Counterfactual definiteness is supposed to underlie the Bell theorem. An old controversy exists among those who reject the theorem implications by rejecting counterfactual definiteness and those who claim that, since it is a direct consequence of locality, it cannot be independently rejected. We propose a different approach for solving this contentious issue by realizing that counterfactual definiteness is an unnecessary and inconsistent assumption. Counterfactual definiteness is not equivalent to realism or determinism neither it follows from locality. It merely reduces to an incongruent application of counterfactual reasoning. Being incompatible with falsifiability, it constitutes an unjustified assumption that goes against the scientific method's rigor. Correct formulations of the Bell theorem's bases show it is absent either as a fundamental hypothesis or as a consequence of something else. Most importantly, we present a coherent Bell inequality derivation carefully devised to show explicitly and convincingly the absence of incompatible experiments or counterfactual reasoning. Thus, even admitting that counterfactual definiteness could be a consistent assumption, the necessary conclusion is that it is irrelevant for the inequality formulation and can be safely ignored when discussing Bell's inequality philosophical and physical implications.
We point out a loophole in the derivation of the Bell inequalities in the form proposed by Clauser, Horne, Shimony and Holt (CHSH). In this derivation it is assumed that statistical independence is a necessary consequence of locality, but this is not a logical necessity.
Assumed data streams from a delayed choice gedanken experiment must satisfy a Bell's identity independently of locality assumptions. The violation of Bell's inequality by assumed correlations of identical form among these data streams implies that they cannot all result from statistically equivalent variables of a homogeneous process. This is consistent with both the requirements of arithmetic and distinctions between commuting and noncommuting observables in quantum mechanics. Neglect of these distinctions implies a logical loophole in the conventional interpretation of Bell's inequalities.
Liquid phase NMR is a general purpose test-bed for developing methods of coherent control relevant to quantum information processing. Here we extend these studies to the coherent control of logical qubits and in particular to the unitary gates necessary to create entanglement between logical qubits. We report an experimental implementation of a conditional logical gate between two logical qubits that are each in decoherence free subspaces that protect the quantum information from fully correlated dephasing.
Two fault tolerant authenticated quantum dialogue (AQD) protocols are proposed in this paper by employing logical Bell states as the quantum resource, which combat the collective-dephasing noise and the collective-rotation noise, respectively. The two proposed protocols each can accomplish the mutual identity authentication and the dialogue between two participants simultaneously and securely over one kind of collective noise channels. In each of two proposed protocols, the information transmitted through the classical channel is assumed to be eavesdroppable and modifiable. The key for choosin…
We discuss a class of proofs of Bell-type inequalities that are based on tables of potential outcomes. These proofs state in essence: if one can only imagine (or write down in a table) the potential outcome of a hidden parameter model for EPR experiments then a contradiction to experiment and quantum mechanics follows. We show that these proofs do not contain hidden variables that relate to time or, if they do, lead to logical contradictions that render them invalid.
In most Bell tests, the measurement settings are specially chosen so that the maximal quantum violations of the Bell inequalities can be detected, or at least, the violations are strong enough to be observed. Such choices can usually associate the corresponding Bell operators to a kind of effective observables, called pseudo Pauli operators, providing us a more intuitive understanding of Bell nonlocality in some sense. Based on that, a more general quantum-to-classical approach for the constructions of Bell inequalities is developed. Using this approach, one can not only derive several kinds o…
In a remarkably insightful pair of papers recently, Sica demonstrated that: dichotomic data taken in any experiment that violates Bell's inequalities ``cannot represent any data streams that could possibly exist or be imagined'' if it is to be consistent with the derivation of the inequalities.\cite{sica} The present writer maintains, however, that corrections in the formulation of Bell's analysis loosen restrictions imposed by Bell inequalities. Moreover, it is argued that the resolution proposed by Sica for the conflict arising from the fact that real data does violate Bell inequalities, nam…
In practical quantum networks, a variety of multi-qubit stabilized states emitted from independent sources are distributed among the agents, and the correlations across the entire network can be derived from each agent's local measurements on the shared composite quantum systems. To reveal the Bell non-locality in such cases as a quantum feature, minimal knowledge of the emitted stabilizer state is required. Here, we demonstrate that knowing the stabilizing and logical operators indeed provides a new way of exploring Bell non-locality in quantum networks. For the qubit distribution in quantum …
Entanglement boosting: Low-volume logical Bell pair preparation for distributed fault-tolerant quantum computation
Nanofiber Quantum Technologies, Inc. (NanoQT)（纳米纤维量子技术有限公司）；Clarendon Laboratory, University of Oxford（牛津大学克拉伦登实验室）；Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学研究系计算机科学专业）
In eliminating the fair sampling assumption, the Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm Bell's historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed. In the present paper, the non-commutative aspects of these unperformed measurements are critically examined. Classical examples and the logic of the GHZ construction are analyzed to demonstrate that combined counterfactual results of non-commuting operations are …
Several emerging quantum technologies, including quantum networks, modular and fusion-based quantum computing, rely crucially on the ability to perform photonic Bell state measurements. Therefore, photon losses and the 50\% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies. Here, we develop protocols that overcome these two key challenges through logical encoding of photonic qubits. Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-dete…
The Greenberger, Horne, Zeilinger (GHZ) theorem is critically important to consideration of the possibility of hidden variables in quantum mechanics. Since it depends on predictions of single sets of measurements on three particles, it eliminates the sampling loophole encountered by the Bell theorem which requires a large number of observations to obtain a small number of useful joint measurements. In evading this problem, the GHZ theorem is believed to have confirmed Bell's historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem …
Bell inequalities play a central role in the study of quantum non-locality and entanglement, with many applications in quantum information. Despite the huge literature on Bell inequalities, it is not easy to find a clear conceptual answer to what a Bell inequality is, or a clear guiding principle as to how they may be derived. In this paper, we introduce a notion of logical Bell inequality which can be used to systematically derive testable inequalities for a very wide variety of situations. There is a single clear conceptual principle, based on purely logical consistency conditions, which und…
Which is the simplest logical structure for which there is quantum nonlocality? We show that there are only three bipartite Bell inequalities with quantum violation associated with the simplest graph of relationships of exclusivity with a quantum-classical gap. These are the most elementary logical Bell inequalities. We show that the quantum violation of some well-known Bell inequalities is related to them. We test the three Bell inequalities with pairs of polarization-entangled photons and report violations in good agreement with the quantum predictions. Unlike other experiments testing nonco…
We show how the efficiency of a logical Bell measurement (BM) can be calculated for arbitrary CSS codes with the experimentally important constraint of using only transversal static linear-optical BMs on the physical single-photon qubit level. For this purpose, we utilize the codes' description in terms of stabilizers in order to calculate general efficiencies for the loss-free case, but also for specific cases including photon loss. These efficiencies can be, for instance, used for obtaining transmission rates of all-optical quantum repeaters. In the loss-free case, we demonstrate that the im…
Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory - nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show some of …
We study the Bell nonlocality of high dimensional quantum systems based on quantum entanglement. A quantitative relationship between the maximal expectation value B of Bell operators and the quantum entanglement concurrence C is obtained for even dimension pure states, with the upper and lower bounds of B governed by C.
We analyze certain aspects of group theoretical approach to Bell inequalities proposed by Güney and Hillery. The general procedure for constructing the relevant group orbits is described. Using Hall theorem we determine the form of Bell inequality in the single orbit case. It is shown that in this case the Bell inequality is not violated for maximally entangled state generating trivial subrepresentation if the representation under consideration is real.
The inequalities proposed in [Phys. Rev. Lett. 105, 170404 (2010)] are not Bell's inequalities in the usual sense: there are local classical models that violate these inequalities. Thus, their violation demonstrates nonlocality only under assumptions about the physical implementation and moreover they cannot be used as a device-independent test of nonlocality.
We discuss supplementary (or hidden) variables in spin-measuring equipments in EPR-Bell experiment. This theme was considered in a Bell's later work. We generalize it. First, we show why the original supplementary variable $λ$ is not to be regarded to include supplementary variables in spin-measuring equipments (why supplementary variables should be introduced additionally in spin-measuring equipments) Next, we show the followings. When the supplementary variables introduced in spin-measuring equipments have local correlations, the Bell inequality is recovered. On the other hand, when they hav…
The logical foundations of Bell's inequality are reexamined. We argue that the form of the reality condition that underpins Bell's inequality comes from the requirement of solving the quantum measurement problem. Hence any violation of Bell's inequality necessarily implies nonlocality because of the measurement problem. The differences in the implications of deterministic and stochastic formulations of Bell's inequality are highlighted. The reality condition used in Bell's inequality is shown to be a generalisation of Einstein's later form of realism.
A proof of Bell's theorem using two maximally entangled states of two qubits is presented. It exhibits a similar logical structure to Hardy's argument of ``nonlocality without inequalities''. However, it works for 100% of the runs of a certain experiment. Therefore, it can also be viewed as a Greenberger-Horne-Zeilinger-like proof involving only two spacelike separated regions.
This text is an introduction to an operational outlook on Bell inequalities, which has been very fruitful in the past few years. It has lead to the recognition that Bell tests have their own place in applied quantum technologies, because they quantify non-classicality in a device-independent way, that is, without any need to describe the degrees of freedom under study and the measurements that are performed. At the more fundamental level, the same device-independent outlook has allowed the falsification of several other alternative models that could hope to reproduce the observed statistics wh…
An explicit model-example is presented to simulate Einstein-Podolsky-Rosen (EPR) experiments without invoking instantaneous influences at a distance. The model-example, together with the interpretation of past experiments by Kwiat and coworkers, uncovers logical inconsistencies in the application of Bell's theorem to actual EPR experiments. The inconsistencies originate from topological-combinatorial assumptions that are both necessary and sufficient to derive all Bell-type inequalities including those of Wigner-d'Espagnat and Clauser-Horne-Shimony-Holt. The model-example circumvents these inc…
We present a detailed analysis of the Bell measurement scheme proposed in [Phys. Rev. Lett. 114, 113603 (2015)] based on a logical qubit using Greenberger-Horne-Zeilinger (GHZ) entanglement of photons. The success probability of the proposed Bell measurement can be made arbitrarily high using only linear optics as the number of photons in a logical qubit increases. We compare our scheme with all the other proposals, using single-photon qubits, coherent-state qubits or hybrid qubits, suggested to enhance the efficiency of the Bell measurement. As a remarkable advantage, our scheme requires only…
With Bell's inequalities one has a formal expression to show how essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a simple experimental arrangement. For the case of entangled pairs of spin-1/2 particles we propose an alternative measurement setup which is consistent to the necessary assumptions corresponding to the derivation of the Bell inequalities. We find that the Bell inequalities are never violated with respect to our suggested measurement process.
Using results of Pitowsky and Gupta, we show in a direct, elementary fashion that, in the case of three spins, Bell's inequalities indeed provide a representation of the tetrahedron of all spin correlation matrices as intersection of half-spaces.
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can…
To Scale Up or To Scale Out: Evaluating Space-Time Costs of Compiled Logical Circuits on Modular Superconducting Quantum Processors
Dark Qore；QuantWare；ETH Zurich（苏黎世联邦理工学院）；Department of Physics, ETH Zurich（苏黎世联邦理工学院物理系）；Quantum Center, ETH Zurich（苏黎世联邦理工学院量子中心）；Delft University of Technology（代尔夫特理工大学）；QuTech, Delft University of Technology（代尔夫特理工大学QuTech）
Most physicists agree that the Einstein-Podolsky-Rosen-Bell paradox exemplifies much of the strange behavior of quantum mechanics, but argument persists about what assumptions underlie the paradox. To clarify what the debate is about, we employ a simple and well-known thought experiment involving two correlated photons to help us focus on the logical assumptions needed to construct the EPR and Bell arguments. The view presented in this paper is that the minimal assumptions behind Bell's inequality are locality and counterfactual definiteness, but not scientific realism, determinism, or hidden …
Cabello has recently (in quant-ph/0210081) observed that ``...an EPR-experiment with a fixed POVM on each particle provides a violation of Bell's inequality without requiring local observers to choose between the alternatives.'' In this note I discuss the implications of this observation for tests of locality.
Bell's theorem shows that the reasonable relativistic causal principle known as "local causality" is not compatible with the predictions of quantum mechanics. It is not possible maintain a satisfying causal principle of this type while dropping any of the better-known assumptions of Bell's theorem. However, another assumption of Bell's theorem is the use of classical logic. One part of this assumption is the principle of "ontic definiteness", that is, that it must in principle be possible to assign definite truth values to all propositions treated in the theory. Once the logical setting is cla…
The purpose of this note is to clarify the logical relationship between joint measurability and contextuality for quantum observables in view of recent developments [1-4].
In a series of very interesting papers [1-7], Joy Christian constructed a counterexample to Bell's theorem. This counterexample does not have the same assumptions as the original Bell's theorem, and therefore it does not represent a genuine disproof in a strict mathematical sense. However, assuming the physical relevance of the new assumptions, the counterexample is shown to be a contextual hidden variable theory. If Bell's theorem's importance is to rule out contextual hidden variable theories obeying relativistic locality, then Joy Christian's counterexample achieves its aim. If however cont…
In this note, we discuss a closed-form necessary and sufficient condition for any two-qubit state to show hidden nonlocality w.r.t the Bell-CHSH inequality. This is then used to numerically compute the relative volume of states showing hidden Bell-CHSH non-locality , among all two-qubit states with one-sided reduction maximally mixed.
There are various non-equivalent definitions of locality. Three of them, impossibility of instantaneous communication, impossibility of action-at-a-distance, and impossibility of faster-than-light travel, while not fully implying each other, have a large overlap. When the term non-locality is used, most physicists think that one of these three conditions is being violated. There is a minority of physicists, however, who uses "locality" with a fourth meaning, the satisfaction of the hypotheses underlying Bell inequality. This definition devoids Bell's theorem of its profoundness, reducing it to…
In a recent paper [arXiv:1703.11003] on this journal Stephen Boughn argued that quantum mechanics does not require nonlocality of any kind and that the common interpretation of Bell theorem as a nonlocality result is based on a misunderstanding. In this note I argue that the Boughn arguments, that summarize views widespread in certain areas of the foundations of quantum mechanics, are based on an incorrect reading of the presuppositions of the EPR argument and the Bell theorem and, as a consequence, are totally unfounded.
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…
This note shows how one can be led from considerations of quantum steering to Bell's theorem. The point is that steering remote systems by choosing between two measurements can be described in a local theory if we take quantum states to be associated many-to-one with the underlying "real states" of the world. Once one adds a third measurement this is no longer possible. Historically this is not how Bell's theorem arose - there are slight and subtle differences in the arguments - but it could have been.
In a recent note, Cavalcanti and Scarani (CS) constructed a counter local-hidden-variable model to explain the violation of our inequalities in Phys. Rev. Lett. 105, 170404 (2010). Here, we briefly discuss some issues in response to the comments raised by CS.
A recent experiment presented, for photons, the first violation of a Bell inequality closing the fair-sampling loophole, i.e., without having to assume that the sample of measured photons fairly represents the entire ensemble. In this note, we discuss a detailed quantum mechanical model for the experimental data. Within experimental error the experiment agrees with quantum mechanical prediction. We also discuss the effects of drifting laser intensity and show that it could not have mimicked a violation of the Bell inequality. Therefore, the experiment was immune to the "production-rate loophol…
We introduce two types of statistical quasi-separation between local observables to construct two-party Bell-type inequalities for an arbitrary dimensional systems and arbitrary number of measurement settings per site. Note that, the main difference between statistical quasi-separations and the usual statistical separations is that the former are not symmetric under exchange of the two local observables, whereas latter preserve the symmetry. We show that a variety of Bell inequalities can be derived by sequentially applying triangle inequalities which statistical quasi-separations satisfy. A s…
The aim of this note is to attract once again attention of the quantum community to statistical analysis of data which was reported as violating Bell's inequality. This analysis suffers of a number of problems. And the main problem is that rough data is practically unavailable. However, experiments which are not followed by the open access to the rough data have to be considered as with no result. The absence of rough data generates a variety of problems in statistical interpretation of the results of Bell's type experiment. One may hope that this note would stimulate experimenters to create t…
The reason for recalling this old paper is the ongoing discussion on the attempts of circumventing certain assumptions leading to the Bell theorem (Hess-Philipp, Accardi). If I correctly understand the intentions of these Authors, the idea is to make use of the following logical loophole inherent in the proof of the Bell theorem: Probabilities of counterfactual events A and A' do not have to coincide with actually measured probabilities if measurements of A and A' disturb each other, or for any other fundamental reason cannot be performed simulaneously. It is generally believed that in the con…
The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden variable theory, has been used as a test to reveal failure of the hidden variable theory and to prove validity of the quantum theory. We note that violation of the BCHSH inequality is caused by noncommutativity of quantum observables and find a systematic method for constructing generalizations of the BCHSH inequality. This method is applied for inventing a new quantity which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum …
By implicitly assuming that all measurements occur simultaneously, Bell's Theorem only applied to local theories that violated Heisenberg's Uncertainty Principle. By explicitly introducing time into our derivation of Bell's theorem, an extra term related to the time-ordering of actual measurements is found to augment (i.e. weaken) the upper bound of the inequality. Since the same locality assumptions hold for this rederivation as for the original, we conclude that only {\em classical} measurement-order independent local hidden variable theories are constrained by Bell's inequality; time depend…
Quantum theory violates Bell's inequality, but not to the maximum extent that is logically possible. We derive inequalities (generalizations of Cirel'son's inequality) that quantify the upper bound of the violation, both for the standard formalism and the formalism of generalized observables (POVMs). These inequalities are quantum analogues of Bell inequalities, and they can be used to test the quantum version of locality. We discuss the nature of this kind of locality. We also go into the relation of our results to an argument by Popescu and Rohrlich (Found. Phys. 24, 379 (1994)) that there i…
The 1964 theorem of John Bell shows that no model that reproduces the predictions of quantum mechanics can simultaneously satisfy the assumptions of locality and determinism. On the other hand, the assumptions of \emph{signal locality} plus \emph{predictability} are also sufficient to derive Bell inequalities. This simple theorem, previously noted but published only relatively recently by Masanes, Acin and Gisin, has fundamental implications not entirely appreciated. Firstly, nothing can be concluded about the ontological assumptions of locality or determinism independently of each other -- it…
In superconducting circuits with interbit untunable (e.g., capacitive) couplings, ideal local quantum operations cannot be exactly performed on individual Josephson qubits. Here we propose an effective dynamical decoupling approach to overcome the "fixed-interaction" difficulty for effectively implementing elemental logical gates for quantum computation. The proposed single-qubit operations and local measurements should allow testing Bell's inequality with a pair of capacitively-coupled Josephson qubits. This provides a powerful approach, besides spectral-analysis [Nature \textbf{421}, 823 (20…
"Bell's theorem" can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of Locality and Predetermination. His 1976 theorem is their incompatibility with the single property of Local Causality. This is contrary to Bell's own later assertions, that his 1964 theorem began with the assumption of Local Causality, even if not by that name. Although the two Bell's theorems are logically equivalent, their assumptions are not. Hence, the earlier and later theorems sugges…
