On generalized probabilities: correlation polytopes for automaton logic and generalized urn models, extensions of quantum mechanics and parameter cheats
Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the Cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Thereby we review and emphasize the use of observables which maximally represent the context. (ii) In the second part, we discuss Pitowsky polytopes for automaton logic as well as for generalized urn models and evaluate methods to find the resulting Boole-Bell type (in)equalities. (iii) Finally, so-called ``parameter cheats'' are introduced, whereby parameters are transformed bijectively and nonlinearly in such a way that classical systems mimic quantum correlations and vice versa. It is even possible to introduce parameter cheats which violate the Boole-Bell type inequalities stronger than quantum ones, thereby trespassing the Tsirelson limit. The price to be paid is nonuniformity.
Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the Cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Thereby we review and emphasize the use of observables which maximally represent the context. (ii) In the second part, we discuss Pitowsky polytopes for automaton logic as well as for generalized urn models and evaluate methods to find the resulting Boole-Bell type (in)equalities. (iii) Finally, so-called ``parameter cheats'' are introduced, whereby parameters are transformed bijectively and nonlinearly in such a way that classical systems mimic quantum correlations and vice versa. It is even possible to introduce parameter cheats which violate the Boole-Bell type inequalities stronger than quantum ones, thereby trespassing the Tsirelson limit. The price to be paid is nonuniformity.
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
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted Q , is fundamental for quantum information theory. We review and systematize what is known about Q , and add many details, visualizations, and complete proofs. In particular, we provide a detailed description of the boundary, which consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, i.e., the application of the sine function to each coordinate. This transforms the classical correlation polytope exactly into the correlation body Q , also identifying the boundary structures. The second principle, self-duality, is an isomorphism between Q and its polar dual, i.e., the set of affine inequalities satisfied by all quantum correlations (“Tsirelson inequalities''). The same isomorphism links the polytope of classical correlations contained in Q to the polytope of no-signalling correlations, which contains Q .We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables, and establish a new non-linear inequality for Q involving the determinant of the correlation matrix.
The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell’s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and “death of reality” which are typically linked to Bell’s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell’s type inequalities statistical data from a number of experiments performed under different experimental contexts.
We discuss different formal frameworks for the description of generalized probabilities in statistical theories. We analyze the particular cases of probabilities appearing in classical and quantum mechanics and the approach to generalized probabilities based on convex sets. We argue for considering quantum probabilities as the natural probabilistic assignments for rational agents dealing with contextual probabilistic models. In this way, the formal structure of quantum probabilities as a non-Boolean probabilistic calculus is endowed with a natural interpretation.