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On entropy production of repeated quantum measurements III. Quantum detailed balance

Tristan Benoist, Noé Cuneo, Vojkan Jakšić, Claude-Alain Pillet

TL;DR

The paper advances the understanding of quantum detailed balance by showing that the KMS-based quantum detailed balance (QDB) for a channel is equivalent to time-reversal invariance and zero entropy production for an informationally complete instrument modeling repeated measurements. It introduces instrumental detailed balance (IQDB) and proves its equivalence to QDB for irreducible channels, with IC instruments guaranteeing implementability of time reversal and a vanishing entropy-production rate. The work leverages Stinespring dilations, IC-POVMs, and finitely correlated states to connect channel reversibility with observational statistics, providing both structural results and explicit channel constructions. It also analyzes how the choice of the Adjacency operator $J$ and its square affects permissible η-values, illustrating the necessity of anti-unitaries in certain cases and the nontrivial constraints arising from channel period and covariance properties.

Abstract

In light of the dynamical-systems approach to entropy production in repeated quantum measurements, proposed and illustrated in Commun. Math. Phys. 357, 77-123 (2018) [arXiv:1607.00162] and J. Stat. Phys. 182, 44 (2021) [arXiv:2012.03885], we characterize the KMS quantum detailed balance condition for quantum channels via time-reversal invariance and the vanishing of the entropy production for the associated informationally complete quantum instruments.

On entropy production of repeated quantum measurements III. Quantum detailed balance

TL;DR

The paper advances the understanding of quantum detailed balance by showing that the KMS-based quantum detailed balance (QDB) for a channel is equivalent to time-reversal invariance and zero entropy production for an informationally complete instrument modeling repeated measurements. It introduces instrumental detailed balance (IQDB) and proves its equivalence to QDB for irreducible channels, with IC instruments guaranteeing implementability of time reversal and a vanishing entropy-production rate. The work leverages Stinespring dilations, IC-POVMs, and finitely correlated states to connect channel reversibility with observational statistics, providing both structural results and explicit channel constructions. It also analyzes how the choice of the Adjacency operator and its square affects permissible η-values, illustrating the necessity of anti-unitaries in certain cases and the nontrivial constraints arising from channel period and covariance properties.

Abstract

In light of the dynamical-systems approach to entropy production in repeated quantum measurements, proposed and illustrated in Commun. Math. Phys. 357, 77-123 (2018) [arXiv:1607.00162] and J. Stat. Phys. 182, 44 (2021) [arXiv:2012.03885], we characterize the KMS quantum detailed balance condition for quantum channels via time-reversal invariance and the vanishing of the entropy production for the associated informationally complete quantum instruments.

Paper Structure

This paper contains 22 sections, 26 theorems, 187 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Detailed-balanced Markov chains
  3. Detailed-balanced quantum channels
  4. Repeated quantum measurements
  5. Time reversal and entropy production
  6. Dilations of channels and instruments
  7. Main result
  8. Instrumental detailed balance
  9. On the choice of $J$ and the possible values of $\eta$
  10. Non-uniqueness of $\eta$ and some forbidden values
  11. A class of channels illustrating various constraints
  12. Proofs of main results
  13. Proof of Proposition \ref{['prop:conv_ep']}
  14. Proof of Proposition \ref{['prop:dilation']}
  15. Proof of Proposition \ref{['prop:prop channel']}
  16. ...and 7 more sections

Key Result

Proposition 1.3

Suppose that ER holds. Then:

Figures (4)

  • Figure 1: Illustration of the construction in Section \ref{['sec:xpl value of c']} for several pairs $(d, a_0)$, with ${\mathbbm{Z}_d}$ represented as $d$ points on a circle. The double arrows indicate points that are exchanged by the involution $\sigma$.
  • Figure 2: Illustration of the proof of Proposition \ref{['prop: no unitary no 1']}(i).
  • Figure 3: Illustration of the values of $p_a$ and $\eta_a$ in Table \ref{['tab:valuespaeta']} with $p=4, d=12,~a_0=10$.
  • Figure 4: Illustration of the proofs of Propositions \ref{['prop:anti U identity nu unitary']} and \ref{['prop:unitary no 1']} with $d=6$, $a_0=5$.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 27 more