Semilocal Thermal Operations

• Semilocal Thermal Operations are a class of thermodynamic maps defined by intermediate locality constraints between fully local operations and global Gibbs-preserving dynamics.
• They underpin catalytic channels and dual-unitary circuits, providing both an information-theoretic framework and experimentally implementable two-level primitives.
• Studies reveal strict hierarchies among channel classes and controlled deviations from ideal settings, clarifying their role in quantum thermodynamic control.

Searching arXiv for the cited papers and related work on semilocal thermal operations. Semilocal thermal operations denote a family of thermodynamic maps defined by locality constraints that are weaker than fully local thermal operations and stronger than unconstrained joint Gibbs-preserving dynamics. In the recent literature, the term appears in several distinct but related senses. In one line of work, it refers to catalytic channels whose environment remains locally invariant for every initial state of the system, providing an information-theoretic idealization of heat-bath behavior (Lie et al., 22 Jul 2025). In another, it denotes experimentally implementable two-level primitives—Level Transformations and Partial Level Thermalizations—that suffice to realize every block-diagonal state transition allowed by Thermal Operations (Perry et al., 2015). A further usage concerns collisional models with inhomogeneous reservoirs, where the induced channel is generated by semilocal interactions with reservoir constituents and deviates perturbatively from ideal Thermal Operations (Shu et al., 2019). In bipartite resource theories, semilocal thermal operations are also defined as joint maps compatible with local Gibbs constraints, either through two baths and weighted-heat conservation (Czartowski et al., 8 Jul 2025) or through Local Thermal Operations and Classical Communication together with marginal Gibbs preservation (Bistroń et al., 2024). These formulations are not identical, but they are linked by a common theme: thermodynamic control mediated by local or semilocal structure rather than arbitrary global access.

1. Information-theoretic origin and relation to thermal operations

A central result of "Thermal operations from informational equilibrium" is that Thermal Operations can be characterized by a dilation property expressed entirely in quantum-information-theoretic terms (Lie et al., 22 Jul 2025). For a system $S$ and bath $B$, an open-system channel is modeled by

$\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$

The defining informational equilibrium condition is that the joint equilibrium state is stationary under the dilation unitary: $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$ equivalently,

$$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$

The paper states an equivalence: a channel $T$ on $S$ admits such an equilibrating dilation if and only if it is exactly a Thermal Operation at temperature $\beta^{-1}$ (Lie et al., 22 Jul 2025). When the fixed point $\gamma_S$ has full rank, the argument proceeds through three steps stated explicitly in the source summary: $[\,U,\gamma_S\otimes\gamma_B\,]=0$, covariance under the induced time-translation on $B$0, and hence energy conservation $B$1. This identifies informational equilibrium with the standard energy-preserving dilation picture of Thermal Operations.

This characterization matters for semilocal notions because it separates two requirements that are often conflated. Thermal Operations demand bath invariance only when the system begins in its Gibbs state. Semilocal or catalytic variants strengthen that condition by imposing environment invariance for all system inputs. This suggests that semilocality is best understood not as a minor implementation detail, but as a structural strengthening of the dilation constraint.

2. Catalytic channels as semilocal heat-bath idealizations

In the same work, a channel $B$2 on $B$3 is called catalytic if there exists a unitary $B$4 on $B$5 and a full-rank catalyst $B$6 such that, for every input $B$7,

$B$8

This is the semilocal condition in the strongest sense discussed in the paper: the environment remains locally invariant for every system input (Lie et al., 22 Jul 2025).

The paper gives two equivalent characterizations. First, $B$9 is catalytic if and only if no correlation ever flows from $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$0 into $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$1, meaning that for any reference $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$2 and joint state $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$3,

$\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$4

Second, $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$5 is catalytic if and only if the partial transpose $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$6 is itself unitary, up to an inconsequential swap (Lie et al., 22 Jul 2025).

The operational distinction from ordinary Thermal Operations is explicit. Thermal Operations demand invariance only when the system starts in equilibrium. Catalytic operations demand invariance of the bath for all system inputs. The source further states that the stronger semilocal invariance allows extra channels beyond random unitaries but still fewer than all doubly stochastic maps in the trivial-Hamiltonian limit (Lie et al., 22 Jul 2025).

A recurrent misconception is that catalytic assistance necessarily enlarges the set of Thermal Operations in a robust way. The source rejects this in the finite-dimensional setting: there is no “robust catalytic advantage” within the set of Thermal Operations, because any attempt to use a finite-dimensional catalyst in a thermal operation can be absorbed into a larger system-plus-bath Thermal Operation (Lie et al., 22 Jul 2025). A plausible implication is that the catalytic notion is not merely an augmentation of Thermal Operations, but a different semilocal idealization with its own geometry.

3. Strict hierarchy in the fully degenerate case

For fully degenerate Hamiltonians, all Gibbs states are maximally mixed and energy conservation reduces to the requirement that channels be doubly stochastic. In this setting, the paper establishes the strict hierarchy

$\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$7

where MU denotes mixed-unitary channels, CAT catalytic channels, $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$8 equilibrating channels that are doubly stochastic and are identified with strongly factorizable maps, $\mathcal{E}(\rho_S)\;=\;\Tr_B\bigl[\,U\bigl(\rho_S\otimes\gamma_B\bigr)U^\dagger\bigr].$9 factorizable maps in the von Neumann-algebra sense, and DS all doubly stochastic maps (Lie et al., 22 Jul 2025).

The strictness arguments are also summarized in the source. Mixed-unitary channels coincide with non-degenerate catalytic or equilibrating dilations: if the catalyst spectrum is nondegenerate, the channel must be mixed unitary. The strict inclusion $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$0 is witnessed by real-entry Schur-multiplier examples that admit a maximally mixed catalytic dilation but are known not to be mixed-unitary. The strict inclusion $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$1 follows because any non-unitary catalytic dilation must decompose as a nontrivial convex combination of doubly stochastic maps, so a doubly stochastic map that is extremal in the convex set but not unitary cannot be catalytic. Finally, the chain $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$2 strongly factorizable $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$3 is taken from operator-algebra theory (Lie et al., 22 Jul 2025).

This hierarchy clarifies the status of semilocal operations in the infinite-temperature limit. They interpolate between random unitaries and broader Gibbs-preserving or doubly stochastic dynamics, but they do not exhaust the latter. This suggests that semilocal invariance is a meaningful restriction even when energetic structure is absent.

4. Dual-unitary circuits and semilocality

The catalytic criterion $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$4 unitary links semilocal thermal structure to the theory of dual-unitary circuits. The source defines a four-leg tensor $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$5 as dual-unitary if both $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$6 and its reshuffling or partial transpose $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$7 are unitary (Lie et al., 22 Jul 2025). It then states that the requirement that $$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$8 be unitary is exactly the dual-unitarity condition, up to a swap of legs. Catalytic unitaries are therefore in one-to-one correspondence with dual-unitary gates.

This connection places semilocal heat-bath idealizations in direct contact with the literature on solvable, maximally chaotic circuits. The source describes this as tying the semilocal heat-bath idealization to “the emergent solvable, maximally chaotic circuits built from dual-unitary gates” (Lie et al., 22 Jul 2025). The significance is conceptual as much as operational: the same algebraic condition that forbids information flow from the system into the catalyst also characterizes a class of exactly tractable many-body gates.

The same source frames the semilocal heat bath as the quantum analogue of a classical ideal bath that never changes under arbitrarily large finite-size interactions. Under that idealization, the catalytic channel is exactly the requirement that the environment remain locally invariant for every system input. As stated in the source, this forces one into the doubly stochastic or Gibbs-preserving framework but still leaves strictly more operations than random unitaries (Lie et al., 22 Jul 2025).

5. Two-level semilocal primitives and constructive reachability

A different use of semilocal thermal operations appears in "A sufficient set of experimentally implementable thermal operations" (Perry et al., 2015). There, semilocality refers to control primitives that act only on one or two energy levels at a time, yet suffice to implement any block-diagonal state transformation permitted by the full resource theory of Thermal Operations.

The two primitives are Level Transformations and Partial Level Thermalizations. For a finite-dimensional system with Hamiltonian

$$U\,\bigl(\,\gamma_S\otimes\gamma_B\bigr)\,U^\dagger \;=\;\gamma_S\otimes\gamma_B,$$9

a Level Transformation $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$0 shifts $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$1 while leaving the state $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$2 unchanged. If $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$3 for all $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$4, the worst-case work cost is

$$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$5

A Partial Level Thermalization $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$6 acts on a chosen two-level subspace $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$7, replacing $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$8 by a partial Gibbs mixture of strength $$U\;\bigl(\,I_S\otimes\gamma_B\bigr)\;U^\dagger \;=\;I_S\otimes\gamma_B.$$9, while leaving other populations fixed (Perry et al., 2015).

The paper states that $T$0 is a CPTP map commuting with the free Hamiltonian and hence a Thermal Operation. The constructive protocol then uses only LT’s, PLT’s on two levels at a time, and at most one Gibbs-ancilla qubit to implement any block-diagonal transition $T$1 satisfying thermo-majorization, equivalently the family of inequalities

$T$2

When $T$3 and $T$4 share the same $T$5-ordering, at most $T$6 PLT’s suffice. When the $T$7-orderings differ, the protocol appends a single thermal qubit and uses Partial Isothermal Reversible Processes, defined as alternating infinitesimal LT’s and PLT’s on two levels, to align thermo-majorization elbows without work cost in the continuum limit (Perry et al., 2015).

The experimental motivation is explicit. Partial Level Thermalization can be implemented by tuning a weak system-bath coupling to be resonant only with a chosen Bohr frequency and waiting for a controlled time, or by a partial SWAP between the system subspace and an identical two-level bath element. Level Transformation is realized by time-dependent control of the system Hamiltonian, such as Stark or Zeeman shifts. The source concludes that one “never needs fine control over the entire bath—only semilocal spectral addressing suffices” (Perry et al., 2015). In this usage, semilocality does not define a new resource-theoretic class of channels; it identifies a sufficient operational toolkit for realizing the standard Thermal Operations resource theory on energy-diagonal states.

6. Inhomogeneous reservoirs and bipartite semilocal frameworks

The term also appears in two further contexts that extend the standard one-bath setting.

In "Almost thermal operations: inhomogeneous reservoirs," semilocal thermal operations arise in a collisional model where a system qudit sequentially interacts with reservoir qudits through two-body partial-swap unitaries

$T$8

The reservoir is allowed small inhomogeneities either in the local Hamiltonians or in the local temperatures, parameterized by i.i.d. random variables $T$9 with mean zero and variance $S$0 (Shu et al., 2019). The induced ensemble-averaged channel is

$S$1

and for diagonal states one has after $S$2 steps

$S$3

with fixed point

$S$4

The paper proves that, under either the inhomogeneous-temperature or inhomogeneous-Hamiltonian model,

$S$5

so the semilocal model differs from the ideal Thermal Operation channel only at second order in the inhomogeneity parameter (Shu et al., 2019). At the same time, the usual generalized free-energy monotones can fail: the source states that $S$6 always after a single collision, but for $S$7 one can find $S$8 such that $S$9, and even $\beta^{-1}$0 when $\beta^{-1}$1 in the inhomogeneous-temperature case. This makes semilocality here a controlled deviation from the ideal one-bath resource theory rather than a mere implementation constraint.

In bipartite settings, semilocal thermal operations are defined as joint maps constrained by local thermodynamic structure. "A Study in Thermal: Advantage framework for resource engines" defines a semilocal thermal operation on $\beta^{-1}$2 using two baths at inverse temperatures $\beta^{-1}$3 and $\beta^{-1}$4, with a joint energy-preserving unitary satisfying both total-energy conservation and a weighted-heat or entropy constraint (Czartowski et al., 8 Jul 2025). In the energy-diagonal basis, the population map $\beta^{-1}$5 is required to satisfy

$\beta^{-1}$6

The same source states the resource-theoretic inclusion

$\beta^{-1}$7

generally strict, and gives a faithful monotone

$\beta^{-1}$8

which cannot increase under bare STOs (Czartowski et al., 8 Jul 2025).

A related but distinct formulation appears in "Local Thermal Operations and Classical Communication," where semilocal thermal maps are bipartite CPTP maps $\beta^{-1}$9 that never disturb either marginal Gibbs state whenever the other subsystem is at equilibrium (Bistroń et al., 2024). In the energy-eigenbasis, these maps are represented by a four-index stochastic tensor $\gamma_S$0 obeying local Gibbs-preserving constraints. The paper introduces thermal tensors and bi-thermal tensors as classical analogues of such maps, establishes a hierarchy

$\gamma_S$1

and derives thermo-majorization-based necessary conditions for transformations of population vectors (Bistroń et al., 2024). These bipartite constructions show that semilocality can encode either access to multiple baths, local Gibbs preservation, or restricted communication-assisted implementations, depending on the operational problem.

Across these usages, semilocal thermal operations consistently mark an intermediate regime between strictly local thermodynamic control and arbitrary global Gibbs-preserving dynamics. The specific mathematical definition changes with context, but the recurring lesson is stable: semilocal constraints are strong enough to generate nontrivial thermodynamic structure, strict hierarchies, and experimentally relevant protocols, yet weak enough to admit transformations inaccessible to the most restrictive local models (Lie et al., 22 Jul 2025).