Signed de Finetti Quasiprobabilities for Finite Fermionic Occupation Statistics: Explicit subset decompositions and exclusion-induced negativity

We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for $α< 0$ . Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter $α$ . Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature $T^{*}$ , at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature $T_{c}$ , the ratio $T^{*}/T_{c}$ depends universally on $α$ . For negative $α$ , corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for $α< 0$. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter $α$. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature $T^{*}$, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature $T_{c}$, the ratio $T^{*}/T_{c}$ depends universally on $α$. For negative $α$, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.

cond-mat.stat-mechCross-listedarXiv:2511.02546Nov 5, 2025

Finite de Finetti Theorem for Infinite-Dimensional Systems

Christian D’Cruz, Tobias J. Osborne, Rüdiger Schack

We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state |Psi/Psi| chosen from a family of subsets {Cn} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {Cn}.We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state |Psi/Psi| chosen from a family of subsets {Cn} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {Cn}.

arXiv:quant-ph/0606139Physical Review LettersApr 19, 200715 citations

Clustering of Fermionic Truncated Expectation Values Via Functional Integration

Manfred Salmhofer B

arXiv:0809.3517Journal of Statistical PhysicsMar 1, 200915 citations

Positive-definite functional integrals for interacting fermions

G. Puddu

Physics Letters ASep 1, 19911 citation

De Finetti Structures in Networks of Quantum Systems

Gavin Hull, Marco Merkli

Consider $N$ identical quantum systems at the vertices of a graph, interacting via pairwise edge couplings. We designate $K$ vertices as a system of interest $\mathrm S$ and the remaining $N-K$ as a reservoir $\mathrm R$ . Under generic connectivity assumptions between $\mathrm S$ and $\mathrm R$ , we show that for $N\to\infty$ , the reduced state of $\mathrm S$ converges to a convex combination of symmetric product states of $K$ single-vertex density matrices -- called a de Finetti representation. Unlike in standard derivations for symmetric (exchangeable) systems, we establish de Finetti representations for quantum networks, which are inherently non-exchangeable. We construct the representations explicitly and quantify the convergence rate. We show that Erdős--Rényi random graphs exhibit the de Finetti structure: as the reservoir grows, the reduced state of any fixed finite subgraph converges almost surely to an explicit deterministic product state. As applications, we establish a general area law for the entanglement entropy of $\mathrm S$ , analyze mean-field and fluctuation scalings of the $\mathrm S$ - $\mathrm R$ interaction, and illustrate our results on qubit networks. We also derive quantitative convergence estimates for information-theoretic quantities, including von Neumann entropy, mutual information, and entanglement within $\mathrm S$ and between $\mathrm S$ and $\mathrm R$ .Consider $N$ identical quantum systems at the vertices of a graph, interacting via pairwise edge couplings. We designate $K$ vertices as a system of interest $\mathrm S$ and the remaining $N-K$ as a reservoir $\mathrm R$. Under generic connectivity assumptions between $\mathrm S$ and $\mathrm R$, we show that for $N\to\infty$, the reduced state of $\mathrm S$ converges to a convex combination of symmetric product states of $K$ single-vertex density matrices -- called a de Finetti representation. Unlike in standard derivations for symmetric (exchangeable) systems, we establish de Finetti representations for quantum networks, which are inherently non-exchangeable. We construct the representations explicitly and quantify the convergence rate. We show that Erdős--Rényi random graphs exhibit the de Finetti structure: as the reservoir grows, the reduced state of any fixed finite subgraph converges almost surely to an explicit deterministic product state. As applications, we establish a general area law for the entanglement entropy of $\mathrm S$, analyze mean-field and fluctuation scalings of the $\mathrm S$-$\mathrm R$ interaction, and illustrate our results on qubit networks. We also derive quantitative convergence estimates for information-theoretic quantities, including von Neumann entropy, mutual information, and entanglement within $\mathrm S$ and between $\mathrm S$ and $\mathrm R$.

math-phCross-listedarXiv:2610.25280Oct 20, 20260 citations

α< 0

. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter

α

. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature

T^{*}

, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature

T_{c}

, the ratio

T^{*}/T_{c}

depends universally on

α

. For negative

α

, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.

Consider

N

identical quantum systems at the vertices of a graph, interacting via pairwise edge couplings. We designate

K

vertices as a system of interest

\mathrm S

and the remaining

N-K

as a reservoir

\mathrm R

. Under generic connectivity assumptions between

\mathrm S

and

\mathrm R

, we show that for

N\to\infty

, the reduced state of

\mathrm S

converges to a convex combination of symmetric product states of

K

single-vertex density matrices -- called a de Finetti representation. Unlike in standard derivations for symmetric (exchangeable) systems, we establish de Finetti representations for quantum networks, which are inherently non-exchangeable. We construct the representations explicitly and quantify the convergence rate. We show that Erdős--Rényi random graphs exhibit the de Finetti structure: as the reservoir grows, the reduced state of any fixed finite subgraph converges almost surely to an explicit deterministic product state. As applications, we establish a general area law for the entanglement entropy of

\mathrm S

, analyze mean-field and fluctuation scalings of the

\mathrm S

\mathrm R

interaction, and illustrate our results on qubit networks. We also derive quantitative convergence estimates for information-theoretic quantities, including von Neumann entropy, mutual information, and entanglement within

\mathrm S

and between

\mathrm S

and

\mathrm R