Local Thermal Operations and Classical Communication

Abstract

In quantum thermodynamics, understanding the interplay between locality, thermal constraints, and communication remains an open challenge. In this manuscript, we introduce Local Thermal Operations and Classical Communication (LTOCC), a novel operational framework that unifies the distant laboratories paradigm with thermodynamic restrictions, defining the fundamental limits on transformations between spatially separated systems. We establish a hierarchy of LTOCC protocols, demonstrating inclusion relations between different levels and revealing their deep connection to semilocal thermal operations. To formalize this framework, we develop thermal tensors and bithermal tensors, extending stochastic and tristochastic tensors to thermodynamic settings and providing new mathematical tools for constrained quantum processes. Finally, we present limitations imposed by LTOCC on single- and multi-copy CHSH scenario, demonstrating no violation in former and a gap between thermal and athermal local operations in the latter with respect to their capability to detect entanglement.

Figures (12)

• Figure 1: Artist's depiction of Alice and Bob using classical communication, in contact with local thermal baths, thus realising the backbone of LTOCC, Local Thermal Operations and Classical Communication (Courtesy of A. de Oliveira Junior).
• Figure 2: Example of majorization and thermomajorization curves for a pair of distributions $\vb{p} = (0.2, 0.33, 0.3, 0.16),\,\vb{q}=(0.45, 0.09, 0.45, 0.01)$ in $d = 4$ with energy levels $E_i = i$ as defined in eq \ref{['cures_def']}. In both examples, the distribution $\vb{p}$ corresponding to the blue curve majorizes the distribution $\vb{q}$ corresponding to the orange curve, $\vb{p}\succ \vb{q},\,\vb{p}\succ_\beta \vb{q}$, since the blue curve lies entirely above the orange one. Notice that in infinite temperature $\beta = 0$, the thermomajorization problem reduces to majorization one since all elbows of the curves happen for the same values of $x$, so it is sufficient to consider only these points.
• Figure 4: Parallel $\text{LTOCC}$ as a special case of $\text{LTOCC}_2+\text{M}$: Parallel protocol can be implemented as a variant of 2-round protocol with memory by limiting first round of the protocol to measurement only and then conditioning the local operations on the memory register of the other party -- Alice on Bob's measurement and Bob on Alice's measurement.
• Figure 5: LTOCC inclusion hierarchy: Each frame corresponds to a different number of rounds in the protocol. In general, variants with shared randomness (+R) contain the ones not including it (Theorems \ref{['thm:strict_sup_LTOCC']} and \ref{['thm:stric_sup_SLTOCC']}), $n$-round protocols contain all $n'$-round protocols for $n'\leq n$ (Observation \ref{['obs:more_rounds_inclusion']}), protocols with memory (+M) contain the ones without it (Observation \ref{['obs:memory_inclusion']}), and protocols without memory (including all one round protocols) form a subset of SLTO (Theorem \ref{['thm:subset_of_SLTO']}).
• Figure 6: Attainable states $\vb{r} = T(\vb{p},\vb{q})$ in dimension $d = 2$ using LTOCC lie between the extreme surfaces generated by $T_1$ (red-orange) and $T_2$ (blue-cyan). Each distribution is characterized by its first coefficient, e.g. $\vb{p} = (p,1-p)$, and the extremal values are denoted by shaded lines. Although fully symmetric for $\beta = 0$, the symmetry is lost for finite temperatures, $\beta > 0$.

Theorems & Definitions (62)

• Definition 1: Majorisation
• Theorem 1: Theorem II.1.10 of Ref. bhatia1996matrix
• Definition 2: $\beta$-ordering
• Theorem 2: Theorem 1.3 of Ref. horodecki2013
• Definition 3
• Definition 4: Bera2021 Definiton 1
• Theorem 3: Bera2021, Supplementary Information, Theorem 11
• Corollary 1: Bera2021, Supplementary Information, Corollary 4
• Corollary 2: Set-closure of SLTO
• Theorem 4