DOCSLIB.ORG

Entanglement Properties of a Measurement-Based Entanglement Distillation Experiment

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Entanglement Properties of a Measurement-Based Entanglement Distillation Experiment PHYSICAL REVIEW A 99, 042304 (2019) Entanglement properties of a measurement-based entanglement distillation experiment Hao Jeng,1 Spyros Tserkis,2 Jing Yan Haw,1 Helen M. Chrzanowski,1,3 Jiri Janousek,1 Timothy C. Ralph,2 Ping Koy Lam,1 and Syed M. Assad1,* 1Centre for Quantum Computation and Communication Technology, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 2601, Australia 2Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, St. Lucia, Queensland 4072, Australia 3Institute of Physics, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany (Received 15 November 2018; published 1 April 2019) Measures of entanglement can be employed for the analysis of numerous quantum information protocols. Due to computational convenience, logarithmic negativity is often the choice in the case of continuous-variable sys- tems. In this work, we analyze a continuous-variable measurement-based entanglement distillation experiment using a collection of entanglement measures. This includes logarithmic negativity, entanglement of formation, distillable entanglement, relative entropy of entanglement, and squashed entanglement. By considering the distilled entanglement as a function of the success probability of the distillation protocol, we show that the logarithmic negativity surpasses the bound on deterministic entanglement distribution at a relatively large probability of success. This is in contrast to the other measures which would only be able to do so at much lower probabilities, hence demonstrating that logarithmic negativity alone is inadequate for assessing the performance of the distillation protocol. In addition to this result, we also observed an increase in the distillable entanglement by making use of upper and lower bounds to estimate this quantity. We thus demonstrate the utility of these theoretical tools in an experimental setting. DOI: 10.1103/PhysRevA.99.042304 I. INTRODUCTION notably, the logarithmic negativity [14] as one can calculate it quite straightforwardly. On the one hand, quantum entanglement is a useful non- In this work, we analyze a continuous-variable classical resource. It can be used for the construction of measurement-based entanglement distillation experiment quantum gates [1] or for the distribution of cryptographic keys [5] using a collection of measures. We present two main in a secure manner [2]. On the other hand, implementations results. First, we show that the logarithmic negativity is of these tasks are usually limited in performance due to ex- distinct from the other measures; it crosses the “deterministic perimental imperfections. Utilizing a variety of methods such bound” before the other measures do, at a probability as photon subtraction [3,4] and noiseless linear amplification of success (of the distillation protocol) that is orders of [5,6], entanglement distillation protocols seek a potential res- magnitude greater. The deterministic bound is the maximum olution to this problem by concentrating weakly entangled entanglement that can be deterministically distributed across a states into subsets that are more entangled. given quantum channel (usually imperfect), assuming that one Here we address the problem of quantifying entanglement had an Einstein-Podolsky-Rosen (EPR) resource state with distillation; this will, in general, depend on the kind of system infinite squeezing. For instance, when the entanglement of that one is working with. In the case of discrete variables, the formation crosses this bound, we can take it to indicate a form fidelity with respect to some maximally entangled state [7] of error correction [15], thus giving an example of how the and nonlocality based on the Bell inequalities [8] are both use- logarithmic negativity can fail to capture important properties ful measures for observing quantum entanglement. However, of distillation protocols. Our results can be regarded as an these methods are not particularly suitable in the case of con- experimental demonstration of such an example. tinuous variables. Maximally entangled continuous-variable Our second result is the certification of an increase in states are unphysical, and a theorem from Bell precludes the distillable entanglement. Currently, there is no known the demonstration of nonlocality using Gaussian states and way to evaluate the distillable entanglement directly, which Gaussian measurements (the standard tools for continuous means that one is only able to look at it through the use variable experiments) [9], unless one introduces additional of upper and lower bounds [16,17]. The upper bound puts a assumptions on one’s system [10,11]. Thus far, the analy- limit on how much distillable entanglement we had prior to ses of continuous-variable entanglement distillation have in- distillation, while the lower bound guarantees at least how stead centered around inseparability criteria [12,13] and, most much we have after performing distillation. By observing a sufficient increase in the lower bound, we could certify that the distillable entanglement has indeed increased. We *[email protected] remark that the minimization of optical loss and the choice 2469-9926/2019/99(4)/042304(11) 042304-1 ©2019 American Physical Society HAO JENG et al. PHYSICAL REVIEW A 99, 042304 (2019) of a sharp upper bound turned out to be important factors in Any given covariance matrix σ can be put into the follow- order to observe the increase in the distillable entanglement. ing standard form: In particular, there are many possible choices for the upper ⎡ ⎤ bound, but the relative entropy of entanglement was found to m 0 c1 0 ⎢ 0 m 0 c ⎥ be the only one that was sufficiently stringent for this task. σ = ⎣ 2⎦, (2) We have organized this paper as follows. In Sec. II,we c1 0 n 0 provide some background in Gaussian quantum optics and 0 c2 0 n establish the notations and conventions that will be used and this can be done using only local Gaussian unitaries [13], throughout this paper. In Sec. III, definitions of the various which does not change the entanglement of the state. entanglement measures are provided, discussing their basic One can always diagonalize the covariance matrix using properties with an emphasis on operational interpretations. In only symplectic matrices, and the corresponding eigenval- Sec. IV, the final section, we briefly describe the experiment ues are called symplectic eigenvalues [20]. Of particular setup and present a discussion of the distillation results based importance are the symplectic eigenvalues of the partially on the measures. transposed state. For two-mode Gaussian states, the partial transpose flips the sign of the phase quadrature, equivalent to II. PRELIMINARIES flipping the sign of the c2 entry in the standard form of the covariance matrix [Eq. (2)]. The symplectic eigenvaluesν ˜± of Gaussian states can be characterized by the first and second the partially transposed state are given by moments of the quadrature operators [18]—also known re- spectively as the mean fields and the covariance matrix. Since ˜ ± ˜ 2 − σ 2 4 det measures of entanglement depend only on the covariance ν˜± = , (3) matrix, we will assume vanishing mean fields without the 2 loss of generality. For two-mode Gaussian states, the covari- where ˜ = det M + det N − 2 det C. ance matrix can be written in block form (using the xpxp For convenience, covariance matrices of the form notation [19]): ⎡ ⎤ m 0 c 0 σ = MC, ⎢0 m 0 −c⎥ T σ = ⎣ ⎦ (4) C N c 0 m 0 0 −c 0 m where M, N, and C are real 2 × 2 matrices. In general, the entries of the covariance matrix depend on the value of will be called symmetric, while those of the form h¯; in this paper, we will normalize to the variance of the ⎡ ⎤ vacuum field, which amounts to settingh ¯ = 2. In order for m 0 c 0 the covariance matrix to represent a physical state, it is also ⎢0 m 0 −c⎥ σ = ⎣ ⎦ (5) required to satisfy the Heisenberg uncertainty principle. c 0 n 0 The density matrix of closed quantum systems evolve 0 −c 0 n under unitaries:ρ ˆ → Uˆ ρˆUˆ †. In general, representations of these unitaries in the Fock basis require infinite-dimensional will be called quadrature symmetric. These states form strict matrices; if we restrict ourselves to Gaussian operations, subsets of general two-mode Gaussian states [Eq. (2)] by then this can be simplified to the evolution of covariance imposing different kinds of symmetries. matrices: σ → Sσ ST . For two-mode states, each S is a 4 × 4 square matrix and is symplectic with respect to the following III. ENTANGLEMENT MEASURES symplectic form: ⎡ ⎤ We present some background on the theory of entangle- 0100 ment measures, including definitions, properties, and formu- ⎢− ⎥ = ⎣ 10 0 0⎦. las that will be useful. We note that the problem of computing 0001 entanglement measures is NP-complete for many measures 00−10 [21], with analytical expressions available only in restricted cases that will generally possess some degree of symmetry. It satisfies the equation SST = . Every Gaussian unitary In the special case of the entanglement of formation, such a Uˆ is associated with a symplectic matrix. We will find the restriction will give rise to a simple operational interpretation following unitary useful: in terms of quantum squeezing and thus a connection to the logarithmic negativity which we briefly discuss. r(ˆabˆ−aˆ†bˆ† )/2 Sˆ2(r) = e , (1) which is known as the two-mode-squeezing operator. As A. Entanglement entropy ˆ usual,a ˆ and b denote the annihilation operators of the The entanglement entropy EV uniquely determines entan- two optical modes. The two-mode-squeezing operator is glement on the set of pure states. Given the von Neumann parametrized by the squeezing parameter r, with r = 0 cor- entropy S, with responding to no squeezing and r →∞to the limit of infinite squeezing. S(ˆρ) =−tr(ρ ˆ lnρ ˆ), (6) 042304-2 ENTANGLEMENT PROPERTIES OF A … PHYSICAL REVIEW A 99, 042304 (2019) wardly.
Load more

Recommended publications
  • {Dоwnlоаd/Rеаd PDF Bооk} Entangled Kindle
    ENTANGLED PDF, EPUB, EBOOK Cat Clarke | 256 pages | 06 Jan 2011 | Hachette Children's Group | 9781849163941 | English | London, United Kingdom Entangled PDF Book A Trick of the Tail is the seventh studio album by English progressive rock band Genesis. Why the Name Entangled? Shor, John A. Annals of Physics. Bibcode : arXiv The Living Years. Suhail Bennett, David P. Part of a series on. Examples of entanglement in a Sentence his life is greatly complicated by his romantic entanglements. Take the quiz Forms of Government Quiz Name that government! These four pure states are all maximally entangled according to the entropy of entanglement and form an orthonormal basis linear algebra of the Hilbert space of the two qubits. The researchers used a single source of photon pairs that had been entangle d, which means their quantum states are intrinsically linked and any change or measurement of one is mirrored in the other. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit which is now a LOCC operation, since both qubits are in B's lab and then teleporting the qubit back to A. Looking for some great streaming picks? In earlier tests, it couldn't be absolutely ruled out that the test result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location. Thesis University of California at Berkeley, The Hilbert space of the composite system is the tensor product.
    [Show full text]
  • Limitations on Protecting Information Against Quantum Adversaries
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School April 2020 Limitations on Protecting Information Against Quantum Adversaries Eneet Kaur Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Quantum Physics Commons Recommended Citation Kaur, Eneet, "Limitations on Protecting Information Against Quantum Adversaries" (2020). LSU Doctoral Dissertations. 5208. https://digitalcommons.lsu.edu/gradschool_dissertations/5208 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. LIMITATIONS ON PROTECTING INFORMATION AGAINST QUANTUM ADVERSARIES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Eneet Kaur B.Sc., University of Delhi, 2012 M.Sc., Indian Institute of Technology, Roorkee, 2014 May 2020 Acknowledgments First and foremost, I would like to thank my advisor, Mark M. Wilde, for offering me the opportunity to work with him. My discussions with him have been instru- mental in shaping my approach towards research and in writing research papers. I am also thankful to Mark for providing me with the much-needed confidence boost, and for being an incredible mentor. I would also like to thank Jonathan P. Dowling for his encouragement, the fun pizza parties, and for all his stories. I would also like to thank Andreas Winter for hosting me in Barcelona and for his collaborations.
    [Show full text]
  • Linear Growth of the Entanglement Entropy for Quadratic Hamiltonians and Arbitrary Initial States
    Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states Giacomo De Palma∗1 and Lucas Hackl†2 1Scuola Normale Superiore, 56126 Pisa, Italy 2School of Mathematics and Statistics & School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia July 26, 2021 Abstract We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the ini- tial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement en- tropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models. 1 Introduction arXiv:2107.11064v1 [quant-ph] 23 Jul 2021 Entanglement is a cornerstone of quantum theory and its dynamics has been extensively studied in a wide range of different systems [1{5]. It also provides an important link be- tween classical chaos and quantum chaos in the context of Lyapunov instabilities [6].
    [Show full text]
  • Quantum Information Chapter 10. Quantum Shannon Theory
    Quantum Information Chapter 10. Quantum Shannon Theory John Preskill Institute for Quantum Information and Matter California Institute of Technology Updated June 2016 For further updates and additional chapters, see: http://www.theory.caltech.edu/people/preskill/ph219/ Please send corrections to [email protected] Contents 10 Quantum Shannon Theory 1 10.1 Shannon for Dummies 2 10.1.1 Shannon entropy and data compression 2 10.1.2 Joint typicality, conditional entropy, and mutual infor- mation 6 10.1.3 Distributed source coding 8 10.1.4 The noisy channel coding theorem 9 10.2 Von Neumann Entropy 16 10.2.1 Mathematical properties of H(ρ) 18 10.2.2 Mixing, measurement, and entropy 20 10.2.3 Strong subadditivity 21 10.2.4 Monotonicity of mutual information 23 10.2.5 Entropy and thermodynamics 24 10.2.6 Bekenstein’s entropy bound. 26 10.2.7 Entropic uncertainty relations 27 10.3 Quantum Source Coding 30 10.3.1 Quantum compression: an example 31 10.3.2 Schumacher compression in general 34 10.4 Entanglement Concentration and Dilution 38 10.5 Quantifying Mixed-State Entanglement 45 10.5.1 Asymptotic irreversibility under LOCC 45 10.5.2 Squashed entanglement 47 10.5.3 Entanglement monogamy 48 10.6 Accessible Information 50 10.6.1 How much can we learn from a measurement? 50 10.6.2 Holevo bound 51 10.6.3 Monotonicity of Holevo χ 53 10.6.4 Improved distinguishability through coding: an example 54 10.6.5 Classical capacity of a quantum channel 58 ii Contents iii 10.6.6 Entanglement-breaking channels 62 10.7 Quantum Channel Capacities and Decoupling
    [Show full text]
  • Entanglement Structures in Qubit Systems Arxiv:1505.03696V3 [Hep-Th]
    Prepared for submission to JHEP DCPT-15/27 Entanglement structures in qubit systems Mukund Rangamani, Massimiliano Rota Centre for Particle Theory & Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. E-mail: [email protected], [email protected] Abstract: Using measures of entanglement such as negativity and tangles we pro- vide a detailed analysis of entanglement structures in pure states of non-interacting qubits. The motivation for this exercise primarily comes from holographic considera- tions, where entanglement is inextricably linked with the emergence of geometry. We use the qubit systems as toy models to probe the internal structure, and introduce some useful measures involving entanglement negativity to quantify general features of entanglement. In particular, our analysis focuses on various constraints on the pat- tern of entanglement which are known to be satisfied by holographic sates, such as the saturation of Araki-Lieb inequality (in certain circumstances), and the monogamy of mutual information. We argue that even systems as simple as few non-interacting qubits can be useful laboratories to explore how the emergence of the bulk geometry may be related to quantum information principles. arXiv:1505.03696v3 [hep-th] 27 Aug 2015 Contents 1 Introduction1 2 Measures of entanglement6 2.1 Bipartite entanglement6 2.2 Multipartite entanglement 10 2.3 Notation 12 3 Warm up: Three qubits 14 4 Four qubits 17 4.1 Generic states of 4 qubits 18 4.1.1 Monogamy of the negativity and disentangling theorem 19 4.1.2 Negativity to entanglement ratio 21 4.1.3 Monogamy of mutual information 25 4.2 SLOCC classification of 4 qubit states 26 5 Large N qubit systems 34 5.1 Negativity versus entanglement 35 5.2 Exploring multipartite entanglement 39 6 Discussion: Lessons for holography 41 A Four qubit states: Detailed analysis of SLOCC classes 47 1 Introduction One of the key features distinguishing quantum mechanics is the presence of entan- glement which is a natural consequence of the superposition principle.
    [Show full text]
  • Classical Simulation of Quantum Many-Body Systems by Yichen Huang Doctor of Philosophy in Physics University of California, Berkeley Professor Joel E
    Classical simulation of quantum many-body systems by Yichen Huang A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Joel E. Moore, Chair Professor Dung-Hai Lee Professor Umesh V. Vazirani Spring 2015 Classical simulation of quantum many-body systems Copyright 2015 by Yichen Huang 1 Abstract Classical simulation of quantum many-body systems by Yichen Huang Doctor of Philosophy in Physics University of California, Berkeley Professor Joel E. Moore, Chair Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation. Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with \little entanglement" are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for \little entanglement" is area law, which has been proved for unique ground states in 1D gapped systems.
    [Show full text]
  • The Squashed Entanglement of a Quantum Channel
    The squashed entanglement of a quantum channel Masahiro Takeoka∗y Saikat Guhay Mark M. Wildez January 22, 2014 Abstract This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed en- tanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Chri- standl, Winter, et al., establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1 + η)=(1 − η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1=(1 − η)). Thus, in the high-loss regime for which η 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal.
    [Show full text]
  • Entanglement of Purification for Multipartite States and Its
    YITP-18-41 Entanglement of Purification for Multipartite States and its Holographic Dual Koji Umemotoa and Yang Zhoub aCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan bDepartment of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China Abstract We introduce a new information-theoretic measure of multipartite corre- lations ∆P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several en- tropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mu- tual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjec- ture a holographic dual of multipartite entanglement of purification ∆W , as a sum of minimal areas of codimension-2 surfaces which divide the entan- glement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of pu- rification. These agreements strongly support the ∆P = ∆W conjecture. arXiv:1805.02625v2 [hep-th] 1 Nov 2018 We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications. Contents 1 Introduction 1 2 Multipartite entanglement of purification 3 2.1 Definition .
    [Show full text]
  • JHEP10(2018)152 Springer June 23, 2018 : October 14, 2018 October 24, 2018 : : Received Accepted Published Conjecture
    Published for SISSA by Springer Received: June 23, 2018 Accepted: October 14, 2018 Published: October 24, 2018 Entanglement of purification for multipartite states and its holographic dual JHEP10(2018)152 Koji Umemotoa and Yang Zhoub aCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan bDepartment of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China E-mail: [email protected], yang [email protected] Abstract: We introduce a new information-theoretic measure of multipartite correlations ∆P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification ∆W , as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this ge- ometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the ∆P = ∆W conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mu- tual information in holographic CFTs and its applications.
    [Show full text]
  • Lecture Notes Live Here
    Physics 239/139: Quantum information is physical Spring 2018 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. THESE NOTES ARE SUPERSEDED BY THE NOTES HERE Schr¨odinger's cat and Maxwell's demon, together at last. Last updated: 2021/05/09, 15:30:29 1 Contents 0.1 Introductory remarks............................4 0.2 Conventions.................................7 0.3 Lightning quantum mechanics reminder..................8 1 Hilbert space is a myth 11 1.1 Mean field theory is product states.................... 14 1.2 The local density matrix is our friend................... 16 1.3 Complexity and the convenient illusion of Hilbert space......... 19 2 Quantifying information 26 2.1 Relative entropy............................... 33 2.2 Data compression.............................. 37 2.3 Noisy channels............................... 43 2.4 Error-correcting codes........................... 49 3 Information is physical 53 3.1 Cost of erasure............................... 53 3.2 Second Laws of Thermodynamics..................... 59 4 Quantifying quantum information and quantum ignorance 64 4.1 von Neumann entropy........................... 64 4.2 Quantum relative entropy......................... 67 4.3 Purification, part 1............................. 69 4.4 Schumacher compression.......................... 71 4.5 Quantum channels............................. 73 4.6 Channel duality............................... 79 4.7 Purification, part 2............................. 84 4.8 Deep
    [Show full text]
  • Entanglement Witnesses with Tensor Networks Characterizing Entanglement in Large Systems
    Entanglement Witnesses with Tensor Networks Characterizing entanglement in large systems Aluno: Orientador: Lucas Vieira Barbosa Prof. Dr. Reinaldo O. Vianna Co-orientador: Thiago O. Maciel Dissertação apresentada à UNIVERSIDADE FEDERAL DE MINAS GERAIS - UFMG como requisito parcial para a obtenção do grau de MESTRE EM FÍSICA Belo Horizonte, MG, Brasil 2019–11–13 Dados Internacionais de Catalogação na Publicação (CIP) B238e Barbosa, Lucas Vieira. Entanglement witnesses with tensor networks: characterizing entanglement in large systems / Lucas Vieira Barbosa. – 2019. 81f., enc.: il. Orientador: Reinaldo Oliveira Vianna. Coorientador: Thiago Oliveira Maciel Dissertação (mestrado) – Universidade Federal de Minas Gerais, Departamento de Física. Bibliografia: f. 74-81. 1. Informação quântica. 2. Teoria quântica. 3. Emaranhamento, teses. I. Título . II. Vianna , Reinaldo Oliveira . III. Universidade Federal de Minas Gerais, Departamento de Física. CDU – 530.145(043) Elaborada pela Biblioteca Professor Manoel Lopes de Siqueira da UFMG. Entanglement Witnesses with Tensor Networks Characterizing entanglement in large systems Lucas Vieira Barbosa 2019–11–13 Resumo Emaranhamento é certamente um dos fenômenos mais fascinantes observados na Natureza, e mesmo após de décadas de pesquisas na teoria de emaranhamento ainda há muito a ser des- coberto e entendido. Em particular, ainda há poucos resultados detalhando e caracterizando a estrutura do emaranhamento em sistemas fortemente correlacionados em larga escala, envol- vendo um número grande de subsistemas. A forma mais comum de se estudar detalhadamente a estrutura de emaranhamento em es- tados quânticos tem sido através da otimização de Testemunhas de Emaranhamento (Entangle- ment Witnesses), otimizadas utilizando-se técnicas algorítmicas como Programação Semidenida (Semidenite Programming) aplicada à matrizes. No entanto, esta descrição matricial torna a téc- nica computacionalmente inviável para sistemas em larga escala devido ao crescimento expo- nencial da dimensão do espaço de parâmetros a serem otimizados.
    [Show full text]
  • Arxiv:0906.1499V2 [Cond-Mat.Stat-Mech] 3 Dec 2009 II Ocuin Nageeta H Are for Barrier the As Entanglement Conclusion: VIII
    A short review on entanglement in quantum spin systems J. I. Latorre and A. Riera Dept. d’Estructura i Constituents de la Matèria, Universitat de Barcelona, 647 Diagonal, 08028 Barcelona, Spain We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation. PACS numbers: 03.67.-a, 03.65.Ud, 03.67.Hk Contents I. INTRODUCTION I. Introduction 1 Quantum systems are ultimately characterized by the ob- servable correlations they exhibit. For instance, an observ- II. An explicit computation of entanglement entropy 2 able such as the correlation function between two spins in a A.EntanglementEntropy 2 typical spin chain may decay exponentially as a function of B.XXmodel 3 the distance separating them or, in the case the system under- C.GroundState 3 goes a phase transition, algebraically. The correct assessment D.Entanglemententropyofablock 4 of these quantum correlations is tantamount to understanding E.Scalingoftheentropy 5 how entanglement is distributed in the state of the system. F. Entanglement entropy and Toeplitz determinant 6 This is easily understood as follows. Let us consider a con- nected correlation III. Scaling of entanglement 6 Ψ O O Ψ (1) A.One-dimensionalsystems 7 | i j | c ≡ B.
    [Show full text]