Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for the outcomes of two binary measurements, where the lower bound on the uncertainty is quantified in terms of the maximum Clauser–Horne–Shimony–Holt value that can be achieved with these measurements. We discuss applications of this uncertainty relation in quantum cryptography, in particular, to certify quantum sources using untrusted devices.
On a related topic, Oppenheim and Wehner @cite_11 ,| ,for a class of generalized physical theories that includes quantum mechanics and classical theory ,| ,showed that the presence of uncertainty, via steering, directly limits the maximally achievable nonlocality. Our result can be seen as complementary to theirs, as we show that in order to achieve a certain nonlocality, at least some specific amount of uncertainty is necessary.
@cite_11
2015389932
Two central concepts of quantum mechanics are Heisenberg’s uncertainty principle and a subtle form of nonlocality that Einstein famously called “spooky action at a distance.” These two fundamental features have thus far been distinct concepts. We show that they are inextricably and quantitatively linked: Quantum mechanics cannot be more nonlocal with measurements that respect the uncertainty principle. In fact, the link between uncertainty and nonlocality holds for all physical theories. More specifically, the degree of nonlocality of any theory is determined by two factors: the strength of the uncertainty principle and the strength of a property called “steering,” which determines which states can be prepared at one location given a measurement at another. Nonlocality refers to correlations between spatially separated parties that are stronger than those explained by the existence of local hidden variables. Quantum mechanics is known to allow some nonlocal correlations between particles in a phenomena known as entanglement. We explore several aspects of nonlocality in general and how they relate to quantum mechanics. First, we construct a hierarchy of theories with nonlocal correlations stronger than those allowed in quantum mechanics and derive several results about these theories. We show that these theories include codes that can store an amount of information exponential in the number of physical bits used. We use this result to demonstrate an unphysical consequence of theories with stronger-than-quantum correlations: learning even an approximate description of states in such theories would be practically impossible. Next, we consider the difficult problem of determining whether specific correlations are nonlocal. We present a novel learning algorithm and show that it provides an outer bound on the set of local states, and can therefore be used to identify some nonlocal states. Finally, we put nonlocal correlations to work by showing that the entanglement present in the vacuum of a quantum field can be used to detect spacetime curvature. We quantify how the entangling power of the quantum field varies as a function of spacetime curvature. I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory tomore » be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.« less We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, it follows that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the feasibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture. This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies. Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions. In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65. PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics Quantum communication schemes rely on cryptographically secure quantum resources to distribute private information. Here, the authors show that graph states—nonlocal states based on networks of qubits—can be exploited to implement quantum secret sharing of quantum and classical information. In this paper, we provide a simple framework for deriving one-shot achievable bounds for some problems in quantum information theory. Our framework is based on the joint convexity of the exponential of the collision relative entropy and is a (partial) quantum generalization of the technique of from classical information theory. Based on this framework, we derive one-shot achievable bounds for the problems of communication over classical-quantum channels, quantum hypothesis testing, and classical data compression with quantum side information. We argue that our one-shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order. The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies. Abstract Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well. We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input. This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows: • the microstresses to depend on ∇E˙p, the gradient of the plastic strain-rate, and • the free energy ψ to depend on the Burgers tensor G=curlEp. The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on ∇E˙p lead to strengthening and weakening effects in the flow rule. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived. I Entropies for Finite Quantum Systems.- 1 Fundamental Concepts.- 2 Postulates for Entropy and Relative Entropy.- 3 Convex Trace Functions.- II Entropies for General Quantum Systems.- 4 Modular Theory and Auxiliaries.- 5 Relative Entropy of States of Operator Algebras.- 6 From Relative Entropy to Entropy.- 7 Functionals of Entropy Type.- III Channeling Transformation and Coarse Graining.- 8 Channels and Their Transpose.- 9 Sufficient Channels and Measurements.- 10 Dynamical Entropy.- 11 Stationary Processes.- IV Perturbation Theory.- 12 Perturbation of States.- 13 Variational Expression of Perturbational Limits.- V Miscellanea.- 14 Central Limit and Quasi-free Reduction.- 15 Thermodynamics of Quantum Spin Systems.- 16 Entropic Uncertainty Relations.- 17 Temperley-Lieb Algebras and Index.- 18 Optical Communication Processes. Work on quantum cryptography was started by S. J. Wiesner in a paper written in about 1970, but remained unpublished until 1983 [1]. Recently, there have been lots of renewed activities in the subject. The most wellknown application of quantum cryptography is the socalled quantum key distribution (QKD) [2–4], which is useful for making communications between two users totally unintelligible to an eavesdropper. QKD takes advantage of the uncertainty principle of quantum mechanics: Measuring a quantum system in general disturbs it. Therefore, eavesdropping on a quantum communication channel will generally leave unavoidable disturbance in the transmitted signal which can be detected by the legitimate users. Besides QKD, other quantum cryptographic protocols [5] have also been proposed. In particular, it is generally believed [4] that quantum mechanics can protect private information while it is being used for public decision. Suppose Alice has a secret x and Bob a secret y. In a “two-party secure computation” (TPSC), Alice and Bob compute a prescribed function f(x,y) in such a way that nothing about each party’s input is disclosed to the other, except for what follows logically from one’s private input and the function’s output. An example of the TPSC is the millionaires’ problem: Two persons would like to know who is richer, but neither wishes the other to know the exact amount of money he she has. In classical cryptography, TPSC can be achieved either through trusted intermediaries or by invoking some unproven computational assumptions such as the hardness of factoring large integers. The great expectation is that quantum cryptography can get rid of those requirements and achieve the same goal using the laws of physics alone. At the heart of such optimism has been the widespread belief that unconditionally secure quantum bit commitment (QBC) schemes exist [6]. Here we put such optimism into very serious doubt by showing We propose a simple yet rich model to extend strategic games to the quantum setting, in which we define quantum Nash and correlated equilibria and study the relations between classical and quantum equilibria. Unlike all previous work that focused on qualitative questions on specific games of very small sizes, we quantitatively address the following fundamental question for general games of growing sizes: How much "advantage" can playing quantum strategies provide, if any? Two measures of the advantage are studied. 1. Since game mainly is about each player trying to maximize individual payoff, a natural measure is the increase of payoff by playing quantum strategies. We consider natural mappings between classical and quantum states, and study how well those mappings preserve equilibrium properties. Among other results, we exhibit a correlated equilibrium p whose quantum superposition counterpart [EQUATION] is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0, 1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases. The result can also be interpreted as in Meyer's comparison [47]: In a state no classical player can gain, one player using quantum computers has an huge advantage than continuing to play classically. 2. Another measure is the hardness of generating correlated equilibria, for which we propose to study correlation complexity, a new complexity measure for correlation generation. We show that there are n-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least log2(n) classical shared random bits plus communication. The randomized lower bound can be improved to n, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations. Simultaneous sparse coding (SSC) or nonlocal image representation has shown great potential in various low-level vision tasks, leading to several state-of-the-art image restoration techniques, including BM3D and LSSC. However, it still lacks a physically plausible explanation about why SSC is a better model than conventional sparse coding for the class of natural images. Meanwhile, the problem of sparsity optimization, especially when tangled with dictionary learning, is computationally difficult to solve. In this paper, we take a low-rank approach toward SSC and provide a conceptually simple interpretation from a bilateral variance estimation perspective, namely that singular-value decomposition of similar packed patches can be viewed as pooling both local and nonlocal information for estimating signal variances. Such perspective inspires us to develop a new class of image restoration algorithms called spatially adaptive iterative singular-value thresholding (SAIST). For noise data, SAIST generalizes the celebrated BayesShrink from local to nonlocal models; for incomplete data, SAIST extends previous deterministic annealing-based solution to sparsity optimization through incorporating the idea of dictionary learning. In addition to conceptual simplicity and computational efficiency, SAIST has achieved highly competent (often better) objective performance compared to several state-of-the-art methods in image denoising and completion experiments. Our subjective quality results compare favorably with those obtained by existing techniques, especially at high noise levels and with a large amount of missing data. In the literature on multiperiod planning under uncertainty, it is generally postulated that preferences may be represented by a von Neumann-Morgenstern utility index that is additive over time. This paper accomplishes two objectives: First, an axiomatic basis is provided for a more general class of non-additive utility indices defined over infinite consumption streams. Second, this class of utility functions is applied to extend existing results (J. Econ. Theory 4 (1972), 479–513; J. Econ. Theory 11 (1975), 329–339) on the nature of optimal growth under uncertainty. Of particular interest are the existence and stability of a stochastic steady state. Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates.