Local Hidden Variable Model Research Articles

Compatibility of Generalized Noisy Qubit Measurements.

It is a crucial feature of quantum mechanics that not all measurements are compatible with each other. However, if measurements suffer from noise they may lose their incompatibility. Here, we consider the effect of white noise and determine the critical visibility such that all qubit measurements, i.e., all positive operator-valued measures (POVMs), become compatible, i.e., jointly measurable. In addition, we apply our methods to quantum steering and Bell nonlocality. We obtain a tight local hidden state model for two-qubit Werner states of visibility 1/2. This determines the exact steering bound for two-qubit Werner states and also provides a local hidden variable model that improves on previously known models. Interestingly, this proves that POVMs are not more powerful than projective measurements to demonstrate quantum steering for these states.

Quantum steering in a star network

In this work, we will consider the star network scenario where the central party is trusted while all the edge parties (with a number of n) are untrusted. Network steering is defined with an n local hidden state model which can be viewed as a special kind of n local hidden variable model. Two different types of sufficient criteria, nonlinear steering inequality and linear steering inequality will be constructed to verify the quantum steering in a star network. Based on the linear steering inequality, how to detect the network steering with a fixed measurement will be discussed.

Quantum Nonlocality: How Does Nature Do It?

In his article in Science, Nicolas Gisin claimed that quantum correlations emerge from outside space-time. We explainthat they are due to space-time symmetries. This paper is a critical review of metaphysical conclusions found in many recent articles. It advocates the importance of contextuality, Einstein -causality and global symmetries. Bell tests allow only rejecting probabilistic coupling provided by a local hidden variable model, but they do not justify metaphysical speculations about quantum nonlocality and objects which know about each other's state, even when separated by large distances. The violation of Bell inequalities in physics and in cognitive science can be explained using the notion of Bohr- contextuality. If contextual variables, describing varying experimental contexts, are correctly incorporated into a probabilistic model, then the Bell-CHSH inequalities cannot be proven and nonlocal correlations may be explained in an intuitive way. We also elucidate the meaning of statistical independence assumption incorrectly called free choice, measurement independence or no- conspiracy. Since correlation does not imply causation, the violation of statistical independence should be called contextuality; it does not restrict the experimenter's freedom of choice. Therefore, contrary to what is believed, closing the freedom-of choice loophole does not close the contextuality loophole.

Continuous and discrete local hidden variable theories are equivalent

In quantum theory, Bell locality of quantum states (generally, of correlations, or boxes) is characterized by local hidden variable models (LHVMs) given by integrals and sums. We call these models continuous and discrete LHVMs (C-LHVMs and D-LHVMs), respectively. Theoretically, this seems leading to two types of local hidden variable theories (LHVTs) given by C-LHVMs and D-LHVMs, respectively. In this paper, we show the equivalence of these two types of LHVTs by mathematically proving that a correlation tensor (equivalently, a box) has a C-LHVM if and only if it has a D-LHVM. As application, we check the Bell locality of some correlation tensors that are defined by integrals.

Bell's theorem in time without inequalities

Bell's theorem revealed that a local hidden-variable model cannot completely reproduce the quantum mechanical predictions. Bell's inequality provides an upper bound under the locality and reality assumptions that can be violated by correlated measurement statistics of quantum mechanics. Greenberger, Horne, and Zeilinger (GHZ) gave a more compelling proof of Bell's theorem without inequalities by considering perfect correlations rather than statistical correlations. In this work, we present a temporal analogue of GHZ argument that establishes Bell's theorem in time without inequalities.

Device-independent and semi-device-independent entanglement certification in broadcast Bell scenarios

It has recently been shown that by broadcasting the subsystems of a bipartite quantum state, one can activate Bell nonlocality and significantly improve noise tolerance bounds for device-independent entanglement certification. In this work we strengthen these results and explore new aspects of this phenomenon. First, we prove new results related to the activation of Bell nonlocality. We construct Bell inequalities tailored to the broadcast scenario, and show how broadcasting can lead to even stronger notions of Bell nonlocality activation. In particular, we exploit these ideas to show that bipartite states admitting a local hidden-variable model for general measurements can lead to genuine tripartite nonlocal correlations. We then study device-independent entanglement certification in the broadcast scenario, and show through semidefinite programming techniques that device-independent entanglement certification is possible for the two-qubit Werner state in essentially the entire range of entanglement. Finally, we extend the concept of EPR steering to the broadcast scenario, and present novel examples of activation of the two-qubit isotropic state. Our results pave the way for broadcast-based device-independent and semi-device-independent protocols.

Graph-theoretic approach to Bell experiments with low detection efficiency

Bell inequality tests where the detection efficiency is below a certain threshold ηcrit can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low ηcrit and relatively low dimension d of the local quantum systems. The method has two steps. First, we show a family of bipartite Bell inequalities for which, for correlations produced by maximally entangled states, ηcrit can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small ηcrit. We present examples in which, for maximally entangled states, ηcrit≤0.516 for d=16, ηcrit≤0.407 for d=28, and ηcrit≤0.326 for d=32. We also show evidence that the upper bound for ηcrit can be lowered down to 0.415 for d=16 and present a method to make the upper bound of ηcrit arbitrarily small by increasing the dimension and the number of settings. All these upper bounds for ηcrit are valid (as it is the case in the literature) assuming no noise. The second step is based on the observation that, using the initial state and measurement settings identified in the first step, we can construct Bell inequalities with smaller ηcrit and better noise robustness. For that, we use a modified version of Gilbert's algorithm that takes advantage of the automorphisms of the graphs used in the first step. We illustrate its power by explicitly developing an example in which ηcrit is 12.38% lower and the required visibility is 14.62% lower than the upper bounds obtained in the first step. The tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.

Quantum computational advantage attested by nonlocal games with the cyclic cluster state

We propose a set of Bell-type nonlocal games that can be used to prove an unconditional quantum advantage in an objective and hardware-agnostic manner. In these games, the circuit depth needed to prepare a cyclic cluster state and measure a subset of its Pauli stabilizers on a quantum computer is compared to that of classical Boolean circuits with the same, nearest-neighboring gate connectivity. Using a circuit-based trapped-ion quantum computer, we prepare and measure a six-qubit cyclic cluster state with an overall fidelity of 60.6% and 66.4%, before and after correcting for measurement-readout errors, respectively. Our experimental results indicate that while this fidelity readily passes conventional (or depth-0) Bell bounds for local hidden-variable models, it is on the cusp of demonstrating a higher probability of success than what is possible by depth-1 classical circuits. Our games offer a practical and scalable set of quantitative benchmarks for quantum computers in the pre-fault-tolerant regime as the number of qubits available increases.

Imitating Quantum Probabilities: Beyond Bell’s Theorem and Tsirelson Bounds

Local hidden-variable model of singlet-state correlations discussed in Czachor (Acta Phys Polon A 139:70, 2021a) is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities can be interpreted as a ‘confusion of languages’ problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.

Witnessing Bell violations through probabilistic negativity

Bell's theorem shows that no local hidden-variable model can explain the measurement statistics of a quantum system shared between two parties, thus ruling out a classical (local) understanding of nature. In this paper we demonstrate that by relaxing the positivity restriction in the hidden-variable probability distribution it is possible to derive quasiprobabilistic Bell inequalities whose sharp upper bound is written in terms of a negativity witness of said distribution. This provides an analytic solution for the amount of negativity necessary to violate the Clauser-Horne-Shimony-Holt inequality by an arbitrary amount, therefore revealing the amount of negativity required to emulate the quantum statistics in a Bell test.

Proposal for a Bell test in cavity optomagnonics

We present a proposal to test Bell inequality in the emerging field of cavity optomagnonics, where a sphere of ferromagnetic crystal supports two optical whispering gallery modes and one magnon mode. The two optical modes are driven by two laser pulses, respectively. Entanglement between magnon mode and one of the two optical modes will be generated by the first pulse, and the state of magnon mode is subsequently mapped into another optical mode via the second pulse. Hence correlated photon-photon pairs is created out of the cavity. A Bell test can be implemented using these pairs, which enables us to test local hidden-variable models at macroscopic optomagnonical system. Our results show that a significant violation of Bell inequality can be obtained in the experimentally relevant weak-coupling regime. The violation of Bell inequality not only verifies the entanglement between magnons and photons, but also implies that cavity optomagnonics is a promising platform for quantum information processing tasks.

Quantum Bell nonlocality as a form of entanglement

Bell nonlocality describes a manifestation of quantum mechanics that cannot be explained by any local hidden variable model. Its origin lies in the nature of quantum entanglement, although understanding the precise relationship between nonlocality and entanglement has been a notorious open problem. In this paper, we resolve this problem by developing a dynamical framework in which quantum Bell nonlocality emerges as special form of entanglement, and both are unified as resources under local operations and classical communication (LOCC). Our framework is built on the notion of quantum processes, which are abstract quantum channels mapping elements between fixed intervals in space and time. Entanglement is then identified as a quantum process that cannot be generated by LOCC while Bell nonlocality is the subset of these processes that have an instantaneous input-output delay time. LOCC pre-processing is a natural set of free operations in this theory, thereby enabling all entangled states to activate some form of Bell nonlocality. In addition, we generalize the CHSH witnesses from the state domain to the domain of entangled quantum measurements, and provide a systematic method to quantify the Bell nonlocality of a bipartite quantum channel.

Signaling between time steps does not allow for nonlocality beyond hidden nonlocality

Hidden nonlocality is the phenomenon that entangled states can be local in the standard Bell scenario but display nonlocality after local filtering. However, there exist entangled states for which all measurement statistics can be described via a local hidden variable model even after local filtering. In this work we consider the scenario that measurement outcomes and settings of Alice can influence measurements of Bob in subsequent time steps (and vice versa), however, there is no signaling among them for measurements at the same time step. We show that in this scenario states that only display local statistics after local filtering remain local even when considering the complete statistics of arbitrary sequences and therefore no advantage can be gained by performing longer sequences in this scenario. We first determine the extreme points of the polytope defined by the no-signaling conditions within the same time step and the arrow-of-time constraints. Based on these results we introduce a notion of locality and provide a complete representation of the corresponding local polytope in terms of inequalities in the simplest scenario. These results imply that in the scenario considered here there is no nonlocality beyond hidden nonlocality. We further propose a device-dependent Schmidt number witness and we compare our finding to known local models in the sequential scenario.

Quantum Theory Cannot Violate a Causal Inequality.

Within quantum theory, we can create superpositions of different causal orders of events, and observe interference between them. This raises the question of whether quantum theory can produce results that would be impossible to replicate with any classical causal model, thereby violating a causal inequality. This would be a temporal analog of Bell inequality violation, which proves that no local hidden variable model can replicate quantum results. However, unlike the case of nonlocality, we show that quantum experiments can be simulated by a classical causal model, and therefore cannot violate a causal inequality.

Can single photon excitation of two spatially separated modes lead to a violation of Bell inequality via weak-field homodyne measurements?

We reconsider the all-optical weak homodyne-measurement based experimental schemes aimed at revealing Bell nonclassicality (‘nonlocality’) of a single photon. We focus on the schemes put forward by Tan et al (TWC, 1991) and Hardy (1994). In our previous work we show that the TWC experiment can be described by a local hidden variable model, hence the claimed nonclassicality is apparent. The nonclassicality proof proposed by Hardy remains impeccable. We investigate which feature of the Hardy’s approach is crucial to disclose the nonclassicality. There are consequential differences between TWC and Hardy setups: (i) the initial state of Hardy is a superposition of a single photon excitation with vacuum in one of the input modes of a 50–50 beamsplitter. In the TWC case there is no vacuum component. (ii) In the final measurements of Hardy’s proposal the local settings are specified by the presence or absence of a local oscillator field (on/off). In the TWC case the auxiliary fields are constant, only phases are varied. We show that in Hardy’s setup the violation of local realism occurs due to the varying strength of the local oscillators. Still, one does not need to operate in the fully on/off detection scheme. Thus, the nonclassicality in a Hardy-like setup cannot be attributed to the single-photon state alone. It is a consequence of its interference with the photons from auxiliary local fields. Neither can it be attributed to the joint state of the single photon excitation and the local oscillator modes, as this state is measurement setting dependent. Despite giving spurious violations of local realism, the TWC scheme can serve as an entanglement indicator, for the TWC state. Nevertheless an analogue indicator based on intensity rates rather than just intensities overperforms it.

Non-Contextual and Local Hidden-Variable Model for the Peres–Mermin and Greenberger–Horne–Zeilinger Systems

A hidden-variable model for quantum-mechanical spin, as represented by the Pauli spin operators, is proposed for systems illustrating the well-known no-hidden-variables arguments by Peres and Mermin (1990) and by Greenberger, Horne, and Zeilinger (1989). Both arguments rely on an assumption of non-contextuality; the latter argument can also be phrased as a non-locality argument, using a locality assumption. The model suggested here is compatible with both assumptions. This is possible because the scalar values of spin observables are replaced by vectors that are components of orientations.

Arithmetic Loophole in Bell's Theorem: Overlooked Threat to Entangled-State Quantum Cryptography

Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations. However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states. Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers, but counterfactual aspects of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and calculus. The model is classical in the sense of Einstein, Podolsky, Rosen, and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variable probaility densities. Probability densities have a Clauser-Horne product form typical of local realistic theories. However, neither the product nor the integral nor the representation of rotations are the usual ones. The integral has all the standard properties but only with respect to the arithmetic that defines the product. Certain formal transformations of integral expressions one finds in the usual proofs \`a la Bell do not work, so standard Bell-type inequalities cannot be proved. The system we consider is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, so is free of all the canonical loopholes discussed in the literature.

The effect on (2, N, 2) Bell tests with distributed measurement dependence

Bell tests, as primitive tools to detect nonlocality in bipartite systems, rely on an assumption, i.e., measurement independence. In practice, it is difficult to ensure measurement independence. It is necessary to investigate how Bell tests are affected by relaxing measurement independence. In the simplest (2, 2, 2) CHSH Bell test which consists of two parties, two measurements per party and two possible outcomes per measurement, the results between the maximal value of CHSH correlation function and distributed measurement dependence (DMD) are given, where DMD is a general measure of relaxing measurement independence. However, in a general Bell scenario of an arbitrary number of measurements per party, i.e., (2, N, 2), pertinent results are still missing. To solve it, we establish the relations between the maximal value of (2, N, 2) Pearle–Braunstein–Caves (PBC) chain correlation function that maintains the locality and the degree of DMD, denoted as DMD-induced PBC chain inequalities. Furthermore, we show the tightness of these derived inequalities via constructing local hidden variable models that fake the upper bounds. Compared with the simplest CHSH Bell test, our derived inequalities need less amount of measurement dependence to fake the quantum prediction with N increasing, which is beneficial to analyze the security of device-independent quantum information processing tasks such as randomness expansion.

Experimental test of Leggett's inequalities with solid-state spins

Bell's theorem states that no local hidden variable model is compatible with quantum mechanics. Surprisingly, even if we release the locality constraint, certain nonlocal hidden variable models, such as the one proposed by Leggett, may still be at variance with the predictions of quantum physics. Here, we report an experimental test of Leggett's nonlocal model with solid-state spins in a diamond nitrogen-vacancy center. We entangle an electron spin with a surrounding weakly coupled $^{13}C$ nuclear spin and observe that the entangled states violate Leggett-type inequalities by more than four and seven standard deviations for six and eight measurement settings, respectively. Our experimental results are in full agreement with quantum predictions and violate Leggett's nonlocal hidden variable inequality with a high level of confidence.

Device-independent secret sharing and a stronger form of Bell nonlocality

Bell nonlocality, the fact that local hidden variable models cannot reproduce the correlations obtained by measurements on entangled states, is a cornerstone in our modern understanding of quantum theory. Apart from its fundamental implications, nonlocality is also at the core of device-independent quantum information processing, which successful implementation is achieved without precise knowledge of the physical apparatus. Here we show that a stronger form of Bell nonlocality, for which even nonlocal hidden variable models cannot reproduce the quantum predictions, allows for the device-independent implementation of secret sharing, a paradigmatic communication protocol where a secret split amidst many possibly untrusted parts can only be decoded if they collaborate among themselves.