Abstract Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra ‘quantum correction’ term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.
Abstract Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra ‘quantum correction’ term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.
Abstract Known results for multi-particle entangled systems that link the Grothendieck formalism with the violation of Bell-like inequalities, are generalised to a single quantum system. The Frechet probabilistic inequalities in a single quantum system, are the analogue of Bell-like inequalities in an entangled multipartite system. Kolmogorov probabilities obey Boole’s inequality and the Frechet probabilistic inequalities, but quantum probabilities violate them. Through examples, a link is made between the violation of these inequalities and the Grothendieck formalism in a single quantum system.
Bell's Inequalites structure can be considered in a logical-probabilistic framework, as outlined by Pitowsky [1], where the origin of the inequality was attributed to George Boole’s “conditions of possible experience”. The inequality has then been often renamed the Bell-Boole inequality. An interesting fact is that G. Boole’s method used to establish his probability inequalities stems directly from his mathematical treatment of logic using idempotent symbols (which he called “elective symbols”). The “Boolean Algebra” we consider today bears some major differences with Boole’s original method as thoroughly discussed by Hailperin [2].We proposed recently the “Eigenlogic” approach [3] where we translate Boole’s original logical method to linear algebra using idempotent and also unitary operators and outline the bridges to quantum theories in particular to quantum information [4]. Eigenlogic can thus be a framework for transposing logic in a quantum context, the probabilistic aspects are then simply derived by measurements on logical operators using the Born rule.Bell Inequalities (BI) can be expressed in an operator form where the measurement observables of an experiment are combined to form Bell operators, the outcomes are then derived by using the quantum mechanical mean value (again the Born rule). The popular form adapted for this method is the CHSH form of the BI for spin 1/2 particles or polarized photons with outcomes ±1. Using the isomorphism between idempotent and self-inverse unitary operators (Householder transform) the CHSH Bell operator transforms in Eigenlogic in a combination of idempotent operators from which by using the Born rule one gets the Fine inequality on probabilities.As stated by Khrennikov [5] the BI violation occurs only when incompatible (non-commuting) observables are used in the BI associated experiments. Particularly in the CHSH-BI experiments on an entangled state, mutually unbiased bases (MUB), representing maximally dissimilar contexts, lead to the maximum quantum violation represented by the number 2√2>2. This is the case for the couple of MUB (z,x) associated to Pauli spin operators σ_x and σ_z. In Eigenlogic these (z,x) contexts represent dual logical systems which are at interplay and have an original syntax-semantic interpretation [4]. It has to be outlined that in many quantum algorithms the MUB (z,x), linked by the Hadamard gate which represents the Fourier Transformation between the two bases, provides a quantum algorithmic advantage.The situations of the BI in the so called post-quantum regime are also analyzed by generalizing the concept of PR-Box which relates outputs (a,b) to inputs (x,y) in a two-party correlation by means of the logical equation: a⊕b=x∧y. Using Eigenlogic, families of boxes with all possible combinations of logical functions are analyzed through the equations: f(a,b)=g(x,y), where f and g correspond to the 16 different Boolean functions, giving 256 possible logical equations. The logical equations are transformed in logical operators and used to define the associated Bell operator. 16 cases of the PR-Box type (considering all the possible complementations of the variables a,b,x,y) violate the CHSH-BI maximally with the number 4>2√2 (Tsirelson’s bound). Another family of 32 logical cases gives a CHSH-BI violation of 3.33 also beyond the quantum limit; this is the case for the logical equation: a∨b=x∧y.[1] Pitowsky, I., George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci., 45:95–125 (1994).[2] Hailperin, T. Boole’s Algebra isn’t Boolean algebra. A description using modern algebra, of what Boole really did create. Math. Mag. 54(4): 172–184 (1981).[3] Toffano, Z. Eigenlogic in the Spirit of George Boole. Log. Univers. 14, 175–207 (2020). https://doi.org/10.1007/s11787-020-00252-3[4] Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139. https://doi.org/10.3390/e22020139[5] Khrennikov, A. Contextuality, Complementarity, Signaling, and Bell Tests. Entropy 2022, 24, 1380. https://doi.org/10.3390/e24101380
In contrast to the widespread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements. In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system. For any separable quantum state, we derive quantum analogs of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended Clauser-Horne-Shimony-Holt inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.
Abstract For a multipartite quantum state, the maximal violation of all Bell inequalities constitutes a measure of its nonlocality (Loubenets 2012 J . Math . Phys . 53 , 022201). In the present article, for the maximal violation of Bell inequalities by a pure bipartite state, possibly infinite-dimensional, we derive a new upper bound expressed in terms of the Schmidt coefficients of this state . This new upper bound allows us also to specify general analytical relations between the maximal violation of Bell inequalities by a bipartite quantum state, pure or mixed, and such entanglement measures for this state as ‘negativity’ and ‘concurrence’. To our knowledge, no any general analytical relations between measures for bipartite nonlocality and entanglement have been reported in the literature though, for a general bipartite state, specifically such relations are important for the entanglement certification and quantification scenarios. As an example, we apply our new results to finding upper bounds on nonlocality of bipartite coherent states intensively discussed last years in the literature in view of their experimental implementations.
We perform a detailed analysis of the possible violation of various Bell-type inequalities for systems of vector boson-antiboson pairs. Considering the general case of an overall scalar state of the bipartite system, we identify two distinct classes of such states, and determine the joint probabilities of spin measurement outcomes for each them. We calculate the expectation values of the CHSH, Mermin and CGLMP inequalities and find that while the generalised CHSH inequality is not expected to be violated for any of the scalar states, in the case of the Mermin and CGLMP inequalities the situation is different – these inequalities can be violated in certain scalar states while they cannot be violated in others. Moreover, the degree of violation depends on the relative speed of the two particles.
Bell inequalities reveal when quantum correlations cannot be explained by any local hidden-variable model, and the CHSH inequality is the most fundamental test of this phenomenon. While entangled two-qubit states can violate CHSH inequalities with suitable local measurements, the situation is more subtle for higher-dimensional systems when the allowed observables are physically restricted. In this work, we study bipartite qutrit systems in which each party is restricted to spin-1 observables S(u)=uxSx+uySy+uzSz, with u∈R3 a unit vector. Hanotel and Loubenets conjectured that nonseparable pure states of two qutrits do not violate the CHSH inequality under this measurement restriction. We prove the stronger result that every bipartite state on C3⊗C3, pure or mixed, satisfies the CHSH inequality for all choices of local spin-1 observables. Thus the absence of CHSH violation is not a property of special states, but a structural limitation of the spin-1 measurement family. Our result clarifies a distinction between two-qutrit entanglement and CHSH nonlocality under natural spin-component measurements.