Bell's Inequality and Quantum Tomography
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A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper
A tutorial introduction to quantum circuit programming in dependently typed Proto-Quipper Peng Fu1, Kohei Kishida2, Neil J. Ross1, and Peter Selinger1 1 Dalhousie University, Halifax, NS, Canada ffrank-fu,neil.jr.ross,[email protected] 2 University of Illinois, Urbana-Champaign, IL, U.S.A. [email protected] Abstract. We introduce dependently typed Proto-Quipper, or Proto- Quipper-D for short, an experimental quantum circuit programming lan- guage with linear dependent types. We give several examples to illustrate how linear dependent types can help in the construction of correct quan- tum circuits. Specifically, we show how dependent types enable program- ming families of circuits, and how dependent types solve the problem of type-safe uncomputation of garbage qubits. We also discuss other lan- guage features along the way. Keywords: Quantum programming languages · Linear dependent types · Proto-Quipper-D 1 Introduction Quantum computers can in principle outperform conventional computers at cer- tain crucial tasks that underlie modern computing infrastructures. Experimental quantum computing is in its early stages and existing devices are not yet suitable for practical computing. However, several groups of researchers, in both academia and industry, are now building quantum computers (see, e.g., [2,11,16]). Quan- tum computing also raises many challenging questions for the programming lan- guage community [17]: How should we design programming languages for quan- tum computation? How should we compile and optimize quantum programs? How should we test and verify quantum programs? How should we understand the semantics of quantum programming languages? In this paper, we focus on quantum circuit programming using the linear dependently typed functional language Proto-Quipper-D. -
Bell Inequalities Made Simple(R): Linear Functions, Enhanced Quantum Violations, Post- Selection Loopholes (And How to Avoid Them)
Bell inequalities made simple(r): Linear functions, enhanced quantum violations, post- selection loopholes (and how to avoid them) Dan Browne: joint work with Matty Hoban University College London Arxiv: Next week (after Matty gets back from his holiday in Bali). In this talk MBQC Bell Inequalities random random setting setting vs • I’ll try to convince you that Bell inequalities and measurement-based quantum computation are related... • ...in ways which are “trivial but interesting”. Talk outline • A (MBQC-inspired) very simple derivation / characterisation of CHSH-type Bell inequalities and loopholes. • Understand post-selection loopholes. • Develop methods of post-selection without loopholes. • Applications: • Bell inequalities for Measurement-based Quantum Computing. • Implications for the range of CHSH quantum correlations. Bell inequalities Bell inequalities • Bell inequalities (BIs) express bounds on the statistics of spatially separated measurements in local hidden variable (LHV) theories. random random setting setting > ct Bell inequalities random A choice of different measurements setting chosen “at random”. A number of different outcomes Bell inequalities • They repeat their experiment many times, and compute statistics. • In a local hidden variable (LHV) universe, their statistics are constrained by Bell inequalities. • In a quantum universe, the BIs can be violated. CHSH inequality In this talk, we will only consider the simplest type of Bell experiment (Clauser-Horne-Shimony-Holt). Each measurement has 2 settings and 2 outcomes. Boxes We will illustrate measurements as “boxes”. s 0, 1 j ∈{ } In the 2 setting, 2 outcome case we can use bit values 0/1 to label settings and outcomes. m 0, 1 j ∈{ } Local realism • Realism: Measurement outcome depends deterministically on setting and hidden variables λ. -
Arxiv:1206.1084V3 [Quant-Ph] 3 May 2019
Overview of Bohmian Mechanics Xavier Oriolsa and Jordi Mompartb∗ aDepartament d'Enginyeria Electr`onica, Universitat Aut`onomade Barcelona, 08193, Bellaterra, SPAIN bDepartament de F´ısica, Universitat Aut`onomade Barcelona, 08193 Bellaterra, SPAIN This chapter provides a fully comprehensive overview of the Bohmian formulation of quantum phenomena. It starts with a historical review of the difficulties found by Louis de Broglie, David Bohm and John Bell to convince the scientific community about the validity and utility of Bohmian mechanics. Then, a formal explanation of Bohmian mechanics for non-relativistic single-particle quantum systems is presented. The generalization to many-particle systems, where correlations play an important role, is also explained. After that, the measurement process in Bohmian mechanics is discussed. It is emphasized that Bohmian mechanics exactly reproduces the mean value and temporal and spatial correlations obtained from the standard, i.e., `orthodox', formulation. The ontological characteristics of the Bohmian theory provide a description of measurements in a natural way, without the need of introducing stochastic operators for the wavefunction collapse. Several solved problems are presented at the end of the chapter giving additional mathematical support to some particular issues. A detailed description of computational algorithms to obtain Bohmian trajectories from the numerical solution of the Schr¨odingeror the Hamilton{Jacobi equations are presented in an appendix. The motivation of this chapter is twofold. -
Introduction to Quantum Information and Computation
Introduction to Quantum Information and Computation Steven M. Girvin ⃝c 2019, 2020 [Compiled: May 2, 2020] Contents 1 Introduction 1 1.1 Two-Slit Experiment, Interference and Measurements . 2 1.2 Bits and Qubits . 3 1.3 Stern-Gerlach experiment: the first qubit . 8 2 Introduction to Hilbert Space 16 2.1 Linear Operators on Hilbert Space . 18 2.2 Dirac Notation for Operators . 24 2.3 Orthonormal bases for electron spin states . 25 2.4 Rotations in Hilbert Space . 29 2.5 Hilbert Space and Operators for Multiple Spins . 38 3 Two-Qubit Gates and Entanglement 43 3.1 Introduction . 43 3.2 The CNOT Gate . 44 3.3 Bell Inequalities . 51 3.4 Quantum Dense Coding . 55 3.5 No-Cloning Theorem Revisited . 59 3.6 Quantum Teleportation . 60 3.7 YET TO DO: . 61 4 Quantum Error Correction 63 4.1 An advanced topic for the experts . 68 5 Yet To Do 71 i Chapter 1 Introduction By 1895 there seemed to be nothing left to do in physics except fill out a few details. Maxwell had unified electriciy, magnetism and optics with his theory of electromagnetic waves. Thermodynamics was hugely successful well before there was any deep understanding about the properties of atoms (or even certainty about their existence!) could make accurate predictions about the efficiency of the steam engines powering the industrial revolution. By 1895, statistical mechanics was well on its way to providing a microscopic basis in terms of the random motions of atoms to explain the macroscopic predictions of thermodynamics. However over the next decade, a few careful observers (e.g. -
Bell's Theorem and Its Tests
PHYSICS ESSAYS 33, 2 (2020) Bell’s theorem and its tests: Proof that nature is superdeterministic—Not random Johan Hanssona) Division of Physics, Lulea˚ University of Technology, SE-971 87 Lulea˚, Sweden (Received 9 March 2020; accepted 7 May 2020; published online 22 May 2020) Abstract: By analyzing the same Bell experiment in different reference frames, we show that nature at its fundamental level is superdeterministic, not random, in contrast to what is indicated by orthodox quantum mechanics. Events—including the results of quantum mechanical measurements—in global space-time are fixed prior to measurement. VC 2020 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-33.2.216] Resume: En analysant l’experience de Bell dans d’autres cadres de reference, nous demontrons que la nature est super deterministe au niveau fondamental et non pas aleatoire, contrairement ace que predit la mecanique quantique. Des evenements, incluant les resultats des mesures mecaniques quantiques, dans un espace-temps global sont fixes avant la mesure. Key words: Quantum Nonlocality; Bell’s Theorem; Quantum Measurement. Bell’s theorem1 is not merely a statement about quantum Registered outcomes at either side, however, are classi- mechanics but about nature itself, and will survive even if cal objective events (for example, a sequence of zeros and quantum mechanics is superseded by an even more funda- ones representing spin up or down along some chosen mental theory in the future. Although Bell used aspects of direction), e.g., markings on a paper printout ¼ classical quantum theory in his original proof, the same results can be facts ¼ events ¼ points defining (constituting) global space- obtained without doing so.2,3 The many experimental tests of time itself. -
Arxiv:Quant-Ph/0504183V1 25 Apr 2005 † ∗ Elsae 1,1,1] Oee,I H Rcs Fmea- Is of Above the Process the Vandalized
Deterministic Bell State Discrimination Manu Gupta1∗ and Prasanta K. Panigrahi2† 1 Jaypee Institute of Information Technology, Noida, 201 307, India 2 Physical Research Laboratory, Navrangpura, Ahmedabad, 380 009, India We make use of local operations with two ancilla bits to deterministically distinguish all the four Bell states, without affecting the quantum channel containing these Bell states. Entangled states play a key role in the transmission and processing of quantum information [1, 2]. Using en- tangled channel, an unknown state can be teleported [3] with local unitary operations, appropriate measurement and classical communication; one can achieve entangle- ment swapping through joint measurement on two en- tangled pairs [4]. Entanglement leads to increase in the capacity of the quantum information channel, known as quantum dense coding [5]. The bipartite, maximally en- FIG. 1: Diagram depicting the circuit for Bell state discrimi- tangled Bell states provide the most transparent illustra- nator. tion of these aspects, although three particle entangled states like GHZ and W states are beginning to be em- ployed for various purposes [6, 7]. satisfactory, where the Bell state is not required further Making use of single qubit operations and the in the quantum network. Controlled-NOT gates, one can produce various entan- We present in this letter, a scheme which discriminates gled states in a quantum network [1]. It may be of inter- all the four Bell states deterministically and is able to pre- est to know the type of entangled state that is present in serve these states for further use. As LOCC alone is in- a quantum network, at various stages of quantum compu- sufficient for this purpose, we will make use of two ancilla tation and cryptographic operations, without disturbing bits, along with the entangled channels. -
Timelike Curves Can Increase Entanglement with LOCC Subhayan Roy Moulick & Prasanta K
www.nature.com/scientificreports OPEN Timelike curves can increase entanglement with LOCC Subhayan Roy Moulick & Prasanta K. Panigrahi We study the nature of entanglement in presence of Deutschian closed timelike curves (D-CTCs) and Received: 10 March 2016 open timelike curves (OTCs) and find that existence of such physical systems in nature would allow us to Accepted: 05 October 2016 increase entanglement using local operations and classical communication (LOCC). This is otherwise in Published: 29 November 2016 direct contradiction with the fundamental definition of entanglement. We study this problem from the perspective of Bell state discrimination, and show how D-CTCs and OTCs can unambiguously distinguish between four Bell states with LOCC, that is otherwise known to be impossible. Entanglement and Closed Timelike Curves (CTC) are perhaps the most exclusive features in quantum mechanics and general theory of relativity (GTR) respectively. Interestingly, both theories, advocate nonlocality through them. While the existence of CTCs1 is still debated upon, there is no reason for them, to not exist according to GTR2,3. CTCs come as a solution to Einstein’s field equations, which is a classical theory itself. Seminal works due to Deutsch4, Lloyd et al.5, and Allen6 have successfully ported these solutions into the framework of quantum mechanics. The formulation due to Lloyd et al., through post-selected teleportation (P-CTCs) have been also experimentally verified7. The existence of CTCs has been disturbing to some physicists, due to the paradoxes, like the grandfather par- adox or the unproven theorem paradox, that arise due to them. Deutsch resolved such paradoxes by presenting a method for finding self-consistent solutions of CTC interactions. -
Studies in the Geometry of Quantum Measurements
Irina Dumitru Studies in the Geometry of Quantum Measurements Studies in the Geometry of Quantum Measurements Studies in Irina Dumitru Irina Dumitru is a PhD student at the department of Physics at Stockholm University. She has carried out research in the field of quantum information, focusing on the geometry of Hilbert spaces. ISBN 978-91-7911-218-9 Department of Physics Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020 Studies in the Geometry of Quantum Measurements Irina Dumitru Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 13.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, and digitally via video conference (Zoom). Public link will be made available at www.fysik.su.se in connection with the nailing of the thesis. Abstract Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally- Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs. This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements. -
A Non-Technical Explanation of the Bell Inequality Eliminated by EPR Analysis Eliminated by Bell A
Ghostly action at a distance: a non-technical explanation of the Bell inequality Mark G. Alford Physics Department, Washington University, Saint Louis, MO 63130, USA (Dated: 16 Jan 2016) We give a simple non-mathematical explanation of Bell's inequality. Using the inequality, we show how the results of Einstein-Podolsky-Rosen (EPR) experiments violate the principle of strong locality, also known as local causality. This indicates, given some reasonable-sounding assumptions, that some sort of faster-than-light influence is present in nature. We discuss the implications, emphasizing the relationship between EPR and the Principle of Relativity, the distinction between causal influences and signals, and the tension between EPR and determinism. I. INTRODUCTION II. OVERVIEW To make it clear how EPR experiments falsify the prin- The recent announcement of a \loophole-free" observa- ciple of strong locality, we now give an overview of the tion of violation of the Bell inequality [1] has brought re- logical context (Fig. 1). For pedagogical purposes it is newed attention to the Einstein-Podolsky-Rosen (EPR) natural to present the analysis of the experimental re- family of experiments in which such violation is observed. sults in two stages, which we will call the \EPR analy- The violation of the Bell inequality is often described as sis", and the \Bell analysis" although historically they falsifying the combination of \locality" and \realism". were not presented in exactly this form [4, 5]; indeed, However, we will follow the approach of other authors both stages are combined in Bell's 1976 paper [2]. including Bell [2{4] who emphasize that the EPR results We will concentrate on Bohm's variant of EPR, the violate a single principle, strong locality. -
Bell's Inequalities and Their Uses
The Quantum Theory of Information and Computation http://www.comlab.ox.ac.uk/activities/quantum/course/ Bell’s inequalities and their uses Mark Williamson [email protected] 10.06.10 Aims of lecture • Local hidden variable theories can be experimentally falsified. • Quantum mechanics permits states that cannot be described by local hidden variable theories – Nature is weird. • We can utilize this weirdness to guarantee perfectly secure communication. Overview • Hidden variables – a short history • Bell’s inequalities as a bound on `reasonable’ physical theories • CHSH inequality • Application – quantum cryptography • GHZ paradox Hidden variables – a short history • Story starts with a famous paper by Einstein, Podolsky and Rosen in 1935. • They claim quantum mechanics is incomplete as it predicts states that have bizarre properties contrary to any `reasonable’ complete physical theory. • Einstein in particular believed that quantum mechanics was an approximation to a local, deterministic theory. • Analogy: Classical statistical mechanics approximation of deterministic, local classical physics of large numbers of systems. EPRs argument used the peculiar properties of states permitted in quantum mechanics known as entangled states. Schroedinger says of entangled states: E. Schroedinger, Discussion of probability relations between separated systems. P. Camb. Philos. Soc., 31 555 (1935). Entangled states • Observation: QM has states where the spin directions of each particle are always perfectly anti-correlated. Einstein, Podolsky and Rosen (1935) EPR use the properties of an entangled state of two particles a and b to engineer a paradox between local, realistic theories and quantum mechanics Einstein, Podolsky and Rosen Roughly speaking : If a and b are two space-like separated particles (no causal connection between the particles), measurements on particle a should not affect particle b in a reasonable, complete physical theory. -
Axioms 2014, 3, 153-165; Doi:10.3390/Axioms3020153 OPEN ACCESS Axioms ISSN 2075-1680
Axioms 2014, 3, 153-165; doi:10.3390/axioms3020153 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms/ Article Bell Length as Mutual Information in Quantum Interference Ignazio Licata 1,* and Davide Fiscaletti 2,* 1 Institute for Scientific Methodology (ISEM), Palermo, Italy 2 SpaceLife Institute, San Lorenzo in Campo (PU), Italy * Author to whom correspondence should be addressed; E-Mails: [email protected] (I.L.); [email protected] (D.F.). Received: 22 January 2014; in revised form: 28 March 2014 / Accepted: 2 April 2014 / Published: 10 April 2014 Abstract: The necessity of a rigorously operative formulation of quantum mechanics, functional to the exigencies of quantum computing, has raised the interest again in the nature of probability and the inference in quantum mechanics. In this work, we show a relation among the probabilities of a quantum system in terms of information of non-local correlation by means of a new quantity, the Bell length. Keywords: quantum potential; quantum information; quantum geometry; Bell-CHSH inequality PACS: 03.67.-a; 04.60.Pp; 03.67.Ac; 03.65.Ta 1. Introduction For about half a century Bell’s work has clearly introduced the problem of a new type of “non-local realism” as a characteristic trait of quantum mechanics [1,2]. This exigency progressively affected the Copenhagen interpretation, where non–locality appears as an “unexpected host”. Alternative readings, such as Bohm’s one, thus developed at the origin of Bell’s works. In recent years, approaches such as the transactional one [3–5] or the Bayesian one [6,7] founded on a “de-construction” of the wave function, by focusing on quantum probabilities, as they manifest themselves in laboratory. -
Quantifying Entanglement in a 68-Billion-Dimensional Quantum State Space
ARTICLE https://doi.org/10.1038/s41467-019-10810-z OPEN Quantifying entanglement in a 68-billion- dimensional quantum state space James Schneeloch 1,5, Christopher C. Tison1,2,3, Michael L. Fanto1,4, Paul M. Alsing1 & Gregory A. Howland 1,4,5 Entanglement is the powerful and enigmatic resource central to quantum information pro- cessing, which promises capabilities in computing, simulation, secure communication, and 1234567890():,; metrology beyond what is possible for classical devices. Exactly quantifying the entanglement of an unknown system requires completely determining its quantum state, a task which demands an intractable number of measurements even for modestly-sized systems. Here we demonstrate a method for rigorously quantifying high-dimensional entanglement from extremely limited data. We improve an entropic, quantitative entanglement witness to operate directly on compressed experimental data acquired via an adaptive, multilevel sampling procedure. Only 6,456 measurements are needed to certify an entanglement-of- formation of 7.11 ± .04 ebits shared by two spatially-entangled photons. With a Hilbert space exceeding 68 billion dimensions, we need 20-million-times fewer measurements than the uncompressed approach and 1018-times fewer measurements than tomography. Our tech- nique offers a universal method for quantifying entanglement in any large quantum system shared by two parties. 1 Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USA. 2 Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA. 3 Quanterion Solutions Incorporated, Utica, NY 13502, USA. 4 Rochester Institute of Technology, Rochester, NY 14623, USA. 5These authors contributed equally: James Schneeloch, Gregory A. Howland. Correspondence and requests for materials should be addressed to G.A.H.