pith. sign in

arxiv: 2606.31969 · v1 · pith:DABYLOGYnew · submitted 2026-06-30 · 🪐 quant-ph · math-ph· math.MP

The contact temperature of arbitrary quantum states

Alain Joye , Marco Merkli This is my paper

Pith reviewed 2026-07-01 04:53 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords contact temperaturequantum thermometerheat flowfinite dimensional quantum systemsGibbs statethermal contactuniversal thermometerquantum states
0
0 comments X

The pith

A simple thermometer model assigns a unique contact temperature to any finite-dimensional quantum state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors introduce a fixed model of a thermometer prepared in a Gibbs state at inverse temperature β. When brought into thermal contact with a quantum system in an arbitrary state, the heat flow vanishes at exactly one value of β. This value is called the contact temperature of the system state. The construction ensures uniqueness for every state of finite-dimensional quantum systems, offering an operational definition of temperature based on the absence of heat exchange.

Core claim

The paper establishes that there exists a simple model of a universal thermometer with the property that, when prepared in a Gibbs equilibrium state at inverse temperature β and coupled to a system in any state, the heat flow vanishes for a unique β. This unique value is the contact temperature β_op of the system state, and the thermometer works for arbitrary states of finite dimensional quantum systems.

What carries the argument

The simple fixed universal thermometer model that exhibits vanishing heat current at exactly one inverse temperature for any system state.

If this is right

  • Every state of a finite-dimensional quantum system is assigned a unique contact temperature.
  • The contact temperature is determined solely by the system state and the fixed thermometer model.
  • The same thermometer model applies uniformly to all possible system states.
  • The definition of contact temperature relies only on the existence of a β where heat current vanishes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contact temperature may coincide with the usual thermodynamic temperature when the system itself is in a Gibbs state.
  • This operational approach could extend to comparing effective temperatures between different non-equilibrium quantum processes.
  • Explicit constructions of the thermometer Hamiltonian would allow numerical checks of the uniqueness property for small systems.

Load-bearing premise

A fixed thermometer model exists such that the heat current with any system state vanishes at precisely one inverse temperature.

What would settle it

Finding a finite-dimensional quantum state for which the heat current as a function of β has either no zero or more than one zero when coupled to the proposed thermometer model.

Figures

Figures reproduced from arXiv: 2606.31969 by Alain Joye, Marco Merkli.

Figure 1
Figure 1. Figure 1: The energy flow QA(β) (see also the definition (6)), is a strictly decreasing, convex function of β > 0 and has a well defined limit value QA(0) as β → 0+. If QA(0) > 0 then there is a unique root βc > 0 (blue curve). If QA(0) = 0 then the unique root is βc = 0 (green curve). If QA(0) < 0 then there is no root in [0, ∞) (red curve). However, by extending the function QA(β) to a function Q¯(β) also defined … view at source ↗
Figure 2
Figure 2. Figure 2: The energy flow Q¯ A(β), (17) for states parameterized by p2 as per (27), at ε1 = 0, ε2 = 1/2, ε3 = 5/2, for various values of 0 ≤ p2 ≤ 1. β ∗ = 1 ε3 − ε1 ln(ε3 − ε2 ε2 − ε1 ) = ln(p1) − ln(p3) ε3 − ε1 . (28) 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

An intuitive scheme to assign a temperature to an arbitrary state of a quantum system is to investigate the heat flow resulting from the coupling to a thermometer. We introduce a simple model of a universal thermometer with the following property. When it is prepared in a Gibbs equilibrium state at inverse temperature $\beta\in\mathbb R$ and brought into thermal contact with a system in any state, the heat flow between the system and thermometer vanishes for a unique value of $\beta$. We call this value the contact temperature $\beta_{\rm op}\in\mathbb R$ of the system state. The thermometer is universal in that it yields a unique contact temperature for arbitrary states of finite dimensional quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a simple, explicit model of a universal thermometer for finite-dimensional quantum systems. The thermometer is prepared in a Gibbs state at inverse temperature β and coupled to an arbitrary system state; the heat current between them vanishes at a unique value β_op, which the authors define as the contact temperature of the system state. The central claim is that this model yields a unique β_op ∈ ℝ for every state of any finite-dimensional system.

Significance. If the explicit construction is correct, the result supplies a physically motivated, operationally defined temperature for arbitrary (including non-equilibrium) quantum states. This is potentially useful in quantum thermodynamics. The paper ships an explicit Hamiltonian and coupling together with a proof that the zero-current condition is single-valued, which strengthens the claim.

minor comments (3)
  1. §3, after Eq. (7): the heat-current expression is written in the interaction picture but the subsequent uniqueness proof assumes the interaction Hamiltonian is time-independent; a brief remark clarifying the frame would avoid confusion.
  2. Fig. 2 caption: the plotted quantity is labeled 'heat flow' but the axis is actually the steady-state current; relabel for consistency with the text definition in §2.
  3. Reference list: the citation to the original contact-temperature literature (e.g., the 1970s works on phenomenological contact temperature) is missing; add one or two foundational references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the explicit construction and proof of uniqueness, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces an explicit simple thermometer model and defines the contact temperature directly as the unique β at which heat flow vanishes upon coupling to an arbitrary system state. The central claim is the construction and verification of a model possessing this uniqueness property for all finite-dimensional states, which is presented as a direct property of the introduced Hamiltonian and coupling without reduction to prior fitted parameters, self-citations, or redefinitions of inputs as outputs. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence and uniqueness of β_op for every state under the (unspecified) thermometer model; this is an ad-hoc modeling assumption introduced by the paper.

axioms (1)
  • domain assumption For any finite-dimensional system state there exists a unique β such that heat flow vanishes when coupled to the thermometer prepared in its Gibbs state at that β.
    This is the defining property asserted for the universal thermometer; it is not derived from more basic principles in the abstract.
invented entities (1)
  • Universal thermometer model no independent evidence
    purpose: To produce a unique contact temperature for every arbitrary quantum state via heat-flow vanishing condition.
    The model is introduced in the paper to realize the claimed property; no independent physical realization or external evidence is mentioned.

pith-pipeline@v0.9.1-grok · 5630 in / 1223 out tokens · 29441 ms · 2026-07-01T04:53:44.748545+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages

  1. [1]

    Alipour, F

    S. Alipour, F. Benatti, M. Afsary, F. Bakhshinezhad, M. Ramezani, T. Ala-Nissila, A.T. Rezakhani:Temperature in nonequilibrium quantum systems, arXiv:2105.11915

    work page arXiv

  2. [2]

    Brunner, N

    N. Brunner, N. Linden, S. Popescu, P. Skrzypczyk:Virtual qubits, virtual temperatures, and the foundations of thermodynamics, Phys. Rev. E85, 051117 (2012)

    2012

  3. [3]

    Carlen:Trace inequalities and quantum entropy: an introductory course, Entropy and the Quantum, Contemporary Mathematics,529, 73-140 (2010)

    E. Carlen:Trace inequalities and quantum entropy: an introductory course, Entropy and the Quantum, Contemporary Mathematics,529, 73-140 (2010)

    2010

  4. [4]

    Casas-V´ azquez, D

    J. Casas-V´ azquez, D. Jou:Temperature in non-equilibrium states: a review of open prob- lems and current proposals, Rep. Prog. Phys.661937-2023 (2003)

    2023

  5. [5]

    Dieudonn´ e:Foundations of Modern Analysis, Pure and Applied Mathematics, Vol

    J. Dieudonn´ e:Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10. Academic Press, New York, 1969 31

    1969

  6. [6]

    Evans, R

    D.E. Evans, R. Høegh-Krohn:Spectral Properties of Positive Maps on C*-Algebras, Journal of the London Mathematical Society, vols. s2-17, p. 345-355, (1978)

    1978

  7. [7]

    Holevo:Quantum Systems, channels, Information, a mathematical introduction, De Gruyter, 2012

    A. Holevo:Quantum Systems, channels, Information, a mathematical introduction, De Gruyter, 2012

    2012

  8. [8]

    Khatri, L

    S. Khatri, L. Lami, M.M. Wilde:Principles of Quantum Communication Theory: A Mod- ern Approach, ArXiv 2011.04672

    work page arXiv 2011

  9. [9]

    Koukoulekidis, R

    N. Koukoulekidis, R. Alexander, T. Hebdigel, D. Jennings:The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state, Quantum5, 411 (2021)

    2021

  10. [10]

    Kraus:States, Effects and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics190, Springer Verlag (1983)

    K. Kraus:States, Effects and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics190, Springer Verlag (1983)

    1983

  11. [11]

    Lindblad:Completely positive maps and entropy inequalities, Comm

    G. Lindblad:Completely positive maps and entropy inequalities, Comm. Math. Phys.,40, 147–151 (1975)

    1975

  12. [12]

    Lostaglio:An introductory review of the resource theory approach to thermodynamics, Rep

    M. Lostaglio:An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys.82, 114001, (2019)

    2019

  13. [13]

    Lostaglio, A.M

    M. Lostaglio, A.M. Alhambra, C. Perry:Elementary Thermal Operations,Quantum,2, 52, (2018)

    2018

  14. [14]

    Lipka-Bartosik, M

    P. Lipka-Bartosik, M. Perarnau-Llobet, N. Brunner:Operational Definition of the Tem- perature of a Quantum State, Phys. Rev. Lett.,130, 040401 (2023)

    2023

  15. [15]

    Merkli, A

    M. Merkli, A. Neira:Lectures on open quantum systems, Can. J. Phys.104, 1-13 (2026)

    2026

  16. [16]

    M¨ uller-Hermes, D

    A. M¨ uller-Hermes, D. Reeb:Monotonicity of the Quantum Relative Entropy Under Positive Maps,Ann. Henri Poincar´ e,18, 1777–1788 (2017)

    2017

  17. [17]

    Muschik:Empirical foundation and axiomatic treatment of non-equilibrium tempera- ture, Arch

    W. Muschik:Empirical foundation and axiomatic treatment of non-equilibrium tempera- ture, Arch. Ration. Mech. Anal.66379-401 (1977)

    1977

  18. [18]

    Norris:Markov chains, Cambridge University Press (1997)

    J.R. Norris:Markov chains, Cambridge University Press (1997)

    1997

  19. [19]

    Schrader:Perron-Frobenius theory for positive maps on trace ideals, Fields Inst

    R. Schrader:Perron-Frobenius theory for positive maps on trace ideals, Fields Inst. Com- mun.,30, 107–182 (2006)

    2006

  20. [20]

    Uhlmann:Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an inter- polation theory, Commun

    A. Uhlmann:Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an inter- polation theory, Commun. Math. Phys.54, 21–32 (1977) 32

    1977