Local Thermal Operations and Classical Communication

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.

💡 Research Summary

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This paper introduces a novel operational framework called Local Thermal Operations and Classical Communication (LTOCC), which merges the concepts of locality, thermodynamic constraints, and classical communication in quantum thermodynamics. The authors begin by reviewing the resource theory of thermal operations (TO) and its infinite‑temperature limit, noisy operations (NO), emphasizing the role of Gibbs‑preserving stochastic matrices and the thermomajorisation order for energy‑incoherent states. They then define semilocal thermal operations (SL TO), a middle ground where distinct local heat baths at different temperatures may interact through arbitrary non‑local (classical or quantum) channels while each local interaction respects energy conservation. The paper proves that SL TO is convex and closed under composition (Theorems 4 and 5) and provides auxiliary results on coherence evolution under SL TO.

Building on SL TO, the authors construct the LTOCC framework by allowing parties to exchange classical messages, share randomness, and use memory across multiple rounds. They organize LTOCC into a hierarchy of protocols—ranging from zero‑communication (LTOCC₀) to one‑round (LTOCC₁) and arbitrarily many rounds (LTOCC_∞)—and prove inclusion relations between these layers (Section IV C, Theorem 7). A conjecture (Conjecture 1) states that, in the infinite‑round limit for energy‑incoherent states, the sets of transformations achievable by LTOCC and SL TO coincide.

A central technical contribution is the introduction of “thermal tensors” and “bithermal tensors.” Thermal tensors generalize stochastic matrices to higher‑order maps that take several probability vectors as input and output a new vector, while preserving the Gibbs state in every hyper‑column. Bithermal tensors are the thermodynamic analogue of tristo‑stochastic tensors; they are symmetric under β‑ordering and form the vertices of a thermal version of the Birkhoff polytope. Section V explores their structural properties, providing preliminary results on the geometry of the thermal Birkhoff polytope.

The paper then investigates non‑local correlations generated under LTOCC. Section VI A shows that, starting from product states that are diagonal in the Hamiltonian basis (energy‑incoherent), LTOCC can create correlations, especially when memory is employed across multiple rounds. A single‑round protocol yields limited correlations, whereas two‑ or more‑round protocols with memory can amplify them significantly.

In the CHSH Bell‑inequality scenario, the authors demonstrate two key limitations. First, in the single‑copy setting, any LTOCC protocol respects the CHSH bound; no violation is possible, highlighting that thermal constraints suppress Bell non‑locality detection for a single copy. Second, in the multi‑copy regime, there exists a gap between the maximal CHSH value attainable by unrestricted LOCC (which reaches the Tsirelson bound) and that attainable by thermally restricted LOCC (LTOCC). This gap evidences that thermal restrictions reduce the power of local operations to reveal entanglement.

Supplementary material includes: (A) a discussion of cooling maps as the zero‑temperature limit of Gibbs‑preserving matrices; (B) an alternative derivation of cooling maps; (C) detailed one‑ and two‑round LTOCC protocols and their placement within composite resource theories; (D) additional proofs of inclusion relations between LTOCC and SL TO; (E) further analysis of coherence dynamics under both frameworks; and (F) an examination of the realizability of classical logic gates (CNOT, SWAP) within LTOCC, showing inherent limitations due to thermal constraints.

Overall, the work establishes LTOCC as a comprehensive resource theory that captures the interplay of locality, thermodynamics, and classical communication. It provides new mathematical tools (thermal and bithermal tensors) for describing constrained quantum processes, clarifies the hierarchy of operational capabilities under different communication resources, and reveals fundamental limits on entanglement detection when thermal laws are enforced. The results open avenues for future research on multipartite systems, continuous thermal flows, and experimental implementation of LTOCC protocols.