bibkey: mazurekhorodecki2018decomposability
authors: Paweł Mazurek; Michał Horodecki
year: 2018
title: "Decomposability and convex structure of thermal processes"
doi: 10.1088/1367-2630/aac057
url: https://arxiv.org/abs/1707.06869v2
claim: "Thermal processes on diagonal states are the column-stochastic matrices preserving the Gibbs vector; the paper describes their extreme points for three levels and conjectures that an extremal process that factors into extremal processes on proper subspaces and is not a direct sum cannot satisfy detailed balance."
strata_touched:
  - D5/S3/StatisticalMechanics/ThermalProcessSelfDualFactorizationRefutation
license: citation-only
triage: anchor

Decomposability and convex structure of thermal processes

P. Mazurek and M. Horodecki, arXiv:1707.06869v2; New J. Phys. 20, 053040 (2018). Quotations are from the v2 TeX source.

Thermal processes as matrices:

From above it is visible that every TP can be represented as a left stochastic (i.e., with elements summing to $1$ within each column), Gibbs preserving matrix $T$: $T\mathbf{g}=\mathbf{g}$, where $\mathbf{g}$: $g_i=q_{i,0}/\sum _j{q}_{j,0}$. Without loss of generality here and in the whole paper we assume that the ground state energy of the system is zero: $E_0=0$.

Processes acting on fewer levels, in the proof that the star process is not decomposable:

Now we show that $P$ cannot be decomposed as a composition of TPs, each acting on at most $d-1$ dimensional subspace. Every such decomposition would take a form $P=AB$, where both $A$ and $B$ are TPs.

The conjugation and detailed balance:

It follows that $\stackrel{i,j=Aj,iTe-\beta (Ei-Ej).\; It\; can\; be\; rewritten\; as\; A}{A}=M_{{\rho }_{\beta }}{A}^T{M}_{{\rho }_{\beta }}^{-1}$

Self-duality with respect to such a scalar product ($\tilde{A}=A$) served as a definition of detailed balance for generator of dynamical semigroup

Conjecture 1:

If an extremal TP $C$ for $d$ dimensional space is decomposable into a sequence of extremal TPs $A$, $B$, each acting non-trivially on at most $d-1$ dimensional space: $C=AB$, and $C$ is not a direct sum of extremal TPs from lower dimensional subspaces, then $C$ is not self-dual with respect to the conjugation associated with the operator scalar product.

Verified locator

Source: https://arxiv.org/abs/1707.06869v2

• URL: https://arxiv.org/abs/1707.06869v2 (source manuscript.tex, md5 69e7001dd049721e95968ea93ad550c2):
  • thermal processes as Gibbs-preserving column-stochastic matrices, l. 159;
  • processes acting on fewer levels, l. 208–211;
  • the conjugation and self-duality, l. 417–418;
  • Conjecture 1, l. 425–427.
• DOI: 10.1088/1367-2630/aac057 (https://doi.org/10.1088/1367-2630/aac057).