Wigner Inequality Research Articles

The Bell Inequality, Inviolable by Data Used Consistently with Its Derivation, Is Satisfied by Quantum Correlations Whose Probabilities Satisfy the Wigner Inequality

It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.

Wigner-approach-enabled detection of multipartite nonlocality using all different bipartitions

Distinct from Bell's approach, Wigner had derived a form of local realist (LR) inequality which is quantum mechanically violated for a bipartite maximally entangled state. Subsequently, this approach was generalized to obtain a multipartite LR inequality. However, the violation of such generalised Wigner's inequality does not guarantee nonlocality between all possible different bipartitions of the multipartite system. In the present work, this limitation has been overcome by formulating a further generalisation of Wigner's approach through the derivation of a set of LR inequalities with respect to all different bipartitions of a N-partite system. Quantum mechanical violations of all individual LR inequalities belonging to such a set would rigorously certify multipartite nonlocality by also providing a finer characterisation of the nature of multipartite nonlocality in the following sense. The quantum mechanical violation of any given inequality of our complete set of LR inequalities would enable identification of the corresponding bipartition which exhibits nonlocality. This is in contrast to other multipartite LR inequalities such as the Svetlichny inequality or its generalisation that cannot be used to detect whether there is any particular bipartition which is nonlocally correlated. The efficacy of the scheme developed in this paper is illustrated for the tripartite and quadripartite states.

The Bell Inequalities: Identifying What Is Testable and What Is Not

The Bell inequalities in three and four correlations are re-derived in general forms showing that three and four data sets, respectively, identically satisfy them regardless of whether they are random, deterministic, measured, predicted, or some combination of these. The Bell inequality applicable to data is thus a purely mathematical result independent of experimental test. Correlations of simultaneously cross-correlated variable pairs do not in general all have the same form, and vary with the physical system considered and its experimental configuration. It is the form of correlations of associated data sets that may be tested, and not whether they satisfy the Bell inequality. In the case of pairs of spins or photons, a third measured or predicted value requires a different experimental setup or predictive computation than is used to obtain data from pairs alone. This is due to the quantum non-commutation of spin and photon measurements when there is more than one per particle of a pair. The Wigner inequality for probabilities, with different probabilities for different variable pairs, may be obtained from the four variable Bell inequality under a simple symmetry condition. Neither the probability or correlation inequality is violated by correlations computed from quantum probabilities based on non-commutation.

Wigner inequalities for testing the hypothesis of realism and concepts of macroscopic and local realism

We propose a new Wigner inequality suitable for test of the hypothesis of realism. We show that this inequality is not identical neither to the well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form. The obtained inequality is suitable for test of realism not only in quantum mechanical systems, but also in quantum field systems. Also we propose a mathematically consistent derivation of the Leggett--Garg inequality in Wigner form, which was recently presented in the literature, for three and $n$ distinct moments of time. Contrary to these works, our rigor derivation uses Kolmogorov axiomatics of probability theory. We pay special attention to the construction and studies of the spaces of elementary outcomes. Basing on the the Leggett--Garg inequality in Wigner form for $n$ distinct moments of time we prove that any unitary evolution of a quantum system contradicts the concept of macroscopic realism. We show that application of the concept of macroscopic realism to any quantum system leads to ``freezing'' of the system in the initial state. It is shown that for a particle with an infinite number of observables the probability to find a pair of the observables in some defined state is zero, even if the operators of these observables commute. This fact might serve as an additional logical argument for the contradiction between quantum theory and classical realism.

Test of the Hypothesis of Realism Using a Modified Version of Wigner Inequality

In the current work we present a modified Wigner inequality for test of the hypothesis of realism. This modified Wigner inequality is suitable for test of realism not only in systems that obey the laws of non-relativistic quantum mechanics, but also in systems that obey the laws of quantum field theory. We show how this inequality can be violated in systems of neutral pseudoscalar mesons.

Maximum violation of Wigner inequality for two-spin entangled states with parallel and antiparallel polarizations

The experimental test of Bell’s inequality is mainly focused on Clauser–Horne–Shimony–Holt (CHSH) form, which provides a quantitative bound, while little attention has been paid to the violation of Wigner inequality (WI). Based on the spin coherent state quantum probability statistics, we in the present paper extend the WI and its violation to arbitrary two-spin entangled states with antiparallel and parallel spin-polarizations. The local part of density operator gives rise to the WI while the violation is a direct result of nonlocal interference between two components of the entangled states. The Wigner measuring outcome correlation denoted by [Formula: see text] is always less than or at most equal to zero for the local realist model ([Formula: see text]) regardless of the specific initial state. On the other hand, the violation of WI is characterized by any positive value of [Formula: see text], which possesses a maximum violation bound [Formula: see text] [Formula: see text]. We conclude that the WI is equally convenient for the experimental test of violation by the quantum entanglement.

Violation of an extended Wigner inequality with high-angular-momentum states

In the study of systems that cannot be described classically, the Wigner inequality, has received only a small amount of attention. In this paper we extend the Wigner inequality---originally derived in 1969---and show how it may be used to contradict local realism with only coincidence detections in the absence of two-outcome measurements---that is, for any system where only one possible result of a pair of potential outcomes can be registered. It thus encapsulates a much broader class of measurement schemes than could previously violate a local-realistic inequality. This is possible due to an assumption of ``extended fairness'' on the measurement outcomes, which we posit is highly plausible. We then apply this inequality to a recently constructed setup with access to entangled pairs of photons with very high angular momenta, in which no previously derived local-realistic inequality could successfully be used without making very broad assumptions. We thus demonstrate the versatility of this inequality under very lossy conditions.

Multipartite Bell-type inequality by generalizing Wigner's argument

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 visibilities for the quadripartite GHZ, Cluster andWstates that determine their respective robustness with respect to the quantum mechanical violation of GWI in the presence of white noise.

Wigner’s inequalities in quantum field theory

We present a relativistic generalization of the Wigner inequality for the scalar and pseudoscalar particles decaying to two particles with spin (fermions and photons.) We consider Wigner's inequality with the full spin anticorrelation (with the nonrelativistic analog), as well as the case with the full spin correlation. The latter case may be obtained by a special choice of the plane of measurement of the spin projections on the direction of propagation of fermions. The possibility for relativistic testing of Bohr's complementarity principle is shown.

QUANTUM CORRELATION BOUNDS FOR WIGNER'S INEQUALITY GENERALIZED PARAMETRIZATION

Bounds of quantum correlation associated to the modified Wigner inequality according to most general possible parametrization are investigated, finding the configuration for its maximal violation. Furthermore we extended this analysis to the implementation of the Ekert's protocol for quantum key distribution based on Wigner

Quantum correlation bounds for optimization of quantum-information experiments: The Wigner inequality case

Violation of modified Wigner inequality in a binary bipartite quantum system allows the discrimination between the quantum world and the classical local-realistic one, and also ensures the security of Ekert-type quantum key distribution protocol. In this paper we study both theoretically and experimentally the bounds of quantum correlation associated to the modified Wigner's inequality finding the optimal experimental configuration for its maximal violation. We also extend this analysis to the implementation of Ekert's protocol.

Experimental eavesdropping attack against Ekert’s protocol based on Wigner’s inequality

We experimentally implemented an eavesdropping attack against the Ekert protocol for quantum key distribution based on the Wigner inequality. We demonstrate a serious lack of security of this protocol when the eavesdropper gains total control of the source. In addition we tested a modified Wigner inequality which should guarantee a secure quantum key distribution.

Modified Wigner inequality for secure quantum-key distribution

In this report we discuss the insecurity with present implementations of the Ekert protocol for quantum-key distribution based on the Wigner Inequality. We propose a modified version of this inequality which guarantees safe quantum-key distribution.

Addendum to “Efficient quantum-key-distribution scheme with nonmaximally entangled states”

The authors call attention to previous work with related result, a quantum-key-distribution scheme based on partial unknown entanglement with Wigner's inequality to establish the security of the quantum channel. The sender chooses any entangled state from a certain set or its equivalent state randomly, and sends a sequence of particles out of each pair to the receiver. The two users each choose between two measurements along the fixed axes. In fact, this is a quantum cryptosystem based on mixed entanglement. The main paper was published in Phys. Rev. A 64, 032305 (2001).

Quantum cryptography with entangled photons

By realizing a quantum cryptography system based on polarization entangled photon pairs we establish highly secure keys, because a single photon source is approximated and the inherent randomness of quantum measurements is exploited. We implement a novel key distribution scheme using Wigner's inequality to test the security of the quantum channel, and, alternatively, realize a variant of the BB84 protocol. Our system has two completely independent users separated by 360 m, and generates raw keys at rates of 400-800 bits/s with bit error rates around 3%.

Scattering theory for quasi-one-dimensional tunneling structures

Quantum theoretical studies of semiconductor microstructures are most naturally done in terms of one-dimensional scattering states, which are characterized far from a structure by k-dependent reflection and transmission amplitudes. We have investigated these states using integral forms such as the Lippmann-Schwinger equation. These allow us to obtain global properties of the states and provide the basis for a formal scattering theory of the kind that has been developed for the conventional problem of three-dimensional potential scattering. We find orthonormality relations for the scattering states, the one-dimensional analogues of Wigner's inequality and Levinson's theorem, and associated properties of the complex-momentum transmission amplitude.

Causality in Nonrelativistic Quantum Scattering

A new formulation of the causality condition, in terms of the total wave function at a finite distance from the scatterer, is given. This condition is employed to derive all known analytic properties of the scattering amplitude in both variables (energy and scattering angle or momentum transfer) for a scatterer of finite radius. It is shown that all the assumptions are valid in the particular case of scattering by a local potential. The relations between causality, Wigner's inequality, and completeness are discussed.

Derivation of the van Kampen's causality condition for complex potentials and related problems

Following a method originally due toSaavedra, van Kampen's causality condition for scattering by complex potentials was derived. Together with this problem, both Wigner inequality and Levinson theorem are also discussed in the context of scattering by regular and singular complex potentials.