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

arXiv:quant-ph/0408193

Distinguishability of non-orthogonal density matrices does not imply violations of the second law

PierGianLuca Porta Mana

Published Aug 31, 2004• 0 citations
arXivHTMLPDF

Abstract

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

DOI
10.48550/arxiv.quant-ph/0408193

Related papers

On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge

PierGianLuca Porta Mana, A. Maansson, Gunnar G. Bjork

Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.

arXiv:quant-ph/0505229•May 31, 2005•1 citation

Entropy in nonlinear quantum mechanics. II

Paul B. Slater

For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.

Journal of Physics A Mathematical and General•Aug 7, 1992•2 citations

Probability, Entropy, and Quantum

Wojciech H. Zurek

The second law of thermodynamics imposes severe constraints on the formal structure of quantum theory. In particular, that law would be violated if it were possible to distinguish non-orthogonal states or if SchrOdinger's equation were nonlinear.

CRC Press eBooks•Mar 8, 2018•0 citations

Entropy of Quantum States

Paolo Facchi A, Giovanni Gramegna, Arturo Konderak

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.

arXiv:2104.12611•Entropy•May 21, 2021•10 citations

The non-conditional nature of the Cerf-Adami inequalities and implications for thermodynamics

Ian T. Durham

We show that the Cerf-Adami inequalities do not necessarily depend on conditional entropies nor any reference to Markov chains. While the latter are not explicit in the original form, they are often implied in certain derivations. We also show that these inequalities are intimately related to at least one interpretation of the second law of thermodynamics. The combination of these results provides added insight into why some quantum systems violate the Cerf-Adami inequalities thereby improving our understanding of the quantum-classical boundary. As a result we suggest that the second law may serve as some type of boundary condition on classical knowledge.

arXiv:0801.0403•Jan 2, 2008•1 citation

Thermodynamics of readout devices and semiclassical gravity

Samuel Fedida, Adrian Kent

We analyze the common claim that nonlinear modifications of quantum theory necessarily violate the second law of thermodynamics. We focus on hypothetical extensions of quantum theory that contain readout devices. These black boxes provide a classical description of quantum states without perturbing them. They allow quantum state cloning, though in a way consistent with the relativistic no-signalling principle. We review the existence of such devices in the context of Møller-Rosenfeld semiclassical gravity, which postulates that the gravitational field remains classical and is sourced by the expectation value of a quantum energy-momentum tensor. We show that the definition of information in the models examined in this paper deviates from that given by von Neumann entropy, and that claims of second law violations based on the distinguishability of nonorthogonal states or on violations of uncertainty principles fail to hold in such theories.

arXiv:2509.07096•Physical Review A•Dec 16, 2025•0 citations
The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.
Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.
For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
We show that the Cerf-Adami inequalities do not necessarily depend on conditional entropies nor any reference to Markov chains. While the latter are not explicit in the original form, they are often implied in certain derivations. We also show that these inequalities are intimately related to at least one interpretation of the second law of thermodynamics. The combination of these results provides added insight into why some quantum systems violate the Cerf-Adami inequalities thereby improving our understanding of the quantum-classical boundary. As a result we suggest that the second law may serve as some type of boundary condition on classical knowledge.
We analyze the common claim that nonlinear modifications of quantum theory necessarily violate the second law of thermodynamics. We focus on hypothetical extensions of quantum theory that contain readout devices. These black boxes provide a classical description of quantum states without perturbing them. They allow quantum state cloning, though in a way consistent with the relativistic no-signalling principle. We review the existence of such devices in the context of Møller-Rosenfeld semiclassical gravity, which postulates that the gravitational field remains classical and is sourced by the expectation value of a quantum energy-momentum tensor. We show that the definition of information in the models examined in this paper deviates from that given by von Neumann entropy, and that claims of second law violations based on the distinguishability of nonorthogonal states or on violations of uncertainty principles fail to hold in such theories.