Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Probabilistic Entanglement of Quantum Observables

Andrei Yu. Khrennikov

Published Nov 28, 2024• 0 citations
Publisher

Abstract

In this chapter contextual probabilistic entanglement is represented withinthe Hilbert space formalism. The notion of entanglement is clarified anddemystified through decoupling it from the tensor product structure andtreating it as a constraint posed by probabilistic dependence of quantum observablesA and B. In this framework, it is meaningless to speak aboutentanglement without pointing to the fixed observables A and B, so thisis AB-entanglement. Dependence of quantum observables is formalized asnon-coincidence of conditional probabilities. Starting with this probabilisticdefinition, we achieve the Hilbert space characterization of the AB-entangledstates as amplitude non-factorisable states. In the tensor productcase, AB-entanglement implies standard entanglement, but not vice versa.AB-entanglement for dichotomous observables is equivalent to their correlation. Finally, observables entanglement is compared with dependence of random variables in classical probability theory.

DOI
10.1017/9781009313469.014

Related papers

Entanglement of Observables: Quantum Conditional Probability Approach

Andrei Yu. Khrennikov, Irina Basieva

Abstract This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it is meaningless to speak about entanglement without pointing to the fixed observables A and B, so this is AB-entanglement. Dependence of quantum observables is formalized as non-coincidence of conditional probabilities. Starting with this probabilistic definition, we achieve the Hilbert space characterization of the AB-entangled states as amplitude non-factorisable states. In the tensor product case, AB-entanglement implies standard entanglement, but not vise verse. AB-entanglement for dichotomous observables is equivalent to their correlation, i.e., $$\langle AB\rangle _{\psi} \not = \langle A\rangle _{\psi} \langle B\rangle _{\psi} .$$ ⟨ A B ⟩ ψ ≠ ⟨ A ⟩ ψ ⟨ B ⟩ ψ . We describe the class of quantum states that are $$A_{u} B_{u}$$ A u B u -entangled for a family of unitary operators (u). Finally, observables entanglement is compared with dependence of random variables in classical probability theory.

arXiv:2303.12393•Foundations of Physics•Sep 15, 2023•12 citations

Introduction to Quantum Theory

Andrei Yu. Khrennikov

This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.

•Jul 30, 2020•0 citations

No extension of the Quantum Tensor Product admits a Superposition principle

Vincenzo Fiorentino, Kuntal Sengupta

Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.

quant-pharXiv:2608.17572•Aug 19, 2026•0 citations

Conditional probability framework for entanglement and its decoupling from tensor product structure

Irina Basieva, Andrei Yu. Khrennikov

Abstract Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B.

arXiv:2205.11510•Journal of Physics A Mathematical and Theoretical•Aug 22, 2022•14 citations

Entanglement beyond tensor product structure: algebraic aspects of quantum non-separability

Łukasz Derkacz, Marek Gwóźdź, Lech Jakóbczyk

An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.

arXiv:1109.0818•Journal of Physics A Mathematical and Theoretical•Dec 8, 2011•12 citations

Extended tensor products and generalization of the notion of entanglement

Andrei Yu. Khrennikov, Elemér Elad Rosinger

Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.

arXiv:1203.0412•AIP conference proceedings•Jan 1, 2012•0 citations
In this chapter contextual probabilistic entanglement is represented withinthe Hilbert space formalism. The notion of entanglement is clarified anddemystified through decoupling it from the tensor product structure andtreating it as a constraint posed by probabilistic dependence of quantum observablesA and B. In this framework, it is meaningless to speak aboutentanglement without pointing to the fixed observables A and B, so thisis AB-entanglement. Dependence of quantum observables is formalized asnon-coincidence of conditional probabilities. Starting with this probabilisticdefinition, we achieve the Hilbert space characterization of the AB-entangledstates as amplitude non-factorisable states. In the tensor productcase, AB-entanglement implies standard entanglement, but not vice versa.AB-entanglement for dichotomous observables is equivalent to their correlation. Finally, observables entanglement is compared with dependence of random variables in classical probability theory.
Abstract This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it is meaningless to speak about entanglement without pointing to the fixed observables A and B, so this is AB-entanglement. Dependence of quantum observables is formalized as non-coincidence of conditional probabilities. Starting with this probabilistic definition, we achieve the Hilbert space characterization of the AB-entangled states as amplitude non-factorisable states. In the tensor product case, AB-entanglement implies standard entanglement, but not vise verse. AB-entanglement for dichotomous observables is equivalent to their correlation, i.e., ⟨AB⟩ψ≠⟨A⟩ψ⟨B⟩ψ.\langle AB\rangle _{\psi} \not = \langle A\rangle _{\psi} \langle B\rangle _{\psi} .⟨AB⟩ψ​=⟨A⟩ψ​⟨B⟩ψ​. ⟨ A B ⟩ ψ ≠ ⟨ A ⟩ ψ ⟨ B ⟩ ψ . We describe the class of quantum states that are AuBuA_{u} B_{u}Au​Bu​ A u B u -entangled for a family of unitary operators (u). Finally, observables entanglement is compared with dependence of random variables in classical probability theory.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.
Abstract Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B.
An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.
Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.