Stefan Filipp, Karl Svozil
Bounds on the norm of quantum operators associated with classical Bell-type inequalities can be derived from their maximal eigenvalues. This quantitative method enables detailed predictions of the maximal violations of Bell-type inequalities.
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The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the "hard" part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.
We consider three parties, A, B, and C, each performing one of two local measurements on a shared quantum state of arbitrary dimension. We characterize the trade-off between the nonlocality of the Bell correlations observed by AB and of those observed by AC. This generalizes Tsirelson's bound on the quantum value of the CHSH inequality, the latter being recovered when C is completely uncorrelated with AB. We also discuss the trade-off between Bell violations and local expectation values of observables that anticommute with the ones used in the Bell test.
We present a simple analytic bound on the quantum value of general correlation type Bell inequalities, similar to Tsirelson's bound. It is based on the maximal singular value of the coefficient matrix associated with the inequality. We provide a criterion for tightness of the bound and show that the class of inequalities where our bound is tight covers many famous examples from the literature. We describe how this bound helps to construct Bell inequalities, in particular inequalities that witness the dimension of the measured observables.
Here we show that, if we insert context dependent unitary evolutions (which can be achieved via post selection) into spatial (i.e., normal) Bell-CHSH test, then it is possible to violate space-time Bell-CHSH inequality maximally (i.e., up to $4$). However this does not contradict Tsirelson quantum bound ($2\sqrt{2}$), as the latter has been derived without taking into consideration context dependent unitary evolutions and/or post selection. As an important application, this leads to a more efficient quantum key distribution protocol.
We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations, which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples: We prove that for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimise some Bell inequalities with respect to the max…
Imitating quantum probabilities: Beyond Bell's theorem and Tsirelson bounds
该论文将单态关联的局域隐变量模型推广为无穷层级演算模型，揭示贝尔不等式违反源于相邻层级混淆，非相邻混合可超越Tsirelson界。
Designing generalized elegant Bell inequalities in higher dimensions from a Tsirelson bound
本文提出从Tsirelson界出发，设计更高维度的广义优雅贝尔不等式，首次在三维中推导出具有更大违反能力的贝尔不等式。
Extremal Tsirelson inequalities
本文从对偶视角研究量子统计量集合的边界，将 CHSH 表达式分解为极值 Tsirelson 不等式，揭示了 (2,2,2) 场景的几何结构，并识别出所有能自检验 Tsirelson 实现的 Bell 表达式。
Refined Tsirelson Bounds on Multipartite Bell Inequalities
针对多方量子关联刻画难题，本文利用类似量子不确定性的基本约束，推导依赖子系统关联的 Svetlichny 与 MK 不等式精细化 Tsirelson 界，并通过实例证明其更紧。
The Bell experiment is a random game with two binary outcomes whose statistical correlation is given by $E_0(Θ)=-\cos(Θ)$, where $Θ\in [-π, π)$ is an angular input that parameterizes the game setting. The correlation function $E_0(Θ)$ belongs to the affine space ${\cal H} \equiv \left\{E(Θ)\right\}$ of all continuous and differentiable periodic functions $E(Θ)$ that obey the parity symmetry constraints $E(-Θ)=E(Θ)$ and $E(π-Θ)=-E(Θ)$ with $E(0)=-1$ and, furthermore, are strictly monotonically increasing in the interval $[0, π)$. Here we show how to build explicitly local statistical models of …
We give strong evidence that divisibility of qubit quantum processes implies temporal Tsirelson's bound. We also give strong evidence that the classical bound of the temporal Bell's inequality holds for dynamics that can be described by entanglement-breaking channels---a more general class of dynamics than that allowed by classical physics.
We classify the extreme points of a polytope of probability distributions in the (2,2,2) CHSH-Bell setting that is induced by a single Tsirelson bound. We do the same for a class of polytopes obtained from a parametrized family of multiple Tsirelson bounds interacting non-trivially. Such constructions can be applied to device-independent random number generation using the method of probability estimation factors [Phys. Rev. A, 98:040304(R), 2018]. We demonstrate a meaningful improvement in certified randomness applying the new polytopes characterized here.
Tsirelson bounds for quantum correlations with indefinite causal order
本文针对确定因果不等式最大量子违反这一开放问题，提出约束任意因果不等式违反的通用方法，建立关联限制并为一类因果不等式确定类Tsirelson界，推动相关物理原理探索。
Maximal violation of the CHSH-Bell inequality is usually said to be a feature of anticommuting observables. In this work we show that even random observables exhibit near-maximal violations of the CHSH-Bell inequality. To do this, we use the tools of free probability theory to analyze the commutators of large random matrices. Along the way, we introduce the notion of "free observables" which can be thought of as infinite-dimensional operators that reproduce the statistics of random matrices as their dimension tends towards infinity. We also study the fine-grained uncertainty of a sequence of f…
Tsirelson showed that $2\sqrt{2}$ is the maximum value that CHSH expression can take for quantum-correlations [B. S.Tsirelson, Lett. Math. Phys, 4 (1980) 93]. This bound simply follows from the algebra of observables. Recently by exploiting the physical structure of quantum mechanics like unitarity and linearity, Buhrman and Massar [H. Buhrman and S.Massar, Phys. Rev. A, 72 (2005) 052103] have established that violation of Tsirelson's bound in quantum mechanics will imply signalling. We prove the same with the help of realistic joint measurement in quantum mechanics and a Bell's inequality whi…
Quantum theory introduces a cut between the observer and the observed system, but does not provide a definition of what is an observer. Based on an informational definition of observer, Grinbaum has recently predicted an upper bound on bipartite correlations in the Clauser-Horne-Shimony-Holt (CHSH) Bell scenario equal to 2.82537, which is slightly smaller than the Tsirelson bound of standard quantum theory, but is consistent with all the available experimental results. Not being able to exceed Grinbaum's limit would support that quantum theory is only an effective description of a more fundame…
Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a new class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice…
We show that, for general probabilistic theories admitting sharp measurements, the exclusivity principle together with two assumptions exactly singles out the Tsirelson bound of the Clauser-Horne-Shimony-Holt Bell inequality.
Modular wedge localization, Majorana fields and the Tsirelson limit of the Bell-CHSH inequality
本文利用1+1维质量Majorana场研究相对论量子场论中Bell-CHSH不等式的违反，通过模楔定位构造明确快速性空间实现，并将真空Bell-CHSH关联子归约为单个谱权重h²(ω)，其谱权重集中于ω≈0时趋近Tsirelson界。
The Bell-Clauser-Horne-Shimony-Holt inequality can be used to show that no local hidden-variable theory can reproduce the correlations predicted by quantum mechanics (QM). It can be proved that certain QM correlations lead to a violation of the classical bound established by the inequality, while all correlations, QM and classical, respect a QM bound (the Tsirelson bound). Here, we show that these well-known results depend crucially on the assumption that the values of physical magnitudes are scalars. The result implies, first, that the origin of the Tsirelson bound is geometrical, not physica…
Systematic derivation of Tsirelson bounds in arbitrary dimensions
提出基于平方和分解的系统方法，推导任意维度两体量子关联的Tsirelson界限，发现新界限并恢复已知结果。
High-fidelity polarization-entangled photons are a powerful resource for quantum communication, distributing entanglement and quantum teleportation. The Bell-CHSH inequality $S\leq2$ is violated by bipartite entanglement and only maximally entangled states can achieve $S=2\sqrt{2}$, the Tsirelson bound. Spontaneous parametric down-conversion sources can produce entangled photons with correlations close to the Tsirelson bound. Sagnac configurations offer intrinsic stability, compact footprint and high collection efficiency, however, there is often a trade off between source brightness and entan…
Two physical principles, Macroscopic Locality (ML) and Information Causality (IC), so far, have been most successful in distinguishing quantum correlations from post-quantum correlations. However, there are also some post-quantum probability distributions which cannot be distinguished with the help of these principles. Thus, it is interesting to see whether consideration of these two principles, separately, along with some additional physically plausible constraints, can explain some interesting quantum features which are otherwise hard to reproduce. In this paper we show that, in a Bell-CHSH …
Quantum correlations resulting in violations of Bell inequalities have generated a lot of interest in quantum information science and fundamental physics. In this paper, we address some questions that become relevant in Bell-type tests involving systems with local dimension greater than 2. For CHSH-Bell tests within 2-dimensional subspaces of such high-dimensional systems, it has been suggested that experimental violation of Tsirelson's bound indicates that more than 2-dimensional entanglement was present. We explain that the overstepping of Tsirelson's bound is due to violation of fair sampli…
Spin operator, Bell nonlocality and Tsirelson bound in quantum-gravity induced minimal-length quantum mechanics
本文研究量子引力预言的最小长度如何影响自旋算符，发现其获得动量依赖修正，可使贝尔不等式违背超过普通量子力学中的 Tsirelson 界，并给出基于中子干涉和量子语境性的实验方案及参数估计。
To answer Wheeler's question "Why the quantum?" via quantum information theory according to Bub, one must explain both why the world is quantum rather than classical and why the world is quantum rather than superquantum, i.e., "Why the Tsirelson bound?" We show that the quantum correlations and quantum states corresponding to the Bell basis states, which uniquely produce the Tsirelson bound for the Clauser-Horne-Shimony-Holt quantity, can be derived from conservation per no preferred reference frame (NPRF). A reference frame in this context is defined by a measurement configuration, just as wi…
We consider the problem of distinguishing between a set of arbitrary quantum states in a setting in which the time available to perform the measurement is limited. We provide simple upper bounds on how well we can perform state discrimination in a given time as a function of either the average energy or the range of energies available during the measurement. We exhibit a specific strategy that nearly attains this bound. Finally, we consider several applications of our result. First, we obtain a time-dependent Tsirelson's bound that limits the extent of the Bell inequality violation that can be…
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden varia…
Many typical Bell experiments can be described as follows. A source repeatedly distributes particles among two spacelike separated observers. Each of them makes a measurement, using an observable randomly chosen out of several possible ones, leading to one of two possible outcomes. After collecting a sufficient amount of data one calculates the value of a so-called Bell expression. An important question in this context is whether the result is compatible with bounds based on the assumptions of locality, realism and freedom of choice. Here we are interested in bounds on the obtained value deriv…
Bell inequalities are mathematical constructs that demarcate the boundary between quantum and classical physics. A new class of multiplicative Bell inequalities originating from a volume maximization game (based on products of correlators within bipartite systems) has been recently proposed. For these new Bell parameters, it is relatively easy to find the classical and quantum, i.e. Tsirelson, limits. Here, we experimentally test the Tsirelson bounds of these inequalities using polarisation-entangled photons for different number of measurements ($n$), each party can perform. For $n=2, 3, 4$, w…
Graph Structured Operator Inequalities and Tsirelson-Type Bounds
本文通过图结构方法建立双分张量和的算子范数界，推广了Tsirelson和CHSH界限的分析结构，提供与交换子和反交换子范数相关的无维度估计。
Further Evidence for Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Haar Wavelets
本文通过Haar小波研究量子场论中真空态中贝尔-CHSH不等式违反，证明其接近Tsirelson边界，并通过数学猜想和数值证据支持最大特征值趋于π的结论。
Superquantum ("PR-box") correlations, though designed to respect relativistic causality, violate relativistic causality in the classical limit. Generalizing to all stronger-than-quantum bipartite correlations, I derive Tsirelson's bound from the axioms of nonlocality, relativistic causality and the existence of a classical limit. This derivation of Tsirelson's bound does not assume quantum mechanics yet suggests how Hilbert space is implicit in quantum correlations.
The CHSH Game, Tsirelson's Bound, and Causal Locality
本文通过不可分割随机过程的公理，重新证明了Tsirelson界限，并展示了其对贝尔不等式违反的限制。
On the representations of Bell's operators in Quantum Mechanics
该研究发现维数大于2的希尔伯特空间中Bell-CHSH不等式的Bell算符存在幺正不等价表示，不同表示违反程度不同，最大违反由Tsirelson界给出，源于空间模式配对机制。
For many protocols, quantum strategies have advantages compared with their classical counter-partners, and these advantages have attracted many interests and applications. One of the famous examples is the Clauser-Horne-Shimony-Holt (CHSH) game, which recasts Bell's theorem~\cite{2} into the framework of a game. In the CHSH game, two space-like separated players, Alice and Bob are each assigned a classical bit $a$ and $b$ respectively. Then they return bits $x$ and $y$ according to some pre-agreed strategies. They will win the game when $x\oplus y= a\cdot b$. In the game, if the players use th…
Robust Quantum Key Distribution Arbitrarily Close to Local Correlations
该研究针对设备无关量子密钥分发，证明贝尔违背接近Tsirelson界时可产生恒定密钥率，还通过半定松弛数值结果显示实用手段可保证非零密钥率。
The well-known Simpson's Paradox, or Yule-Simpson Effect, in statistics is often illustrated by the following thought experiment: A drug may be found in a trial to increase the survival rate for both men and women, but decrease the rate for all the subjects as a whole. This paradoxical reversal effect has been found in numerous datasets across many disciplines, and is now included in most introductory statistics textbooks. In the language of the drug trial, the effect is impossible, however, if both treatment groups' survival rates are higher than both control groups'. Here we show that for qu…
In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regul…
We describe a new technique for obtaining Tsirelson bounds, or upper bounds on the quantum value of a Bell inequality. Since quantum correlations do not allow signaling, we obtain a Tsirelson bound by maximizing over all no-signaling probability distributions. This maximization can be cast as a linear program. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violati…
The famous Clauser-Horne-Shimony-Holt (CHSH) inequality certifies a quantum violation, by a factor $\sqrt{2}$, of correlations predicted by the classical view of the world in the simplest possible nontrivial measurement setup (two systems with two dichotomic measurements each). In such setting, this is the largest possible violation, which is known as the \emph{Tsirelson bound}. In this paper we calculate the exact values of quantum violations for the other Bell correlation inequalities that appear in the setups involving up to four measurements; they are all smaller than $\sqrt{2}$. While var…
Entangled coherent states and violations of Bell-CHSH inequalities
本文利用三类纠缠相干态和无限维赝自旋算符构造Bell-CHSH不等式中的二分算符，通过计算关联函数分析参数区域，揭示了可导致不等式违背并饱和Tsirelson界的条件。
Activation of post-quantum steering
本文展示了如何通过更大网络激活后量子引导，使其在贝尔场景中表现为后量子相关性，同时展示了在假设后量子资源下如何自我测试双方面量子集合。
We give a analytic quantitative relation between Hardy's non-locality and Bell operator. We find that Hardy's non-locality is a sufficient condition for violation of Bell inequality, the upper bound of Hardy's non-locality allowed by information causality just correspond to Tsirelson bound of Bell inequality, and the upper bound of Hardy's non-locality allowed by the principle of no-signaling just correspond to the algebraic maximum of Bell operator. Then we study the Cabello's argument of Hardy's non-locality (a generalization of Hardy's argument) and find a similar relation between it and vi…
Wigner's argument inferring Bell-type inequality for the EPR-Bohm entangled state is generalized here for any N-partite state. This is based on assuming for the relevant dichotomic observables the existence of the overall joint probability distributions, satisfying the locality condition, that would yield the measurable marginal probabilities. For any N, such Generalized Wigner's Inequality (GWI) is violated by quantum mechanics for all pure entangled states. The efficacy of GWI is probed, comparing with the Seevinck-Svetlichny multipartite Bell-type inequality, by calculating threshold visibi…
Understanding the limits of quantum theory in terms of uncertainty and correlation has always been a topic of foundational interest. Surprisingly this pursuit can also bear interesting applications such as device-independent quantum cryptography and tomography or self-testing. Building upon a series of recent works on the geometry of quantum correlations, we are interested in the problem of computing quantum Bell inequalities or the boundary between quantum and post-quantum world. Better knowledge of this boundary will lead to more efficient device-independent quantum processing protocols. We …
(Almost-)Quantum Bell Inequalities and Device-Independent Applications
本文提出多族（几乎）量子贝尔不等式，用于刻画量子关联边界，并应用于弱源随机性提取、自测试及非局域计算无优势原理，给出迄今最精确的量子边界刻画。
Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particl…
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with eac…
Finite Weyl polynomials and the approach to Tsirelson's bound in relativistic scalar quantum field theory
该研究在1+1维自由有质量标量场中构造有限外尔多项式形式的Bell-CHSH不等式算符，证明其族上确界为2√2，且有限成员可达2.80027，突破了高斯表示下结果低于2的限制。
Bosonization, vertex operators and maximal violation of the Bell-CHSH inequality in wedge regions
研究通过一维chiral玻色子的顶点算符，揭示了在真空态下满足贝尔-CHSH不等式上界的方法和贡献。
The correlations in quantum networks have attracted strong interest with new types of violations of the locality. The standard Bell inequalities cannot characterize the multipartite correlations that are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for general quantum networks. In this work, we show a significant improvement by presenting new, explicit Bell-type inequalities for general networks including cyclic networks. These nonlinear inequalities are related to the matching problem of an e…
Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound
本文研究自由马约拉纳场真空中的Bell-CHSH不等式，通过模种子构造双线性可观测量，精确计算Bell参数，证明Tsirelson界不可达，并阐明其与Summers-Werner机制的联系。
Full-correlation Bell-like inequalities represent an important subclass of Bell-like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science. Loosely speaking, these are inequalities where only measurement statistics involving all parties play a role. In this paper, we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties. As an illustration of the power of our framework, we derive (i) a new family of Svetlichny inequalities fo…
Bell nonlocality with a single shot
本文提出将贝尔不等式转化为非局域博弈并最大化局域界与Tsirelson界间隙的方法，证明间隙接近1时可在单次实验中以任意小p值拒绝局域隐变量，并给出高效计算局域界的算法。
Information causality was proposed as a physical principle to put upper bound on the accessible information gain in a physical bi-partite communication scheme. Intuitively, the information gain cannot be larger than the amount of classical communication to avoid violation of causality. Moreover, it was shown that this bound is consistent with the Tsirelson bound for the binary quantum systems. In this paper, we test the information causality for the more general (non-binary) quantum communication schemes. In order to apply the semi-definite programming method to find the maximal information ga…
Information Causality Characterizes the Set of Quantum Correlations in the Simplest Bell Scenario
本文通过结合一般化信息因果性与新通信协议，推导出精确刻画最简单贝尔场景量子关联的贝尔不等式，并证明一般化IC严格强于宏观局域性，强化了其在解释量子非局域性极限中的作用。
The perturbative method for quantum correlations
本文提出利用李群工具分析量子策略下贝尔函数的微扰响应，揭示了经典确定性点附近贝尔算子的分解特性，并探讨了Ansatz维度在分布式学习中的关键作用。
On Removing the Classical-Quantum Boundary
本文指出 Bell 不等式违反源于反事实确定性而非定域性或实在论，该假设在经典力学中不成立，因此经典理论也可违反 Bell 不等式，并可能在 Hamilton 平稳原理下突破 Tsirelson 界。
We present a characterization of the set of non-signaling correlations in terms of a two dimensional representation that involves the maximal value of a Bell functional and the mutual information between the parties. In particular, we apply this representation to the bipartite Bell scenario with two measurements and two outcomes. In terms of these physically meaningful quantities and through numerical optimization methods and some analytical results, we investigate the frontier between the different subsets of the non-signaling correlations, focussing on the quantum and post-quantum ones. Fina…
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