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The von Neumann entropy and information rate for integrable quantum Gibbs ensembles 2

Oliver Johnson, Yuri Suhov

Published Jan 1, 2002• 6 citations
OpenAlex

Abstract

In this and the subsequent paper under the same title a number of exactly sovlable models of quantum statistical mechanics are studied in a context of quantum information theory. The basic model in this paper is a lattice system of ideal (free) particles (bosons or fermions), in the infinite-volume lmit of the grand canonical Gibbs ensemble. We interpet the (limiting) von Neumann entropy rate h as the information rate and use the classical Lempel-Ziv universal coding algorithm to assess the value of h from a single eigenvector W of the Gibbs ensemble density matrix p. We also show how to apply this scheme to the Calogero model.

Related papers

The von Neumann entropy and information rate for integrable quantum Gibbs ensembles, 2

Oliver Johnson, Yuri Suhov

This paper considers the problem of data compression for dependent quantum systems. It is the second in a series under the same title. As in the previous paper, we are interested in Lempel--Ziv encoding for quantum Gibbs ensembles. Here, we consider the canonical ideal lattice Bose- and Fermi-ensembles. We prove that as in the case of the grand canonical ensemble, the (limiting) von Neumann entropy rate $h$ can be assessed, via the classical Lempel--Ziv universal coding algorithm, from a single eigenvector of the density matrix.

arXiv:math-ph/0305016•May 8, 2003•4 citations

The von Neumann entropy and information rate for ideal quantum Gibbs ensembles

Oliver Johnson, Yuri Suhov

A model of a quantum information source is proposed, based on the Gibbs ensemble of ideal (free) particles (bosons or fermions). We identify the (thermodynamic) von Neumann entropy as the information rate and establish the classical Lempel--Ziv universal coding algorithm in Grassberger's form for such a source. This generalises the Schumacher theorem to the case of non-IID qubits.

arXiv:math-ph/0109023•Sep 21, 2001•0 citations

Quantum Fisher Information for Entropy of Gibbs States

Francis J. Headley

We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.

quant-pharXiv:2603.16456•Mar 18, 2026•0 citations

Entropy minimization for many-body quantum systems

Romain Duboscq, Olivier Pinaud

The problem considered here is motivated by a work by B. Nachtergaele and H.T. Yau where the Euler equations of fluid dynamics are derived from manybody quantum mechanics, see [10]. A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here R d , d $\ge$ 1). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.

arXiv:2103.07310•Mar 12, 2021•0 citations

Exact finite reduced density matrix and von Neumann entropy for the Calogero model

Omar Osenda, Federico M. Pont, Anna Okopińska +1 more

The information content of continuous quantum variables systems is usually studied using a number of well known approximation methods. The approximations are made to obtain the spectrum, eigenfunctions or the reduced density matrices that are essential to calculate the entropy-like quantities that quantify the information. Even in the sparse cases where the spectrum and eigenfunctions are exactly known the entanglement spectrum, {\em i.e.} the spectrum of the reduced density matrices that characterize the problem, must be obtained in an approximate fashion. In this work, we obtain analytically a finite representation of the reduced density matrices of the fundamental state of the N-particle Calogero model for a discrete set of values of the interaction parameter. As a consequence, the exact entanglement spectrum and von Neumann entropy is worked out.

arXiv:1510.03644•Journal of Physics A Mathematical and Theoretical•Oct 29, 2015•9 citations

A Short Note on Quantum Entropies

Amelia Carolina Sparavigna

In quantum statistical mechanics, the extension of the classical Gibbs entropy is the von Neumann entropy, obtained from a quantum-mechanical system described by means of its density matrix. Here we shortly discuss this entropy and the use of generalized entropies instead of it

•Jan 11, 2016•1 citation
In this and the subsequent paper under the same title a number of exactly sovlable models of quantum statistical mechanics are studied in a context of quantum information theory. The basic model in this paper is a lattice system of ideal (free) particles (bosons or fermions), in the infinite-volume lmit of the grand canonical Gibbs ensemble. We interpet the (limiting) von Neumann entropy rate h as the information rate and use the classical Lempel-Ziv universal coding algorithm to assess the value of h from a single eigenvector W of the Gibbs ensemble density matrix p. We also show how to apply this scheme to the Calogero model.
This paper considers the problem of data compression for dependent quantum systems. It is the second in a series under the same title. As in the previous paper, we are interested in Lempel--Ziv encoding for quantum Gibbs ensembles. Here, we consider the canonical ideal lattice Bose- and Fermi-ensembles. We prove that as in the case of the grand canonical ensemble, the (limiting) von Neumann entropy rate hhh can be assessed, via the classical Lempel--Ziv universal coding algorithm, from a single eigenvector of the density matrix.
We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.
The problem considered here is motivated by a work by B. Nachtergaele and H.T. Yau where the Euler equations of fluid dynamics are derived from manybody quantum mechanics, see [10]. A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here R d , d ≥\ge≥ 1). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.
The information content of continuous quantum variables systems is usually studied using a number of well known approximation methods. The approximations are made to obtain the spectrum, eigenfunctions or the reduced density matrices that are essential to calculate the entropy-like quantities that quantify the information. Even in the sparse cases where the spectrum and eigenfunctions are exactly known the entanglement spectrum, {\em i.e.} the spectrum of the reduced density matrices that characterize the problem, must be obtained in an approximate fashion. In this work, we obtain analytically a finite representation of the reduced density matrices of the fundamental state of the N-particle Calogero model for a discrete set of values of the interaction parameter. As a consequence, the exact entanglement spectrum and von Neumann entropy is worked out.