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Connecting classical finite exchangeability to quantum theory

Alessio Benavoli, Alessandro Facchini, Marco Zaffalon

TL;DR

The paper builds a bridge between classical finite exchangeability and quantum theory by presenting two representation schemes based on quasi-probabilities: a signed-measure form and a Bernstein-polynomial/ quasi-expectation form. It proves that finitely exchangeable probabilities can be captured by a quantum-like representation using boson-symmetric density matrices, enabling simulation of finite exchangeable sequences with bosonic systems and linking de Finetti-type results to second-quantized formalisms. A concrete application to entanglement for indistinguishable bosons demonstrates how the exchangeability framework clarifies (and sometimes challenges) proposed locality and distinguishability properties, and motivates reformulations that avoid paradoxes. The work highlights a conceptual convergence between probability theory and quantum information, offering tools to study uncertainty measures and potentially extending to fermions, with practical implications for entanglement notions in IPs and for quantum-inspired representations of classical probabilistic models.

Abstract

Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.

Connecting classical finite exchangeability to quantum theory

TL;DR

The paper builds a bridge between classical finite exchangeability and quantum theory by presenting two representation schemes based on quasi-probabilities: a signed-measure form and a Bernstein-polynomial/ quasi-expectation form. It proves that finitely exchangeable probabilities can be captured by a quantum-like representation using boson-symmetric density matrices, enabling simulation of finite exchangeable sequences with bosonic systems and linking de Finetti-type results to second-quantized formalisms. A concrete application to entanglement for indistinguishable bosons demonstrates how the exchangeability framework clarifies (and sometimes challenges) proposed locality and distinguishability properties, and motivates reformulations that avoid paradoxes. The work highlights a conceptual convergence between probability theory and quantum information, offering tools to study uncertainty measures and potentially extending to fermions, with practical implications for entanglement notions in IPs and for quantum-inspired representations of classical probabilistic models.

Abstract

Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
Paper Structure (16 sections, 8 theorems, 120 equations, 4 figures)

This paper contains 16 sections, 8 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The classical de Finetti's representation theorem
  3. First representation
  4. Second representation
  5. Making the connection with QT
  6. Application
  7. Local operators
  8. Analogy with finite/infinite exchangeability
  9. Effective distinguishability
  10. Analogy with finite/infinite exchangeability
  11. Conclusions
  12. Quasi-expectation operator
  13. Proofs
  14. Proof of Theorem \ref{['th:repr41']}
  15. Equivalent definitions of Bose-symmetric density matrix
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\Omega$ be the possibility spaces given by the six faces of a dice, and let $(t_{1}, \dots, t_{r})$ be a sequence of random variables that is infinitely exchangeable with probability measure $P$. Then there exists a distribution function $q$ such that where ${\boldsymbol \theta}=[\theta_1,\theta_2,\dots,\theta_6]^\top$ are the probabilities of the corresponding faces and whose values belong

Figures (4)

  • Figure 1: Value of $v_n$ as a function of $n$. The red line corresponds to $v_{\infty}$.
  • Figure 2: The worst-case marginal expectation for two rolls, which are part of a longer finitely exchangeable sequence of additional $0,1,2,\dots$ rolls, converges to the classical probability value.
  • Figure 3: Value of $v_n$ as a function of $n$. The red line corresponds to $v_{\infty}$.
  • Figure 4: The worst-case marginal expectation for two bosons, which are part of a longer indistinguishable sequence of additional $0,1,2,\dots$ bosons, converges to the classical probability value.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Proposition 1: finetti1937
  • Example 1
  • Example 2
  • Proposition 2: kerns2006definetti
  • Proposition 3: zaffalon2019b
  • Remark 1
  • Example 3
  • Example 4
  • ...and 16 more