Karl Hess, Walter Philipp
We show that Mermin's reasoning against our refutation of his non-technical proof for Bell-type inequalities is of limited significance or contains mathematical inconsistencies that, when taken into account, do not permit his proof to go forward. Our refutation therefore stands.
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We investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems.
Mermin states that his nontechnical version of Bell's theorem stands and is not invalidated by time and setting dependent instrument parameters as claimed in one of our previous papers. We identify deviations from well-established protocol in probability theory as well as mathematical contradictions in Mermin's argument and show that Mermin's conclusions are therefore not valid: his proof does not go forward if certain possible time dependencies are taken into account.
Mermin states in a recent paper that his nontechnical version of Bell's theorem stands and is not invalidated by time and setting dependent instrument parameters as claimed in one of our previous papers. We identify a number of misinterpretations (of our definitions) and mathematical inconsistencies in Mermin's paper and show that Mermin's conclusions are therefore not valid: his proof does not go forward if certain possible time dependencies are taken into account.
Bell's and Mermin's inequalities, entangled coherent states and unitary operators
该研究利用幺正位移算符分析纠缠相干态的Bell与Mermin不等式，发现Mermin不等式接近最大违背，结果与既有文献一致。
The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown, in particular, that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentially by a factor of 2^{-m/2} whenever any m among the n single-particle commutators happen to vanish.
We consider a two-particle/two-setting Bell experiment to visualize the conflict between Bell-Żukowski inequality and Bell-Mermin inequality. The experiment is reproducible by local realistic theories which are not rotationally invariant. We found that the average value of the Bell-Żukowski operator can be evaluated only by the two-particle/two-setting Bell experiment in question. The Bell-Żukowski inequality reveals that the constructed local realistic models for the experiment are not rotationally invariant. That is, the two-particle Bell experiment in question reveals the conflict between B…
The detection of nonlocal correlations in a Bell experiment implies almost by definition some intrinsic randomness in the measurement outcomes. For given correlations, or for a given Bell violation, the amount of randomness predicted by quantum physics, quantified by the guessing probability, can generally be bounded numerically. However, currently only a few exact analytic solutions are known for violations of the bipartite Clauser-Horne-Shimony-Holt Bell inequality. Here, we study the randomness in a Bell experiment where three parties test the tripartite Mermin-Bell inequality. We give tigh…
Bell-Mermin-Klyshko Inequalities and One-way Information Deficit of Dirac Fields in Noninertial Frames
本研究针对非惯性系中狄拉克场，结合乌伦贝克效应下惯性与加速观察者的量子关联，推导特定量子态单向信息赤字的部分解析结果并给出贝尔-默明-克吕什科不等式的数值结果，揭示了贝尔非定域性及单向信息赤字在相对论量子信息中的意义。
It is shown how a proof of the Bell-Kochen-Specker (BKS) theorem given by Kernaghan and Peres can be experimentally realized using a scheme of measurements derived from a related proof of the same theorem by Mermin. It is also pointed out that if this BKS experiment is carried out independently by two distant observers who repeatedly make measurements on a specially correlated state of six qubits, it provides an inequality-free demonstration of Bell's theorem as well.
We critically analyze a recent paper by D. Mermin and we compare his statements with Bell's position on the problems he is discussing.
It is well known that Bell inequality supporting the local realism can be violated in quantum mechanics. Numerous tests of such a violation have been demonstrated with bipartite entanglements. Using spectral jointmeasurements of the qubits, we here propose a scheme to test the tripartite Mermin inequality (a three-qubit Bell-type inequality) with three qubits dispersively-coupled to a driven cavity. First, we show how to generate a three-qubit Greenberger-Horne-Zeilinger (GHZ) state by only one-step quantum operation. Then, spectral joint-measurements are introduced to directly confirm such a …
Generalized Mermin Inequalities for Benchmarking Large-Scale GHZ States
研究针对多体贝尔测试在大规模应用的障碍，引入广义Mermin不等式，通过实验在可编程超导处理器上对多达80个量子比特的GHZ态测试，结果表明增加测量设置数可提高贝尔违反率和噪声鲁棒性，为大规模GHZ态提供更优贝尔基准。
Probing Mermin's inequalities violations through pseudospin operators
本文利用赝自旋算符构建Bell装置，基于压缩态和相干态纠缠态计算n=3和n=4时Mermin多项式的期望值，确定了违背Mermin不等式并饱和量子力学界限的参数条件。
Revisiting the experimental test of Mermin's inequalities at IBMQ
本文在IBM五量子位量子计算机上实验检验并展示了Mermin不等式的违背，从而否定局域实在论，且数值结果较以往实现显著改进，并提出该电路实现可用于评估量子计算机的可靠性。
Bell and Mermin inequalities in Quantum Field Theory from vacuum projectors and Weyl operators
本文利用真空投影子和 Weyl 算符在标量量子场论中构造二分算符，并通过菱形区域测试函数的数值方法研究了 Bell 和 Mermin 不等式，发现了两者的显著违背并验证了团簇性质。
We introduce a two-observer all-versus-nothing proof of Bell's theorem which reduces the number of required quantum predictions from 9 [A. Cabello, Phys. Rev. Lett. 87, 010403 (2001); Z.-B. Chen et al., Phys. Rev. Lett. 90, 160408 (2003)] to 4, provides a greater amount of evidence against local realism, reduces the detection efficiency requirements for a conclusive experimental test of Bell's theorem, and leads to a Bell's inequality which resembles Mermin's inequality for three observers [N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990)] but requires only two observers.
We study the explicit relation between violation of Bell inequalities and bipartite distillability of multi-qubit states. It has been shown that even though for $N\ge 8$ there exist $N$-qubit bound entangled states which violates a Bell inequality [Phys. Rev. Lett. {\bf 87}, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [Phys. Rev. Lett. {\bf 88}, 027901 (2002)]. We here prove that for all $N$-qubit states violating the inequality the number of distillable biparti…
The Mermin-Klyshko inequality for n spin-1/2 particles and two dichotomic observables is generalized to n spin-s particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any s. For a fixed n, the magnitude of the violation is constant for any s, which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-s particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed s, the violation grows e…
Randomness is a potential resource for cryptography, simulations and algorithms. Non-local correlations violating Bell's inequality certify the generation of bit strings whose randomness is guaranteed in a device-independent manner. We provide interesting semi-device independent randomness certification protocols by Kochen-Specker (KS) contextuality. For this, we first cast the Mermin's magic-square proof of KS contextuality for two-qubit system as a semi-device independent communication game in prepare-measure scenario. This provides a semi-device independent certification of two-bit of rando…
We present a generalized quantum scattering circuit which can be used to perform non-invasive quantum measurements, and implement it on NMR qubits. Such a measurement is a key requirement for testing temporal non-contextual inequalities. We use this circuit to experimentally demonstrate the violation of the Peres-Mermin inequality (which is the temporal analog of a Klyachko-Can- Binicioglu-Shumovsky (KCBS) inequality), on a three-qubit NMR quantum information processor. Further, we experimentally demonstrate the violation of a transformed Bell-type inequality (the spatial equivalent of the tem…
It is shown that the entanglement-structure of 3- and 4-qubit states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure 2-qubit entanglements and higher entanglements. A comparison with a global entanglement measure and the i-concurrence is made.
Effects of coherent and incoherent measurement imperfections on multipartite quantum nonlocality and quantum key distribution
研究相干和非相干测量缺陷对多体量子非局域性及量子密钥分发的影响，基于MABK等不等式分析其对贝尔值等的作用，还通过与密钥率界限联系得出认证约束，为相关认证和估计提供定量基准。
The quantum Fisher information (QFI) of certain multipartite entangled quantum states is larger than what is reachable by separable states, providing a metrological advantage. Are these nonclassical correlations strong enough to potentially violate a Bell inequality? Here, we present evidence from two examples. First, we discuss a Bell inequality designed for spin-squeezed states which is violated only by quantum states with a large QFI. Second, we relax a well-known lower bound on the QFI to find the Mermin Bell inequality as a special case. However, a fully general link between QFI and Bell …
A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.
We study the effects of Lorentz transformations on three-particle non-local system states (GHZ and W) of spin 1/2 particles, using the Pauli spin operator and a three-particle generalization of Bell's inequality, introduced by Svetlichny. In our setup, the moving and laboratory frames used the (same) set of measurement directions that maximally violate Svetlichny's inequality in the laboratory frame. We also investigate the behavior of Mermin's and Collins' inequalities. We find that, regardless of the particles' type of entanglement, violation of Svetlichny's inequality in the moving frame is…
In the Greenberger-Horne-Zeilinger-Mermin (GHZM) proof of Bell's theorem, a source periodically emits an entangled state of three particles whose properties are analyzed by three distant observers and used to prove Bell's nonlocality theorem. This paper analyzes a somewhat different gedanken experiment involving only two observers that nevertheless makes indirect use of the GHZ states to prove Bell's theorem. The relationship of the GHZM proof to the present one is discussed, and it is pointed out that the latter provides an interesting new view of the connection between the "two theorems of J…
Non-locality of the type first elucidated by Bell in 1964 is a difficult concept to explain to non-specialists and undergraduates. Here we attempt this by showing how such non-locality can be used to solve a problem in which someone might find themselves as the result of a collection of normal, even if somewhat unlikely, events. Our story is told in the style of a Sherlock Holmes mystery, and is based on Mermin's formulation of the "paradoxical" illustration of quantum non-locality discovered by Greenberger, Horne and Zeilinger.
Bell's theorem contains the proposition that the Einstein-Podolsky-Rosen (EPR) theory (hypothesis) of the existence of elements of reality together with Einstein locality permits a mathematical description of EPR experiments by functions that are all defined on one common probability space. This proposition leads in turn to restrictions for possible experimental outcomes that Bell expressed in terms of his well known inequalities and that Vorob'ev and others had investigated before Bell. Summarizing several previous publications and adding new material, the above proposition is refuted by Eins…
We investigate the violation of local realism in Bell tests involving homodyne measurements performed on multimode continuous-variable states. By binning the measurement outcomes in an appropriate way, we prove that the Mermin-Klyshko inequality can be violated by an amount that grows exponentially with the number of modes. Furthermore, the maximum violation allowed by quantum mechanics can be attained for any number of modes, albeit requiring a quantum state that is rather unrealistic. Interestingly, this exponential increase of the violation holds true even for simpler states, such as multip…
Greenberger, Horne, Shimony and Zeilinger gave a new version of the Bell theorem without using inequalities (probabilities). Mermin summarized it concisely; but Bohm and Hiley criticized Mermin's proof from contextualists' point of view. Using the Branching Space-time language, in this paper a proof will be given that is free of these difficulties. At the same time we will also clarify the limits of the validity of the theorem when it is taken as a proof that quantum mechanics is not compatible with a deterministic world nor with a world that permits correlated space-related events without a c…
Bell-type inequalities for systems of relativistic vector bosons
本文分析了矢量玻色子-反玻色子对系统中各类贝尔型不等式的违反情况，发现Mermin和CGLMP不等式在特定标量态下可被违反且违反程度依赖于粒子相对速度，而广义CHSH不等式则不被违反。
Beyond the Magic Square Game: Widening the Gap for Two Bell States
研究通过构造非局域游戏，证明使用两个贝尔态时，纠缠值与经典值的最大差距至少为4/35，优于魔方阵游戏的1/9。
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_β$ obtained from the Mermin scenario, parametrized by a function $β$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $β$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\text{MP}_0$ turn out to be deterministic. This result provides a new topological p…
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.
In 1990, Mermin presented a n player game that is won with certainty using n spin-1/2 particles in a GHZ state whilst no classical strategy (or local theory) can win with probability higher than ${1/2} + \frac{1}{2^{\lceil n/2 \rceil}}$ (which is larger than 1/2). This article first introduces a class of arithmetic games containing Mermin's and gives a quantum algorithm based on a generalized n party GHZ state that wins those games with certainty. It is then proved for a subclass of those games where each player is given a single bit of input that no classical strategy can win with a probabili…
An elegant proof of self-testing for multipartite Bell inequalities
本工作提出了一种简单且广泛适用的自测试论证，用于N体关联贝尔不等式，推广了Mayer-Yao公式并突破了线性不等式限制，为MABK、WWWŻB和Uffink不等式族提供了自测试证明。
The combination of various physically plausible properties, such as no signaling, determinism, and experimental free will, is known to be incompatible with quantum correlations. Hence, these properties must be individually or jointly relaxed in any model of such correlations. The necessary degrees of relaxation are quantified here, via natural distance and information-theoretic measures. This allows quantitative comparisons between different models in terms of the resources, such as the number of bits, of randomness, communication, and/or correlation, that they require. For example, measuremen…
Bi-Contextuality: A Novel Non-Classical Phenomenon in Bipartite Quantum Systems
提出并实验演示了双源量子系统中的新型非经典现象bi-contextuality，该现象作为反转的Bell场景，证实了单节点测量的量子性，并对PBR定理及经典psi-ontic模型提出了新挑战。
For eight-dimensional quantum systems there is a Kochen-Specker (KS) set of 40 quantum yes-no tests that is related to the Greenberger-Horne-Zeilinger (GHZ) proof of Bell's theorem. Here we experimentally implement this KS set using an eight-dimensional Hilbert space spanned by the transverse momentum of single photons. We show that the experimental results of these tests violate a state-independent noncontextuality inequality. In addition, we show that, if the system is prepared in states that are formally equivalent to a three-qubit GHZ and $W$ states, then the results of a subset of 16 test…
Contrary to what Lambare [arXiv:2008.00369] assumes, in non-Newtonian calculus (a calculus based on non-Diophantine arithmetic) an integral is typically given by a nonlinear map. This is the technical reason why all the standard proofs of Bell-type inequalities fail if non-Newtonian hidden variables are taken into account. From the non-Newtonian perspective, Bell's inequality is a property of a limited and unphysical class of hidden-variable models. An explicit counterexample to Bell's theorem can be easily constructed.
The field of device-independent quantum cryptography has seen enormous success in the past several years, including security proofs for key distribution and random number generation that account for arbitrary imperfections in the devices used. Full security proofs in the field so far are long and technically deep. In this paper we show that the concept of the mirror adversary can be used to simplify device-independent proofs. We give a short proof that any bipartite Bell violation can be used to generate private random numbers. The proof is based on elementary techniques and is self-contained.
In 1960, the mathematician Ernst Specker described a simple example of nonclassical correlations which he dramatized using a parable about a seer who sets an impossible prediction task to his daughter's suitors. We revisit this example here, using it as an entree to three central concepts in quantum foundations: contextuality, Bell-nonlocality, and complementarity. Specifically, we show that Specker's parable offers a narrative thread that weaves together a large number of results, including: the impossibility of measurement-noncontextual and outcome-deterministic ontological models of quantum…
Loophole-free Bell inequality violations cannot disprove local realism
该研究指出贝尔不等式推导中误将局域实在论的充分条件当作必要条件，通过构建三个违反贝尔不等式的局域实在论模型，证明无漏洞的贝尔不等式违反也无法证伪局域实在论。
'Locality' is a fraught word, even within the restricted context of Bell's theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantu…
In this article, a procedure called Probabilistic Swapped-Bell-States Analysis (PSBA) is proposed. Using this procedure two communication partners can transmit (binary-encoded) information over large spatial distances. This procedure is unusual insofar as no classical communication channels are used either during or after information encoding. To make this possible, entanglement swapping is used as a transport channel. The encoding of bits is realized by the execution (or non-execution, as the case may be) of entanglement swapping on multiple photon pairs, while decoding is realized by a stati…
Colbeck (Thesis, 2006) proposed using Bell inequality violations to generate certified random numbers. While full quantum-security proofs have been given, it remains a major open problem to identify the broadest class of Bell inequalities and lowest performance requirements to achieve such security. In this paper, working within the broad class of spot-checking protocols, we prove exactly which Bell inequality violations can be used to achieve full security. Our result greatly improves the known noise tolerance for secure randomness expansion: for the commonly used CHSH game, full security was…
Another Triumph of Locality: Colliding Histories Skew Handshakes
本文探讨了贝尔定理对局部现实的影响，指出传统隐藏变量策略的缺陷，并通过量子力学中的酉量子力学和海森堡-picture方法，提出局部解释的可能。
Violation of no-signaling on a public quantum computer
该研究在IBM量子设备上以大规模统计测试无信号原则，发现显著违反，其程度与贝尔测试相当，需排查技术缺陷并重复无漏洞测试以最终确认。
Bell's Theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions -- e.g. locality or measurement independence -- have to be relaxed in order to allow for a classical description of such experiments. Here, we develop a conceptual and computational framework for treating this problem. We employ the language of Bayesian networks to systematically construct alternative causal structures and bound the degree of relaxation using q…
In 2007, and in a series of later papers, Joy Christian claimed to refute Bell's theorem, presenting an alleged local realistic model of the singlet correlations using techniques from Geometric Algebra (GA). Several authors published papers refuting his claims, and Christian's ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell's theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian …
There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here we define the strength of such nonlocality proofs in terms of the amount of evidence against local realism provided by the corresponding experiments. This measure tells us how many trials of the experiment we should perform in order to observe a substantial violation of local realism. Statistical considerations show that the amount of evidence should be measured by the Kullback-Leibler or relative entropy divergence between the probability distributions ov…
This contribution to the book in honour of J.S. Bell will probably differ from the remaining ones, in particular since only a part of it will be devoted to specific technical arguments. In fact I have considered appropriate to share with the community of physicists interested in the foundational problems of our best theory the repeated interactions I had with him in the last four years of his life, the deep discussions in which we have been involved in particular in connection with the elaboration of collapse theories and their interpretation, the contributions he gave to the development of th…
Loophole-Robust Certification of Quantum Advantage
研究量子优势声明的鲁棒性，将经典策略与基准相关性形式化为基准依赖性。给出有界奖励任务等的相关边界及扩展框架，应用于IBM硬件数据，还分析非贝尔量子核基准，将量子 - 经典分数分离转化为经典信息定量下限。
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