degree of freedom, we mark the unambiguous which-path information of particles' past on its internal degrees of freedom. In photon's case it is polarization state. We consider the scheme presented in the figure. M-1, M-2, M-3, M-4 and M-5 are mirrors. PBS1, PBS2, PBS3 and PBS4 are partial beam splitters acting on spatial modes of photon. PR1, PR2, PR3, PR4 and PR5 are time dependent unitary operators acting on polarization state of the photon given by: 1− 훿 (푡) −훿 (푡) 푃푅푖 = 훿(푡) 1− 훿(푡) Where 푖 = 1,2,3,4,5 and 훿 (푡) = (1+cos2휔 푡) with 훿 ≈0. Unitary operators for beam splitters are given by: 1 1 2휄 1 1 휄 푃퐵푆1= 푃퐵푆4= √ , 푃퐵푆2= 푃퐵푆3= √3 √2휄 1 √2 휄 1

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Photon enters the interferometer at time 푡 with spatial mode I and horizontal polarization. A projective measurement M is performed in horizontal polarization basis of photon in mode I at time 푡. All the detected photons form a pre and post selected ensemble before time 푡.Condition 훿 ≈0 allows us to obtain a destructive interference near mirror M-5 as it was in Vaidman’s vibrating mirror scheme.

In our scheme, it is shown that the predictions of TSVF is also the same as those of Vaidman’s scheme i.e. weak values of projection operators near M-2 and M-5 are almost zero while near M-1, M-3 and M-4, they are significantly large (order of one). The probability of photon being detected in M is time dependent according to standard formalism of quantum mechanics. Our results show that this probability is a superposition of cos2휔푡, cos2휔푡, cos2휔푡, cos(휔 ± 휔)푡, cos(휔 ± 휔)푡, cos(휔 ± 휔)푡, cos(휔 ± 휔)푡 with non- negligible contributions. This clearly shows that photon has been near M-1 and M-5 otherwise it could have never known the frequencies 휔 and 휔.These theoretical predictions are experimentally verifiable. Inconsistency between the predictions of TSVF and the results of our scheme leads us to the conclusion that the weak value being zero for a projection operator is a necessary but not a sufficient condition for quantum particle not being in its eigenstate. References: [1] L. Vaidman, Past of a quantum particle, Phys. Rev. A 87, 052104 (2013). [2] A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Asking Photons Where They Have Been, Phys. Rev. Lett. 111, 240402 (2013). [3] D. Sokolovski, Asking Photons Where They Have Been in Plain Language, Phys. Lett. A 381, 227 (2017). [4] R. B. Griffiths, Particle Path Through a Nested Mach-Zehnder Interferometer, Phys. Rev. A 94, 032115 (2016). [5] Berthold-Georg Englert, Kelvin Horia, Jibo Dai, Ying Loong Len and Hui Khoon Ng, Past of a quantum particle revisited, Phys. Rev. A 96, 022126 (2017)

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Ravi Kamal Pandey CONTROLLED ENTANGLEMENT DIVERSION USING GHZ TYPE ENTANGLED COHERENT STATE

Ranjana Prakash1, Ravi Kamal Pandey1, *, and Hari Prakash1 1 Physics Department, University of Allahabad, Allahabad, India, 211002. *Corresponding author: [email protected]

Abstract: We consider a scheme of controlled entanglement diversion (CED) of a bi-partite entangled coherent state using GHZ type entangled coherent state as a resource. Average fidelity of diversion is found to be almost unity for appreciable coherent amplitude.

Summary: INTRODUCTION: An important aspect of quantum entanglement is entanglement swapping [1, 2] in which we can entangle two particles, say, A and B which have never interacted before. The scheme is having entangled bi-partite states of particles A and C and of Band D and making a bell state measurement on particles C and D. A similar scheme of Entanglement diversion was introduced by Xin hua [3] in which we consider three remote players say Alice, Bob and Charlie. Initially, Alice and Bob and, independently, Alice and Charlie are connected by sharing perfectly entangled states.

Alice, if she wants, can perform a local measurement on two particles on her side to make a direct connection between Bob and Charlie. Xin-Hua et al scheme of entanglement diversion of entangled coherent state has a success probability of . This scheme was further modified [4, 5], to give an almost perfect entanglement diversion, even for small coherent amplitude. We present a scheme of controlled entanglement diversion involving three remote partners and a controller, using a GHZ type entangled coherent state [6]. The scheme has the advantage of changing the role of controller and one of the other players to whom we wish to entangle. As we find, the diversion is almost perfect for appreciable mean photons in the coherent state.

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CALCULATIONS: We consider four different parties Alice, Bob, Charlie and David at various locations. Alice and Bob share a maximally entangled bi-partite coherent state given 2  by |휓⟩, = [2 (1− 푥 )] [|훼, 훼⟩, − |− 훼,−훼⟩,; where x  e and modes 1 and 3 are with Alice and Bob respectively. Also, Alice, Charlie and David shares a GHZ type maximally entangled coherent state given by

|휓⟩,, = [2 (1− 푥 )] [|훼, 훼, 훼⟩,, − |− 훼,−훼,−훼⟩,, ;

Where, modes 2, 4 and 5 are with Alice, Charlie and David respectively. State of the system consisting of these five modes is the direct product|휓⟩, ⨂ |휓⟩,,. Alice passes mode 2 through a  /2 phase shifter which transform it to mode 6 as |훼⟩ → |−푖훼⟩. She then mixes modes 1 and 6 through a 50:50 symmetric beam splitter which transforms modes 1 and 6 to 7 () () and 8 with transformation|훽, 훿⟩ → | , ⟩ . She again passes mode 8 through a , √ √ ,    i  2 Phase shifter which transforms it to mode 9 with 8 9 . We expand states in terms of vacuum (0), non-zero even (NZE) and odd number states (ODD) [7] as |±훼 ⟩ = √푥|0⟩ + (1− 푥)|푁푍퐸, 훼⟩ ± 2(1− 푥)|푂퐷퐷, 훼⟩, using which the final state of the √ system becomes: 푥 |휓⟩ = [(1 − 푥)(1 − 푥)] |0⟩ |0⟩ (|훼, 훼, 훼⟩ − |훼,−훼,−훼⟩ − |−훼, 훼, 훼⟩ ,,,, 2 ,, ,, ,, + |−훼,−훼,−훼⟩,,) 1 + [2(1+ 푥)(1+ 푥 + 푥)] 푁푍퐸, √2훼 |0⟩ |훼, 훼, 훼⟩ + |−훼,−훼,−훼⟩ 2 ,, ,, + |0⟩ 푁푍퐸, √2훼 − |훼,−훼,−훼⟩ − |−훼, 훼, 훼⟩ ,, ,, 1 + [2(1− 푥)] 푂퐷퐷, √2훼 |0⟩ |훼, 훼, 훼⟩ − |−훼,−훼,−훼⟩ 2 ,, ,, + |0⟩ 푂퐷퐷, √2훼 − |훼,−훼,−훼⟩ + |−훼, 훼, 훼⟩ ,, ,,

Alice now performs photon number measurement in modes 7 and 9 in her possession and sends her measurement result to Bob via. 2 bit classical channel. At this point, we have the freedom of diverting the entanglement of Bob with either Charlie or David, the other acting as a controller. If Charlie is the controller, then to complete the diversion protocol, he makes a photon count measurement in mode 4 in his possession and discriminates between even and odd counts and sends the measurement result to Bob using 1 bit classical channel. Bob on getting the classical information’s from Alice and Charlie makes a suitable unitary transformation, if required, to get entangled with David.

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DISCUSSION OF PHOTON COUNT RESULTS: Alice and Charlie photon count measurement in modes 7, 9 and 3 results in one of 10 different possible outcomes, (1) 0, 0, even (2) 0, 0, odd (3) NZE, 0, Even (4) NZE, 0, odd (5) 0, NZE, Even (6) 0, NZE, odd (7) odd, 0, even (8) odd, 0, odd (9) 0, odd, even (10) 0, odd, odd. For cases 1 and 2, the collapsed state is separable and hence Figure 1. Showing variation of probability of occurrence for each of cases specified diversion is not possible however its probability of occurrence by numerals with corresponding colors. vanishes for appreciable coherent amplitude (FIG. 1 purple curve). Cases 4, 6, 7 and 9 leads to perfect diversion while cases 3, 5, 8 and

10 gives almost perfect diversion for |  |2  2.5 (fig. 2), if Bob applies appropriate unitary transformations. We define average fidelity of diversion as the sum of the product of probability for occurrence of a

case (in modes 7, 9 and 5) and the corresponding fidelity. Figure2. Variation of Fidelity of The average fidelity turns out to be, diversion, the average fidelity is shown in blue while for almost perfect cases it is shown in red. 2+푥 +푥 퐹 = 퐹 푃 = 2(1+푥 +푥 )

Which, is nearly unity for mean photons as low as 3 (fig. 2). Hence, as the mean number of photons increases the average fidelity sharply converges to unity. Acknowledgement: One of the authors (RKP) acknowledges the UGC for providing financial support.

References: [1] Zukowski, Marek, et al. "" Event-ready-detectors" Bell experiment via entanglement swapping." Physical Review Letters 71.26 (1993): 4287-4290. [2] Pan, J. W., Bouwmeester, D., Weinfurter, H., & Zeilinger, A. (1998). Experimental entanglement swapping: entangling photons that never interacted. Physical Review Letters, 80(18), 3891. [3] Xin-Hua, C., Jie-Rong, G., Jian-Jun, N., & Jin-Ping, J. (2006). Entanglement diversion and quantum teleportation of entangled coherent states. Chinese Physics, 15(3), 488. [4] Prakash, H. (2009, December). Quantum teleportation. In Emerging Trends in Electronic and Photonic Devices & Systems, 2009. ELECTRO'09. International Conference on (pp. 18-23). IEEE. [5] Prakash, H., Chandra, N., Prakash, R., & Kumar, S. A. (2010). Improving the entanglement diversion between two pairs of entangled coherent states. International Journal of Modern Physics B, 24(17), 3331-3339. [6] Jeong, H., & An, N. B. (2006). Greenberger-Horne-Zeilinger–type and W-type entangled coherent states: Generation and Bell-type inequality tests without photon counting. Physical Review A, 74(2), 022104. 116

[7] Prakash, H., Chandra, N., & Prakash, R. (2007). Improving the teleportation of entangled coherent states. Physical Review A, 75(4), 044305.

Rengaraj Measuring the deviation from the superposition principle in interference experiments

G. Rengaraj, Prathwiraj U, Surya Narayana Sahoo, R. Somashekhar, Urbasi Sinha

Raman Research Institute, Bangalore, India. (email: [email protected])

Abstract: We experimentally quantify the deviation from the superposition principle in interference experiments by measuring a non zero Sorkin parameter for the first time in the microwave domain.

Summary: Superposition principle is one of the fundamental principles in Quantum mechanics. However, it is usually incorrectly applied in slit based interference experiments. This has been investigated through numerics based on Finite Difference Time Domain (FDTD) methods [1] as well as the Feynman path integral formalism [2,3]. Recently, we have done an experiment [4] in which we have been able to measure the deviation from the common application of the Superposition Principle and reported the first non-zero measurement of the Sorkin parameter κ (of the order 10^(-2)) which quantifies this deviation. As this is a precision experiment, years of work has gone in for an optimal choice of experimental parameters as well as painstaking error analysis. In this poster, we will discuss such details of the experiment, complete theoretical simulations as well as systematic and random error analysis.

References: [1] Raedt, H. D., Michielsen, K. & Hess, K. Analysis of multipath interference in three-slit experiments. Phys. Rev. A. 85, 012101 (2012) [2] Sawant, R., Samuel, J., Sinha, A., Sinha, S. & Sinha, U. Non classical paths in quantum interference experiments. Phys.Rev.Lett. 113, 120406 (2014). [3] A. Sinha, Aravind H.Vijay and U.Sinha. On the Superposition principle in interference experiments, Scientific Reports 5, 10304 (2015). [4] G. Rengaraj, Prathwiraj U, Surya Narayana Sahoo, R. Somashekhar, Urbasi Sinha, Measuring the deviation from the superposition principle in interference experiments (https://arxiv.org/pdf/1610.09143)

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S Lakshmibala Entangled quantum states: Nonclassical effects and entanglement indicators

Pradip Laha, B Sharmila, S Lakshmibala and V Balakrishnan Department of Physics, IIT Madras, Chennai 600036, India

Summary: Quantum optics provides an ideal framework for investigating nonclassical effects such as quadrature and entropic squeezing of subsystem states during the temporal evolution of atomic systems interacting with the radiation field. A generic factored-product of subsystem initial states in both bipartite and tripartite systems inevitably gets entangled during dynamics. Both entanglement and squeezing are useful tools in various aspects of quantum information processing such as dense coding and teleportation of information. Extensive work has been carried out on protocols for exploiting squeezed states in dense coding with Gaussian state quantum channel, and also in exploiting properties of photon pairs with entanglement of spin and angular momentum, (see for instance, [1, 2]). Continuous variable quantum information and computation has been an active area of research in recent years [3, 4]. We have examined the extent of squeezing as also the entanglement of relevant quantum states using a tomographic approach. A wide range of squeezing properties are estimated directly from appropriate optical tomograms [5], without carrying out explicit quantum state reconstruction. Optical tomograms are essentially histograms from homodyne measurements of a quorum of observables [6]. For a single-mode radiation field with annihilation and creation operators ɑˉ and ɑ+, consider the family of quadrature operators − −푖휃 + 푖휃 푋휃= (ɑ 푒 + ɑ 푒 )/√2 where θ(0<θ<π), is the phase of the single mode in the homodyne detection setup. It is evident that for θ=0 and π/2, respectively, we have the two field quadrature operators analogous to position and momentum, respectively. The optical tomogram W(푋, θ) corresponding to a density matrix ρ is given by w (X θ ,θ)=⟨ X θ ,θ|ρ|X θ ,θ⟩ . These definitions can be extended in a straightforward manner to include multipartite systems as well. We first demonstrate the use of this approach for output states of a beamsplitter. For this purpose, experimentally relevant optical states such as the cat states are used as input states. We have quantified both quadrature squeezing and entropic squeezing directly from the optical tomograms also, for bipartite systems evolving in time with inherent nonlinearity present in the atom-field system. Further, the rich entanglement dynamics of tripartite systems such as a three-level atom interacting with radiation fields is also examined in a general setting with intensity-dependent field-atom couplings. The bipartite system examined is a multilevel atom interacting with a non-linear medium [7]. We investigate the roles played by classical and quantum correlations in determining the extent of entanglement at various instants during dynamical evolution of this system. (A similar system in a somewhat different setting is the double-well Bose-Einstein condensate). For our purpose, we compare standard measures of entanglement such as subsystem von

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Neumann entropy (SVNE) and subsystem linear entropy (SLE) with two tomographic entropy-based indicators that we have developed. We have shown that in the two systems considered, one of these indicators, namely, tomographic entanglement indicator (TEI) is in fair qualitative agreement with SVNE and SLE [8]. Inspired by [9], we have improved the TEI to obtain the other indicator which we refer to as the ‘dominant tomographic entanglement indicator’ (DTEI). We emphasize that while both these indicators primarily capture the classical correlations, for certain regions of nonlinearity and interaction strength parameters, and for certain initial states, they compare very favourably with SVNE and SLE. Lessons learnt from our preliminary tomographic analysis (in the case of the beamsplitter) of a wide class of states including ‘Janus-faced’ partner states [10] are used in adopting this approach to a tripartite quantum system [11]. This is a Λ atom interacting with two radiation fields, with nonlinearities in the two fields and an intensity-dependent coupling of the form √1+ 휅푎푎. Here, the intensity parameter 휅 varies from 0 to 1. This form of the coupling [12] is a straightforward generalisation of the two-photon realisation of the SU (1, 1) algebra in Jaynes-Cummings models with intensity dependence. We investigate the roles played by field nonlinearity and intensity-dependent coupling in determining the extent of quadrature and entropic squeezing of relevant subsystem states, during temporal evolution. The extent of entanglement and the time duration over which the entanglement collapses to a constant non- zero value (in contrast to sudden death) are sensitive to the extent of nonlinearity, the precise nature of the initial unentangled state and the extent of decoherence.

References: [1] J. Lee, Se-Wan Ji, J. Park, and H. Nha, Phys. Rev. A 90 022301 (2014). [2] J. T. Barreiro, Tzu-Chieh Wei, and P. G. Kwiat, Nature Physics 4 282 (2008). [3] S. L. Braunstein, A. K. Pati, Quantum Information with Continuous Variables, ISBN 978- 94-015-1258-9, Springer Netherlands (2003). [4] C. Weedbrook et. al., Rev. Mod. Phys. 84 621 (2012). [5] A. Wunsche, Phys. Rev. A 54 5291 (1996). [6] S. D. Nicola, R. Fedele, M. A. Man’ko and V. I. Man’ko, J. Phys.: Conf. Ser. 70012007 (2007). [7] G. S. Agarwal and R. R. Puri, Phys. Rev. A 39 2969 (1989). [8] B. Sharmila, K. Saumitran, S. Lakshmibala, and V. Balakrishnan, J. Phys. B: At. Mol. Opt. Phys. 50 045501 (2017). [9] S. Ghose and B. C. Sanders, Phys. Rev. A 70 062315 (2004). [10] P. Shanta et. al., Phys. Rev. Lett. 72 1447 (1994). [11] Pradip Laha et. al., Int. J. Theor. Phys. 55 4044 (2016); arXiv:1705.08190 (2017). [12] S. Sivakumar, J. Phys. A: Math. Gen. 35, 6755 (2002).

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Deterministic Quantum Dense Coding Networks

Saptarshi Roy, Titas Chanda, Tamoghna Das, Aditi Sen(De) and Ujjwal Sen Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211019, India (https://arxiv.org/abs/1707.02449)

Keywords: DC – Dense Coding, DDC - Deterministic Dense Coding, GHZ - Greenberger- Horne-Zeilinger, gGHZ - Generalised GHZ, gW - Generalised W Abstract: We consider deterministic transfer of classical information using non-maximally entangled pure multiparty states. We prove impossibility of DDC with gGHZ states and demonstrate quantum advantage of DDC for gW states.

Summary: The original DC protocol describes the advantage to send the classical information of N possible outcomes of a classical random variable, say X, when encoded in a quantum state from a single sender (Alice) to a single receiver (Bob). It was shown that if Alice and Bob share the maximally entangled state, the singlet,| 휓 ⟩ = (|01 ⟩ − |10 ⟩), √ Alice can transform the state into four possible orthogonal states by acting local unitaries on her part and can sendlog 4=2 bits of classical information by sending only a single spin- 1/2 particle, i.e., a two-dimensional system. If the initially shared state is non maximally entangled pure state,|휙⟩ = 푎|00⟩ + 푏|11⟩, where a and b are complex numbers with |푎| + |푏| =1, or an arbitrary state, 휌, Alice can no longer create orthogonal output states by performing usual DC protocol and hence the receiver gets less information. In the asymptotic limit, when many copies of휌are provided, the amount of maximal classical information transferred on an average is the densecoding capacity (C), given by퐶(퐴퐵) = log 푑 + max {푆(휌) − 푆(휌),0}; where 푑 is the dimension of the Hilbert space of the sender’s subsystem, 푆(휌) = −Tr(휌 log 휌) is the von Neumann entropy of 휌, and 휌 =-TrB(휌) is the reduced density matrix of the receiver’s subsystem. The first term is the classical limit for information transfer, while the remaining terms quantify the quantum advantage in DC. Instead of considering an asymptotic way of transferring classical bits which also is probabilistic in nature, we deal with a DC scheme in a single-copy level where Alice encodes the information ‘deterministically’ by acting unitary operators on her parts in such a way that upon receiving the entire system, Bob possess an orthogonal ensemble and hence can always distinguish the output states without any error i.e., deterministically by performing global measurements. Such protocol for a single sender and a single receiver was introduced in Ref. [19] of https://arxiv.org/abs/1707.02449, and referred to as the deterministic dense coding (DDC) protocol. Since the protocol is at the single-copy level, it is also important from an experimental point of view. In DDC, Alice’s aim is to find out unitary operators,{푈 }, such that mutually orthogonal states can be created by applying {푈 },on her part of|휙⟩.Bob can easily distinguish them by using global measurements. If |휙⟩ is a product state, the maximal number of unitary operators, 푁 = 푑 is the classical limit, while an arbitrary state |휙⟩is said to be deterministically dense codeableif푁 > 푑. It was proven that the entire family of pure states of 2 qubits except the maximally entangled state, is useless for deterministic dense coding. Till now, all the studies on DDC is restricted to a single sender and a single receiver, although the importance of building communication protocol between several senders and several receivers is unquestionable. In this work, we address the question of building a DDC network between several senders and a single receiver. Interestingly, we show that DDC is possible even with two-level systems already if one increases the number of senders to two. We first prove that the DDC

120 protocol with quantum advantage is not possible when the shared state is a generalized GHZ state with two or more than two senders and a single receiver except when it is a GHZ state for which DDC and DC attain the maximum capacities. We show that the DDC scheme can be executed by using the generalized W states beyond the classical limit. We also perform a comparison between the states from the GHZ- and the W-classes according to their usefulness in DDC. Results: Suppose the shared state is un-entangled in the sender and the receiver bipartition, then the maximum amount of information that the two senders, each having two dimensional systems can send to the receiver is two bits, i.e., 푁 = 4. We investigate 푁 for entangled 3-qubit states. No-go theorem for gGHZ states:|푔퐺퐻푍⟩ = √훼|000⟩ + 푒 √1− 훼|111⟩ For the GHZ state, the capacity of dense coding reaches its maximum value, implying successful implementation of DDC protocol with 푁 = 8. However when we consider DDC by using 3-qubit gGHZ states with 훼 ≠1⁄ 2, we find 푁 =4. For several reasons, the GHZ state is considered to be the “maximally entangled” among generalized GHZ state. We therefore refer generalized GHZ states with 훼 ≠1⁄ 2 as “non-maximally entangled” generalized GHZ states. We found that non-maximally entangled 4-qubit gGHZ states also show the same feature. We encapsulate these results in the following theorem. Theorem: Non-maximally entangled generalized Greenberger-Horne-Zeilinger states are not useful for deterministic dense coding with two/three senders and a single receiver.

DDC with gW states:|푔푊⟩ = √훼|001⟩ + 훽|010⟩ + 1− 훼 − 훽|100⟩ We show for the gW states, for certain ranges of state parameters, we find,푁 >4.It was known that there exists gW state which shows perfect dense coding,푁 = 8,for 훼 =1⁄ 2and for all 훽 as seen in the figure alongside. By numerical simulations, we find that for certain ranges of 훼and훽, for which 푁 > 4. In the figure alongside, we map numerically-obtained values of 푁 with the state parameters which clearly depict the quantum advantage of DDC. For the well-known W state, i.e.,훼 = 훽 = 1/3, 푁 =6. Surprisingly, we were unable tofind any states from the set of gW states, that can have 푁 = 7. GHZ-class vs W-class: We Haar uniformly generate states from both the 3-qubit GHZ and W-classes and find: 1. For GHZ class: 8.25% states have 푁 =5 and 9.77% states have 푁 = 6. 2. For W class: 1.57% states have 푁 =5 and 1.08% states have 푁 = 6. Results for some 4-qubit states: Apart from the no-go theorem for 4-qubit gGHZ states, we quote percentages of 4-qubit Dicke states for which 푁 > 8.

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The Margenau-Hill quasi-probabilities for symmetric two qubit system and joint measurability

Seeta Vasudevrao, H. S. Karthik, and A. R. Usha Devi Department of Physics, Bangalore University, Jnanabharathi Campus, Bengaluru-560056, India

Abstract: We construct quasi-probability mass functions for the non-commuting spin components of a permutation symmetric two qubit system. Using coarse-grained unsharp measurements we bring out how the quasi-probabilities make a transition to legitimate positive probabilities.

Summary: In the classical framework, physical observables are all compatible and they can be measured jointly. In contrast, measurement of observables, which do not commute, are declared to be incompatible in the quantum scenario. However, the notion of compatibility of measurements is limited to that of commutativity of the observables only when one restricts to projective valued (PV) measurements. An extended notion of compatibility emerges in a generalized framework of unsharp measurements using positive operator valued measures (POVMs). Measurement incompatibility [1, 2] plays an important role in exploring the contrasting features of classical and quantum worlds. In this context, it may be recalled that Wigner's quasi-classical phase-space distribution function approach [3] highlights how a quantum system deviates from its corresponding classical counterpart. Wigner's approach proposes to evaluate the quantum expectation values of physical observables in terms of a phase-space integration involving classical functions. The quantum phase-space distribution differs from its classical counterpart in that it can assume negative values and hence, cannot be treated as genuine probability distribution function. For this reason, Wigner function is referred to as quasi-probability distribution. The non-positivity of the Wigner function is attributed to the non-commutativity of position and momentum operators’푞 and푝̂ Besides Wigner distribution function, which makes use of Weyl's operator correspondence rule [4] to associate classical functions with quantum operators, various other distribution functions have also been developed by designing suitable operator correspondence rules[5]. The phase- space distribution formalism, initiated by Wigner for canonical position and momentum observables, has also been extended to develop distribution functions for spin [6]. In this work we investigate the impact of joint-measurability (compatibility) of non-commuting spin observables 퐽 = (휎 ⊗ 퐼 + 퐼 ⊗ 휎 ) and 퐽 = (휎 ⊗ 퐼 + 퐼⊗휎 ) of a symmetric two-qubit system on the discrete Margenau-Hill quasi-probability mass functions푃(푋, 푍), 푋, 푍 = 1,0,−1. This work is a continuation of our earlier work [10] where we had shown that the quasi-probabilities 푃(푋, 푍), 푋, 푍 = 1,−1 of a single qubit system transform to a set of legitimate positive-definite probabilities, when the unsharpness parameter associated with joint measurability of휎 and휎 lies within the compatibility region η ≤ 1/√2. It may be noted that joint measurability of non-commuting single qubit observables 122 is studied extensively and necessary and sufficient conditions on the unsharpness parameters has been established already [79]. On the other hand, there are no established results to discern joint measurability of non-commuting observables in higher dimensional systems. Our present work is thus useful in identifying the joint measurability region of observables of a two-qubit system. By imposing that the Margenau-Hill discrete probabilities be positive, we identify the impact of generalized local measurements of the pauli observables 휎and 휎(employed by both the parties sharing the two-qubit state) on the region of joint measurability. Furthermore, we discuss the role of correlations between the qubits on joint measurability of 퐽 and 퐽 and also on the positivity of the associated Margenau-Hill probabilities.

ACKNOWLEDGEMENTS: ARU is supported by the University Grants Commission (UGC) Major Research Project (Grant No. MRPMAJOR- PHYS-2013-29318), Government of India.

References: [1] P. Busch, Phys. Rev. D33, 2253 (1986). [2] T. Heinosaari, D. Reitzner, and P. Stano, Found. Phys. 38, 1133 (2008) [3] E. Wigner, Phys. Rev. 40, 749 (1932). [4] H. Weyl, Z. Phys. 46, 1 (1927). [5] M. Hillery, R. F. O'Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106, 121 (1984). [6]H. Margenau and R. N. Hill, Prog. Theoret. Phys. 26, 722 (1961); J. P. Dowling, G. S. Agarwaland W. P. Schleich, Phys. Rev. A49. 4101 (1994); G. Ramachandran, A. R. Usha Devi, P. Devi and SwarnamalaSirsi, Foun. Phys. 26, 401 (1996); R.R. Puri, J. Phys. A: Math. Gen. 28, 6227 (1995); J. Phys. A: Math. Gen. 29, 5719 (1996). [7] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman, Phys. Rep. 506, 1 (2011); 666, 110 (2017). [8]R. Kunjwal and S. Ghosh, Phys. Rev. A 89, 042118 (2014). [9] H. S. Karthik, A. R. Usha Devi, J. PrabhuTej, A. K. Rajagopal, Sudha and A. Narayanan, Phys. Rev. A 95, 052105 (2017).[10] SeetaVasudevarao, H. S. Karthik and A. R. Usha Devi, Poster presentation at the International Conference on Quantum Foundations, November 28 -December 4, 2015, NIT, Patna.

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Shamiya Javed IMPROVEMENT OF PROBABILISTIC QUANTUM TELEPORTATION BY REPEATED GENERALIZED BELL-STATE MEASURENEMTS

Rnajan Prakash1, Shamiya Javed2 and Hari Prakash3

Physics Department, University of Allahabad, Allahabad-211002 ([email protected], [email protected], [email protected] )

Abstract: In probabilistic quantum teleportation using non-maximally entangled resource, success probability < ½ is reported. We show here how Generalized Bell-State Measurements can be repeated and any desired success rate obtained. Summary: Introduction: Quantum Teleportation (QT) scheme was first proposed by Bennett et al [1] in 1993, in which sender Alice teleports an unknown quantum information to a distant party Bob using a maximally entangled EPR channel [2] and a classical 2-bit channel. Recently, many authors showed [3-7] that it is possible to get perfect QT with non-maximally entangled resource, although with reduced success probability. For achieving this, authors made generalized Bell state measurements (GBSM) instead of using the usual maximally entangled Bell states. For resource with concurrence C less than one, two out of four possible GBSM 1 results lead to perfect QT with success probability C 2 [4, 5]. The other two GBSM's destroy 2 information and hence no repetition of GBSM has been regarded possible.

Improved PQT using 3-qubit resource or suitably prepared Ancilla have been proposed [3, 7- 9]. Another way of getting perfect QT with as high success as desired was proposed by Mishra and Prakash [10], where entangled coherent states and repeated BSM were used. BSM was a two-stage process, the first indicating whether QT will be successful or not, and not messing up with the qubit-state, thus permitting repeated BSM's.

We propose here a scheme involving repeated GBSM's to improve success probability of PQT. Here, Alice possesses both entangled qubits and makes repeated generalized Bell-state measurements (GBSM) on the pair of qubits, consisting of (1) the qubit having information and (2) one of the two entangled qubits, chosen alternately, until perfect QT is indicated. If perfect QT is not indicated, Alice repeats GBSM with the entangled qubit not used in the GBSM replacing the one used. If perfect QT is indicated, Alice sends the qubit not used for last GBSM and the information of the last GBSM result to Bob, who then makes unitary transformation on his qubit to get the exact replica of information. If perfect QT is not indicated, Alice continues with exchange of the initially entangled qubits and the repetition of GBSM.

C 2 Results: Probability for the first GBSM giving successful QT is P(0)  [4, 5]. Success 2

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If failure is indicated in first GBSM, Alice exchanges the qubit used for GBSM out of entangled pair with the other in the entangled pair and repeats GBSM. If success is indicated, 6 4 (1) (0) C C the process finishes. Success probability for this is PSuccess  PSuccess   . 32  24C 2 8 If in the first repeated GBSM, failure is indicated, another GBSM is done by Alice after exchanging the two qubits entangled initially. If success is indicated this time the process ends. Probability for PQT thus obtained, is C18 C 12 C 8 C 6 P (2)  P (1)     Success Success 32(4  3C 2 )3[4(4  3C 2 ) 2  3C 6 ] 32(4  3C 2 )3 32(4  3C 2 ) 32

This process of exchanging the initially entangled qubits and performing repeated GBSM's may be continued till success is obtained. (0) (1) (2) (3) We show variations of PSuccess , PSuccess , PSuccess and PSuccess with the concurrence C of the entangled resource in the diagram (fig-1).

Conclusions: We proposed a scheme of repeated generalized BSM's. We calculated the success of perfect QT up to 3rd repeated attempt of GBSM and plot it with the concurrence (C) of the entangled resource. From the graph, it is concluded that the success can be increased by repeating GBSM's.

We are thankful to the Physics Department, University of Allahabad for giving facility of doing our research work. One of the authors SJ is also thankful to UGC for providing financial support.

References: [1] C.H. Bennett, H.G. Brassard, C. Crepeau, R. Joza, A. Peres and W.K. Wootters, Phys. Rev. Lett. 70, (1993) [2] A Einstein, B. Pdolsky and N. Rosen, Phys. Rev. 47, (1935) [3] Wan-Li Li, Chuan Feng Li and Guang-Can Guo, Phys. Rev. A 61, (2000)

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[4] P. Agrawal, A.K. Pati, Phys. Lett. A 305, (2002) [5] P. Agrawal, A.K. Pati, J. Opt. B: Quantum Semiclass. Opt. 6, (2004) [6] A.K. Pati and P. Agrawal, Phys. Lett. A 371, (2007) [7] Xu Chun-Jie, LIU Yi-Min, ZHANG Wen and ZHANG Zhan Jun, Chinese Physical Society 54 (2010) [8] Fengli Yan, Dong Wang, Phys. Lett. A 316, (2003). [9] Fengli yan, Tao Yan, Chinese Science Bulletin 55, (2010) [10] M.K. Mishra and H. Prakash, Annals of Physics 360, (2015)

Role of Simultaneous Correlations in Complementary Bases for Quantum Advantage in Random Access Code with Separable States

Som Kanjilal1, Aiman Khan2, C. Jebaratnam3

1Centre forAstroparticle Physics and Space Science (CAPSS), Bose Institute, Kolkata 700 091, India 2Indian Institute of Technology, Roorkee, India 3S. N. Bose National Centre for Basic Sciences, Kolkata 700 098, India

Abstract: A suitable measure of simultaneous correlations in complementary bases is identified as the necessary and sufficient quantum resource for 2→1 Random Access Code scenarios where certain separable states are also useful.

Summary of the work: Random Access Code (RAC) scenario introduced by Ambainis et al. [1] and Pawlowski et al. [2] is a two-party information transfer protocol defined as follows: Definition: A 푛 → 푚 RAC from Alice to Bob involves Alice’s encoding n bit string to m bits (m < n) and sending to Bob in such a way that Bob can guess any of the n bits with probability at least p. We study 2 → 1-quantum RAC scenarios in the presence of a limited amount of shared classical randomness. It has been recently demonstrated that for such quantum RAC protocols, even separable states may provide quantum advantage [3]. Though all separable states that have been shown to be useful for quantum RAC have nonzero discord, nonzero discord does not necessarily imply that the separable state has quantum advantage in the given situation. This raises the question of what is the appropriate resource for quantum RAC scenarios in the presence of limited amount of shared classical randomness. We answer this question by showing that nonzero value of a suitable measure of simultaneous correlations in complementary bases bears a quantitative relationship with the minimum success probability in any given 2→1 Random Access Code scenario.

In the classical RAC scheme that we consider, Alice and Bob share a random source that sends binary bits to Alice and Bob. Apart from her part of the shared bit, Alice has a two-bit string (S). The protocol involves Alice’s communication of 1 bit to Bob (classical encoding of 2-bit string to 1 bit). Using that 1 bit and his part of shared bit Bob has to guess correctly the bit value of either the first place or the second place of Alice’s two bit string s(the choice of the place whose bit value Bob guesses is not predecided). It can be shown that if the

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Alice’s encoding strategy is deterministic, p is upper bounded by half. This is in contrast to Bobby and Patarek’s result [3] that showed that for non-deterministic strategies, p is upper bounded by two third.

In the quantum version of the RAC protocol, Alice and Bob share a bipartite qubit state and Alice chooses a measurement direction based on the specific bit values of her two-bit string (encoding measurement). Then she encodes the measurement outcomes to binary bit values and sends it Bob (classical encoding). Bob then chooses a measurement direction depending on whether he wants to guess the bit value of first place or the second place (decoding measurement). After that, depending on the outcomes of the measurement Bob guesses either correctly or wrongly.

We identify the conditions on the shared state for which Quantum mechanical RAC will outperform Classical RAC for any choice of encoding and decoding operations. The choice of encoding and decoding operations either can be random or can be decided beforehand.

Furthermore and more importantly, we also identify that for the Bell-diagonal states, quantifiers of quantum correlations in mutually unbiased bases introduced by Guo et. al. [4] is the necessary and sufficient ingredient for quantum advantage in 2 → 1RAC protocol using any set of encoding-decoding measurement.

References: [1] AndrisAmbainis, Debbie Leung, Laura Mancinska and Maris Ozols, Quantum Random Access Codes with Shared Randomness, Arxiv.0810.2939 (2009). [2] Marcin Pawłowski and Marek Żukowski Phys. Rev. A 81, 042326 (2010) [3] T. K. C. Bobby and T. Paterek, New J. Phys16, 093063(2014). [4] Y. Guo and S. Wu Scientific Reports 4, 7179 (2014)

Souradeep Sasmal Exploring linear steering inequalities from state space structure

SOURADEEP SASMAL1, DEBARSHI DAS1, ARUP ROY2

1Centre for Astroparticle Physics and Space Science (CAPSS), Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700 091, India. 2Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata- 700108, India.

Abstract: We find out that maximally entangled state will give maximum quantum violation of n-settings linear steering inequality among all possible two-qubit states for any given set of spin- 1/2 observables at trusted and untrusted party’s side.

Summary of the work: The task of quantum steering, has been introduced recently [1], is to prepare different ensembles at one part of a bipartite system by performing local quantum measurements on another part of the bipartite system in such a way that these ensembles cannot be explained by a local hidden state (LHS) model though respect no-signal can

127 transmit. This implies that the steerable correlations cannot be reproduced by a local hidden variable local hidden state (LHV-LHS) model. In recent years, investigations related to quantum steering have been acquiring considerable significance, as evidenced by a wide range of studies.

In [2] the authors have developed a series of “linear steering inequalities” which are necessary conditions to check whether a bipartite state is steerable when both the parties are allowed to perform n dichotomic measurements on his or her part. The maximum magnitude of the left hand side of 2-settings linear steering inequality for any two qubit state under projective measurements of spin-1/2 observables have been investigated [3,4]. Furthermore, it has been shown that a given two qubit state violates 2-settings linear steering inequality if and only if the given state violates Bell-CHSH inequality [5].In all these studies, maximum magnitudes of the left hand sides of 2-settings linear steering inequality and Bell-CHSH inequality for a given two qubit state have been studied by performing the maximization over all possible measurement settings.

Motivated by the above results we investigate the maximum magnitude (maximized over all possible two-qubit states) of the left hand side of 2-settings linear steering inequality for two spin-1/2 observables at untrusted party’s side and for any two spin-1/2 observables in mutually orthogonal directions at trusted party’s side in the present study. For any given set of spin-1/2 observables, the maximum magnitude of the left hand side of Bell-CHSH inequality has already been studied [6]. Using this result we show that a given set of spin-1/2 observables violate 2-settings linear inequality if and only if the given set of spin-1/2 observables violate Bell-CHSH inequality.

It was argued in [6] that for any set of spin-1/2 observables at the two spatially separated party’s side, maximum magnitude the left hand side of Bell-CHSH inequality is always achieved by a pure maximally entangled state. Since, EPR steering has a vast application in semi device independent scenario, it is thus important to study which entangled state is the most effective resource for witnessing EPR steering contingent upon using a specific set of observables. In particular, we address the following question: given a set of spin-1/2 observables, which two-qubit state achieve the maximum magnitude of the left hand side of the 2-settings, 3-settings, and finally n-settings (n can have arbitrary integer positive values) linear steering inequality. Finding out from which point, non-locality and steering differ from each other, it is point of interest, and we are trying to address this question from state space structure rather than correlation aspect.

It was earlier shown that any given two-qubit state violates 2-settings linear steering inequality if and only of that state violates Bell-CHSH inequality [3]. In this present study we have shown that a given set of spin-1/2 observables violate 2-settings linear inequality if and only if the given set of spin-1/2 observables violate Bell-CHSH inequality. Hence, the result presented in this study complements the result obtained in the previous studies. Moreover, we have also shown that, given any n spin-1/2 observables at Alice’s side and given any n spin- 1/2 observables at Bob’s side, pure maximally entangled two-qubit state gives the maximum magnitude of left hand side of the n-settings linear steering inequality among all possible two-qubit states.

References:

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[1] H. M. Wiseman, S. J. Jones and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007). [2] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Phys. Rev. A. 80, 032112 (2009). [3] A. C. S. Costa, and R. M. Angelo, Phys. Rev. A 93, 020103(R) (2016). [4] S. Mal, D. Das, S. Sasmal, and A. S. Majumdar, arXiv: 1711.00872 (2017). [5] P. Girdhar, and E. G. Cavalcanti, Phys. Rev. A 94, 032317 (2016). [6] G. Kar, Phys. Lett. A 204, 99 (1995).

Experimental certification and quantification of entanglement in a novel spatially correlated bipartite qutrit system

Subhajit Bhar1, Debadrita Ghosh1, Thomas Jennewein2,3 , Piotr Kolenderski4 and Urbasi Sinha1

1 Raman Research Institute, Sadashivanagar, Bangalore 560080, India. 2 Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L3G1, Canada. 3 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada. 4 Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5, 87-100 Torun, Poland.

Abstract: We present an experimental demonstration of spatially correlated bipartite qutrits using novel pump beam modulation techniques and certification and quantification of its entanglement using a novel measure based on Pearson Coefficients.

Summary: One of the long standing open problems for quantum correlation in higher dimensional systems is to certify and quantify entanglement in the experimental scenario. Study of higher dimensional systems is interesting because we can access larger Hilbert space and hence computational power by using such systems than qubits, which are two-level quantum systems. In qubit systems, Negativity is one of the most used quantities to measure entanglement but it provides necessary and sufficient condition to certify entanglement for 2 ⊗ 2 and 2 ⊗ 3 dimensions only. However, this approach involves full state tomography and hence the number of required measurements increases with the size of the system. In a recent paper, Maccone et. al. [1] suggested the use of Pearson correlation coefficient (PCC) for entanglement certification. In our work [2], we extend these criteria for PCC based certification upto dimension 5. We also propose a scheme to quantify entanglement using PCCs. This is interesting particularly because PCCs are directly measurable quantities and it requires a very small number of measurements. In another recent experimental work [3], we develop a novel spatially correlated bipartite qutrit source using pump beam modulation techniques, and successfully measure and quantify the amount of entanglement using the PCCs.

In this poster, I will primarily discuss this experiment on the preparation of spatially correlated bipartite qutrit source and certification and quantification of entanglement using

129 our newly developed theoretical scheme based on PCCs [4]. Our experimental setup is shown below.

Figure 1: Experimental setup

References: [1] Lorenzo Maccone, Dagmar Bruß, and Chiara Macchiavello. Complementarity and correlations. Phys. Rev. Lett., 114:130401, Apr 2015. [2] C.Jebarathinam, Subhajit Bhar, Urbasi Sinha, and Dipankar Home. On certifying and quantifying higher dimensional entanglement using the pearson correlation coefficient as a measure, work in preparation, 2018. [3] Debadrita Ghosh, Thomas Jennewein, Piotr Kolenderski, and Urbasi Sinha. Correlated photonic qutrit pairs for quantum information and communication, 2017, arXiv:1702.02581. [4] Subhajit Bhar, Debadrita Ghosh, Thomas Jennewein, Piotr Kolenderski, and Urbasi Sinha. Experimentally certifying and quantifying entanglement of spatially correlated qutrits, work in preparation, 2018.

INFERRING EXPECTATION VALUE OF NON-HERMITIAN OPERATOR FROM VISIBILITY

Surya Narayan Sahoo1, Gaurav Nirala2, Dr.Arun K Pati3, Dr.Urbasi Sinha1,4,5 1Raman Research Institute, Bengaluru 2The University of Oklahama, USA 3Harish-Chandra Research Institute,Allahabad 4Institute for Quantum Computation, Waterloo, Ontario, Canada 5Centre for Quantum Information and Quantum Computing, Toronto, Canada

Abstract: We experimentally infer the expectation value of a non-Hermitian operator by measuring visibility and phase-shift in a Mach-Zehnder interferometer when the operator’s polar decomposed elements are placed in each arm.

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Summary: Experimentally realizable outcomes are described by real numbers [1] and in quantum mechanics we demand that all observables to be represented by Hermitian operators since their eigenvalues are real. However, it has been argued that, demanding a) eigenvalues to be real and b) validity of Spectral theorem (existence of complete orthonormal eigenbasis) , the general class of operators that may be used to describe observables are normal operators [2]. It has been shown that [3] the expectation value of a non-Hermitian operator can be inferred by measuring the weak value of the Hermitian operator into which the non- Hermitian operator can be polar decomposed. Consider the operator퐴, which in general can be non-Hermitian, which can be polar decomposed as: 퐴 = 푈푅, where 푈 is Unitary operator and 푅 is the Hermitian operator obtained as = √퐴퐴. The expectation value of 퐴 can be expressed in terms of the weak value of 푅 as follows ⟨휙|푅|휓⟩ ⟨퐴⟩ = ⟨휓|퐴|휓⟩ = ⟨휓|푈푅|휓⟩ = ⟨휙|휓⟩ ⟨휙|휓⟩ where, ⟨휙|=⟨휓|푈 In this presentation, we experimentally obtain the quantity ⟨휓|푈푅|휓⟩from visibility and phase shift in a Mach Zehnder interferometer where 푈and 푅are operators corresponding to physically realizable optical components placed in each arm. From this, we infer the expectation value of 퐴 and weak value of 푅. In particular, we choose the spin ladder operator as our non-Hermitian operator 0 0 퐴 = 1 0 which are realized in lab through different optical components. We also infer weak value of 푅 by measuring visibility in two different experiments to have ⟨휙|푅|휓⟩and⟨휙|휓⟩.

References: [1] The Principles of Quantum Mechanics, Paul Dirac [2] Hu, MJ., Hu, XM. & Zhang, YS. Quantum Stud.: Math. Found. (2017) 4: 243. [3] Measuring non-Hermitian operators via weak values, Phys. Rev. A 92, 052120

Noise tolerance of N-qubit symmetric states in discriminating rotations

Syed Raunaq Ahmed, H.S.Karthik, Sudha‡, A. R. Usha Devi

Department of Physics, Jnanabharathi, Bangalore University, Bangalore-560056, India ‡ Department of Physics, Jnanasahyadri, Kuvempu University, Shankaraghatta – 577451, India

Abstract: Any two rotations are shown to be discriminated using N-qubit symmetric state, represented on the Bloch sphere via N distinct Majorana spinors. Noise tolerance of this discrimination is explored.

Summary: Suppose we have to distinguish two unknown spatial rotations 푅 and 푅. We may consider a unit vector 푛on which the rotations 푅 are 푅 are applied. Can therotated unit

131 vectors 푛and 푛 be resolved as distinct? The problem gets translated to finding 훼 = cos (푛 ∙ 푛)(i.e., the angle between the rotated unit vectors 푛 and 푛) with greater precision. This can be done by maximizing the Euclidean distance 푑(푛, 푛) = 2 sin over the set of all initial unit vectors푛. Smaller the angle 훼, lessresolvable are the two rotations.

In a similar spirit, given any two discrete probability distributions (푝 ≥ 0, ∑ 푝 =1) and (푞 ≥ 0, and∑ 푞 = 1) the statistical distance, given by 푑(푝, 푞) = cos ( ∑ 푝푞),serves as a natural measure of distinguishability of the two probability distributions. Wooters [1] extended the notion of statistical distance between two probability distributions to the quantum domain. More specifically, suppose we have to identify an unknown quantum state, given a set of two quantum states |휓 ⟩ and |휑 ⟩ with minimum error. When a measurement

{퐸, 푖 = 1,2…} (with the elements satisfying 퐸 ≥ 0, ∑ 퐸 = 퐼, with 퐼 denoting the identity operator) is carried out, one obtains probabilities of outcome i as 푝 = ⟨휓|퐸|휓⟩ and

푞 = ⟨휑|퐸|휑⟩. Thus, the problem of distinguishing |휓⟩ and |휑⟩ gets translated into finding the statistical distance 푑(푝, 푞)between the probability distributions {푝} and {푞}. Interestingly, the statistical distance 푑(푝, 푞),with respect to a quantum measuring device {퐸}is equal to the Hilbert space distance 푑(|휓⟩,|휑⟩) = cos (|휓⟩,|휑⟩)=|⟨휓|휙⟩| between two pure states [1]:

( ) ⟨ | | ⟩⟨ | | ⟩ 푑 푝, 푞 = min{} cos 휓 퐸 휓 휑 퐸 휑

=퐹(|휓⟩,|휑⟩) = |⟨휓|휙⟩| = 푑(|휓⟩,|휑⟩) where 퐹(|휓 ⟩,|휑 ⟩)= |⟨휓|휙⟩| denotes the fidelity between two pure states. This remarkable connection exhibited between the statistical distance 푑(푝, 푞) and the Hilbert space distance

푑(|휓⟩,|휑⟩) plays a foundational role in the framework of information geometry [2]. Acin[3] explored the statistical distinguishability of two unitary matrices 푈, 푈 휖 푆푈(2) and showed thatby optimizing over the input states|휓 ⟩ of a single qubit, the Hilbert space distance is given by Tr[푈 푈] max 푑 (푈 |휓 ⟩, 푈 |휓 ⟩) = cosퟏ 퐹(푈 , 푈 ) = cos {|⟩} 2 Where, the fidelity between two unitary operations is given by 퐹(푈, 푈) = min{| ⟩} cos 휓푈 푈휓 = |cos 휃|

The input state which leads to the optimal fidelity is given by |휓⟩ = a|0⟩ + 푏 |1⟩ with |푎| = 1/2 ; here |0 > and |1 > are the eigenstates of 푈 = 푈 푈 corresponding to the eigenvalues 푒±. In other words, it is not possible to discriminate two unitary operations perfectly unless 휗 = . However, there always exists [3] a finite integer 푁 such     that 푈 ,푈 are perfectly distinguishable i.e., 퐹푈 , 푈 =0. The input state which   leads to perfect discrimination between 푈 and 푈 turns out to be a 푁-qubit entangled state. In this paper we paper we address the issue of distinguishing two unknown rotations 푅,푅 using the (2푗 +1)dimensional irreducible representations 퐷 (푅)of rotations, which is

132 characterized by the index푗 = 1/2,1,3/2...Based on previous works [2,3] we show that there exists an irreducible representation with 푗 = 푁/2for which퐷 (푅), 퐷 (푅)can be distinguished perfectly. We identify that the Majorana geometric representation [4, 5, 6,7] of the permutation invariant 푁-qubit input state – which results in the zero-error discrimination of two rotations – corresponds geometrically to a constellation 푁 equally spaced distinct points on the surface of the Bloch sphere. This provides an elegant geometric picture for the task of discrimination of any two unknown rotations. While discrimination fidelity is zero when N qubit symmetric pure state, corresponding to which the Majorana geometric constellation has N distinct points on the Bloch sphere, we extend our investigation on the noise robustness of discrimination by evaluating the fidelity between the noisy mixed state obtained after rotations. We also discuss the possibility of distinguishability of rotations symmetric states of 푁 qubits consisting of Majorana geometric constellation of 푘 < 푁 distinct points. Our work provides an intrinsic geometric connection between the Majorana constellation and statistical distinguishability of rotations. Acknowledgements: SRA and ARU acknowledge the support of UGC MRP (Ref. MRP- MAJOR-PHYS-2013-29318).

References: [1] W. K. Wootters, “Statistical distance and Hilbert space,” Phys. Rev. D 23, 357–362 (1981). [2] K. R. Parthasarathy, “On consistency of the maximum likelihood method in testing multiple quantum hypothesis,”in Stochastics in finite and infinite dimensions, T.Hida et. al., eds. (Springer, 2001), pp. 361–377. [3] A. Acin, “ Statistical distinguishability between unitary operations,” Phys. Rev. Lett. 87, 177901 (2001). [4] E.Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento 9, 43–50 (1932). [5] A.R.Usha Devi, Sudha and A.K.Rajagopal, “Majorana representation of symmetric multiqubit states” Quantum Inf. Proc. 11, 685–710 (2012). [6] D. Markham, “Entanglement and symmetry in permutation-symmetric states” , Phys. Rev. A83, 042332 (2011). [7] Z. Wang and D. Markham, “Nonlocality of Symmetric States” Physical Review Letters, 108, 210407 (2012)

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Quantum key distribution in a noise-restricted-adversary model

U. Shrikant1,2 , Vishal Sharma3, R. Srikanth1 and Subhashish Banerjee3

1Poornaprajna Institute of Scientific Research, Bengaluru- 560080, India 2Graduate Studies, Manipal Acedemy of Higher Education, Manipal - 576104. 3IIT Jodhpur, Rajasthan, India.

Abstract: We find that the information security of the quantum key distribution scheme based on the Ping-Pong protocol against a noise-restricted adversary improves, surprisingly, under a non-unital noisy channel.

Summary: The Ping-pong protocol leverages the idea behind dense coding for sending secure deterministic bits of information [1], where Bob sends Alice one half of a singlet, on which Alice encodes her bit by applying Pauli I or Z operations. The security arises, intuitively, from the fact that the travelling particle is in a maximally mixed state. Surprisingly, Wojcik [2] found an attack strategy on the Ping-pong protocol, the subtlety being that the channel is attacked before and after Alice’s action, on the onward and return legs of Bob’s photon, and relying on Alice and Bob not monitoring channel losses.

Here we analyze the performance of the Ping-pong protocol (modified to a key distribution, rather than direct communication, scheme) under a noisy channel. Conservatively, one assumes that all noise is due to eavesdropper Eve’s action. Here we introduce the model of a noise-restricted Eve, wherein she can’t replace Alice-Bob’s noisy channel with an ideal one, but her own probes are noiseless.

In our work, the inclusion of noise is as per the idealized scheme depicted in fig(1). In the case of a unital channel, the results are straightforward, in that noise imparts no advantage. But the case of a non-unital channel (such as AD) is interesting, and reported below.

In this work, we have used a typical noise scheme as shown in fig(1). Communication and noise scenario: Bob sends to Alice one half of a Bell state. Alice retransmits the particle after applying operations Ior σZ .The effect of noise is idealized as once in the onward leg before Eve’s action Q, and once afterwards in the return leg after her attack action Q −1.

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The mutual information between Alice and Bob is found to be:

( ) ( ) 퐼(퐴: 퐵) = 푝 푝 log + 푝 log + (푝 −2) log () () + (푝 −2) log −2푝(푝 +2) log(2) () () () −4(푝 −1)푙표푔 + 2log + 2 푙표푔 +4 () () () and that of Alice and Eve to be,

퐼(퐴: 퐸) = 6+2log + (1− (푝 −2)푝) log () . ()()

These two quantities are represented as a function of noise p in Figure 2. The surprising conclusion is that the key rate κ ≡ I (A:B) − I( A:E) is positive for all finite noise below 1. Since this quantity vanishes at zero noise, noise can be considered as breaking the symmetry that exists between Alice and Eve under this attack.

The obvious implication is that noise bestows an advantage for Alice and Bob.

FIG. 2. Performance of the modified Ping-pong protocol under AD noise: The bold, dashed and dotted plots represents I(A:B) , I(A:E) and the Holevo bound for Alice- Bob. That I(A:B)> I( A:E) for 0 < p ≤ 1 implies that noise is beneficial to the legitimate users. In the noiseless limit, the Holevo bound coincides with I(A:B), implying that the measurement strategy is optimal.

Similarly it has been shown in Ref [6], that Depolarizing noise turns out to be disadvantageous to legitimate parties but advantageous to eavesdropper. Conclusion: Noise is usually detrimental to quantum information processing. We report a contrary example to this expectation here. In quantum key distribution, conservatively one assumes that all noise is due to Eve. In the more realistic scenario considered here, environmentally induced decoherence bounds Eve, too. Nontrivially, Eve’s probes are noiseless. Her only limitation is that she can’t replace the noisy channel of Alice and Bob by an ideal one. This noise-induced advantage is exemplified here in a quantum key distribution scheme based on the Pingpong protocol for the case of a non-unital channel (amplitude damping), but not for unital channels.

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References: [1] Kim Boström and Timo Felbinger. Deterministic secure direct communication using entanglement. Physical Review Letters, 89(18):187902, 2002. [2] Antoni Wójcik, Eavesdropping on the “ping-pong” quantum communication protocol. Physical Review Letters, 90(15):157901, 2003. [3] Kim Boström and Timo Felbinger. On the security of the ping-pong protocol. Physics Letters A, 372(22):3953–3956, 2008. [4] M.A. Nielsen and I.L. Chuang, Quantum computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [5] R Srikanth and Subhashish Banerjee. Squeezed generalized amplitude damping channel. Physical Review A, 77(1):012318, 2008. [6] Vishal Sharma, U. Shrikant, R. Srikanth, Subhashish Banerjee, Decoherence can help quantum cryptographic security, arXiv:1712.06519v1[quant-ph]

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