Local Hidden Variables Research Articles

NOVEL BELL INEQUALITIES FOR n QUBITS DISTRIBUTED BETWEEN m < n PARTIES

We describe a method for obtaining m-partite Bell inequalities that are maximally violated by n-qubit states by an amount that grows exponentially with n (n > m). These inequalities, derived for states with perfect correlations, are, however, valid for all local hidden variable theories.

Application of the 1H(d,2He)n reaction to the EPR paradox

The test of the local hidden variable theories involving strongly interacting pairs of massive spin-\(\frac{1}{2}\) hadrons, in a nearly loophole-free experimental setting is reported. The spin correlation function is deduced to be Cexp(π/4) = −0.73 ± 0.08. This result is in agreement with non-local quantum mechanical prediction and it violates the Bell’s inequality of CBell(π/4) ≥ −0.5 at a confidence level of 99.3%.

On the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site

We consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature. This allows us (1) to specify in probabilistic terms the difference between nonsignaling, the Einstein–Podolsky–Rosen (EPR) locality and Bell's locality; (2) to introduce the notion of a local hidden variable (LHV) model for an S1 × ⋅ ⋅ ⋅ × SN-setting N-partite correlation experiment with outcomes of any spectral type, discrete or continuous, and to prove both general and specifically ‘quantum’ statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for a multipartite quantum state, in particular, to show that any N-partite quantum state, pure or mixed, admits an arbitrary S1 × 1 × ⋅ ⋅ ⋅ × 1-setting LHV description; (4) to evaluate a threshold visibility for an arbitrary bipartite noisy quantum state to admit an S1 × S2-setting LHV description under any generalized quantum measurements of two parties.

Strong violations of Bell-type inequalities for Werner-like states

We investigate the violation of Bell-type inequalities for two-qubit Werner-like states parametrized by the positive parameter 0 1/3. However, the improvement over the Clauser-Horne inequality is achieved at the price of restricting the class of possible local hidden variable theories.

Experimental test of nonlocal realism using a fiber-based source of polarization-entangled photon pairs

We describe an experimental test of local realistic and nonlocal realistic theories using polarization-entangled two-photon singlet states created using a fiber-based polarization Sagnac interferometer. We show a violation of Bell's inequality in the Clauser-Horne-Shimony-Holt form by 15 standard deviations, thus excluding local hidden-variable theories, and a violation of a Leggett-type nonlocal hidden-variable inequality by more than three standard deviations, thus excluding a class of nonlocal hidden-variable theories.

Nonlocality for graph states

The possibility of preparing two-photon entangled states encoding three or more qubits in each photon leads to the following problem: If n quabits were distributed between two parties, which quantum pure states and qubit distributions would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell’s theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs and provide all existing proofs up to n = 7 qubits. On the other hand, the possibility of preparing n-photon n-qubit graph states leads to the following problem: If n qubits were distributed between n parties, which would be the optimal Bell inequalities? We show all optimal n-party Bell inequalities for the perfect correlations of any graph state of n < 6 qubits. Optimal means that the ratio between the quantum violation and the bound for local hidden-variable theories is the maximum over all possible combinations of perfect correlations. This implies that the required detection efficiencies for loophole-free Bell tests are minimal.

New Bell inequalities for the singlet state: Going beyond the Grothendieck bound

Contemporary versions of Bell’s argument [Physics (Long Island City, N.Y.) 1, 195 (1964)] against local hidden variable (LHV) theories are based on the Clauser-Horne-Shimony-Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)] inequality and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper, we derive two-particle Bell inequalities for traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constants both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations and applying a recent result of Alon et al. [Invent. Math. 163, 499 (2006)], we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to infinity as the dimension and number of measurements grow. Technically, these result are possible because perfect correlations (or anticorrelations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%.

Analysis of a recent experiment seemingly favouring local realism against quantum mechanics

A recent experiment by Brida et al. (arXiv:0705.0439 [quant-ph]) is analyzed with the conclusion that is shows a violation of standard quantum predictions. A simple local hidden variables model is studied which is compatible with the empirical results and fits fairly well the deviation from the quantum predictions.

Towards a Local Hidden Variable Theory

A local hidden variable theory of quantum mechanics is formulated by adapting Gell-Mann and Hartle’s many-histories formulation. The resulting theory solves the measurement problem by exploiting the independence loophole in Bell’s theorem; it violates the independence of hidden variable values and measuring device settings. Although the theory is problematic in some respects, it provides a concrete example via which the tenability of this approach can be better evaluated.

Bell inequalities for a sensible family of local hidden variable theories testable at low detection efficiency

A family of local models containing two angles as hidden variables is defined for experiments measuring polarization correlation of optical photons. Searching for the best model of the family, that is giving predictions most close to quantum mechanics, allows deriving Bell-type inequalities which may be tested with relatively low detection efficiency.

QUANTUM TELEPORTATION WITH PAIR-COHERENT STATES

Recently, it has been argued that all presently performed continuous variable quantum teleportation experiments could be explained using a local hidden variable theory. In this paper, we study a modification of the original protocol which requires a fully quantum mechanical explanation even when coherent states are teleported. Our calculations of the fidelity of teleportation using a pair-coherent state under ideal conditions suggest that fidelity above the required limit of 1/2 may be achievable in an experiment also.

Local Hidden Variables Underpinning of Entanglement and Teleportation

Entangled states whose Wigner functions are non-negative may be viewed as being accounted for by local hidden variables (LHV). Recently, there were studies of Bell's inequality violation (BIQV) for such states in conjunction with the well known theorem of Bell that precludes BIQV for theories that have LHV underpinning. We extend these studies to teleportation which is also based on entanglement. We investigate if, to what extent, and under what conditions may teleportation be accounted for via LHV theory. Our study allows us to expose the role of various quantum requirements. These are, e.g., the uncertainty relation among non-commuting operators, and the no-cloning theorem which forces the complete elimination of the teleported state at its initial port.

Classical Versus Quantum Probability in Sequential Measurements

We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider a quantum mechanical measurement device or the presence of an environment. We then examine the same issues in alternative interpretations of quantum theory. We first show that multi-time probabilities cannot be naturally defined in terms of a frequency operator. We next prove that local hidden variable theories cannot reproduce the predictions of quantum theory for sequential measurements, even when the degrees of freedom of the measuring apparatus are taken into account. Bohmian mechanics, however, does not fall in this category. We finally examine an alternative proposal that sequential measurements can be modelled by a process that does not satisfy the Kolmogorov axioms of probability. This removes contextuality without introducing non-locality, but implies that the empirical probabilities cannot be always defined (the event frequencies do not converge). We argue that the predictions of this hypothesis are not ruled out by existing experimental results (examining in particular the "which way" experiments); they are, however, distinguishable in principle.

Feasibility of testing local hidden variable theories in a Charm factory

It is commonly believed that the local hidden variable theories (LHVTs) can be tested through measuring the Bell inequalities. This scheme, for the massive particle system, was originally set up for the entangled ${K}^{0}{\overline{K}}^{0}$ pair system from the $\ensuremath{\phi}$ factory. In this paper we show that the $J/\ensuremath{\Psi}\ensuremath{\rightarrow}{K}^{0}{\overline{K}}^{0}$ process is even more realistic for this goal. We analyze the unique properties of $J/\ensuremath{\Psi}$ in the detection of basic quantum effects, and find that it is possible to use $J/\ensuremath{\Psi}$ decay as a test of LHVTs in the future $\ensuremath{\tau}$-charm factory. Our analyses and conclusions are generally also true for other heavy onium decays.

Comment on “All quantum observables in a hidden-variable model must commute simultaneously”

Malley discussed {[Phys. Rev. A {\bf 69}, 022118 (2004)]} that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related with the LHV model for quantum events, in general.

Reply to “Comment on ‘All quantum observables in a hidden-variable model must commute simultaneously’” [Phys. Rev. A73, 066101 (2006)

Nagata [Phys. Rev. A 73, 066101 (2006)] questions whether a general no-go theorem of Malley [Phys. Rev. A 69, 022118 (2004)] applies to local hidden variables and outlines a ``counterexample.'' In fact this is not a counterexample at all, but in seeing why it fails we clarify the significance of Malley's result and its relation to other no-go theorems.

The Wigner Function as Distribution Function

Some entangled states have nonnegative Wigner representative function. The latter allow being viewed as a distribution function of local hidden variables. It is argued herewith that the interpretation of expectation values using such distribution functions as local hidden variable theory requires restrictions pertaining to the observables under study. The reasoning lead to support the view that violation of Bell’s inequalities that is always possible for entangled states hinges not only on the states involved but also whether the dynamical variables have their values defined even when they cannot be measured.

Separability of Quantum States vs. Original Bell (1964) Inequalities

All separable states satisfy all Bell-type inequalities, which involve as their assumption only existence of local realistic (local hidden variable) models of the correlations of spatially separated systems, observed by two or more observers making independent decisions on what to measure (free will). The recent observation by Loubenets, that some separable states do not satisfy the original Bell inequality (1964) has no consequences whatsoever for the studies of the relation of separability with local realism. The original Bell inequality was derived using an additional assumption that the local results for a certain pair of local settings reveal perfect Einstein–Podolsky–Rosen (EPR) correlations. Therefore violation of this inequality by some quantum predictions implies that either (i) the predictions do not allow a local realistic model, or (ii) the predictions do not have the required EPR correlations, or finally both (i) and (ii).

Multipartite nonlocal quantum correlations resistant to imperfections

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of superluminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in Einstein-Podolsky-Rosen (EPR) experiments in the presence of noise. We apply our method to an example involving $n$ parties sharing a Greenberger-Horne-Zeilinger-type state on which they carry out local measurements. For this example, we show that for local hidden-variable theories to reproduce the quantum correlations, the amount of superluminal classical communication $c$ and the detector efficiency $\ensuremath{\eta}$ are constrained by $\ensuremath{\eta}{2}^{\ensuremath{-}c∕n}\ensuremath{\leqslant}O({n}^{\ensuremath{-}1∕6})$. This result holds even if the classical models are allowed to make an error with constant probability.

The incompatibility between local hidden variable theories and the fundamental conservation laws

I discuss in detail the result that the Bell’s inequalities derived in the context of local hidden variable theories for discrete quantized observables can be satisfied only if a fundamental conservation law is violated on the average. This result shows that such theories are physically nonviable, and makes the demarcating criteria of the Bell’s inequalities redundant. I show that a unique correlation function can be derived from the validity of the conservation law alone and this coincides with the quantum mechanical correlation function. Thus, any theory with a different correlation function, like any local hidden variable theory, is incompatible with the fundamental conservation laws and space-time symmetries. The results are discussed in the context of two-particle singlet and triplet states, GHZ states, and two-particle double slit interferometry. Some observations on quantum entropy, entanglement, and nonlocality are also discussed.