Universal thermodynamic implementation of a process with a variable work cost

TL;DR

This work develops a universal, variable-work-cost thermodynamic framework for implementing time-covariant quantum processes. By combining thermal operations with a large information battery and a conditional erasure protocol, the authors realize $\\mathcal{E}^{\\otimes n}$ with per-copy work given by $W[\\mathcal{E}, \\sigma] = \\beta^{-1}[ D(\\mathcal{E}(\\sigma_X) \\| \\Gamma_X) - D(\\sigma_X \\| \\Gamma_X) ]$, while preserving output states for time-covariant inputs up to a work-cost dephasing that encodes the spent work. The approach unifies work extraction and process implementation, recovers recent results on optimal work extraction, and shows how conditional erasure can drive universal, cost-efficient implementations of quantum channels. It also analyzes how revealing work expenditure induces decoherence and discusses implications for extending the framework to Gibbs-preserving maps and broader dynamical regimes. The results provide a principled link between thermodynamic costs and information-processing tasks in quantum thermodynamics with potential impacts on quantum computation and state preparation under energetic constraints.

Abstract

The minimum amount of thermodynamic work required in order to implement a quantum computation or a quantum state transformation can be quantified using frameworks based on the resource theory of thermodynamics, deeply rooted in the works of Landauer and Bennett. For instance, the work we need to invest in order to implement $n$ independent and identically distributed (i.i.d.) copies of a quantum channel is quantified by the thermodynamic capacity of the channel when we require the implementation's accuracy to be guaranteed in diamond norm over the $n$-system input. Recent work showed that work extraction can be implemented universally, meaning the same implementation works for a large class of input states, while achieving a variable work cost that is optimal for each individual i.i.d. input state. Here, we revisit some techniques leading to derivation of the thermodynamic capacity, and leverage them to construct a thermodynamic implementation of $n$ i.i.d. copies of any time-covariant quantum channel, up to some process decoherence that is necessary because the implementation reveals the amount of consumed work. The protocol uses so-called thermal operations and achieves the optimal per-input work cost for any i.i.d. input state; it relies on the conditional erasure protocol in our earlier work, adjusted to yield variable work. We discuss the effect of the work-cost decoherence. While it can significantly corrupt the correlations between the output state and any reference system, we show that for any time-covariant i.i.d. input state, the state on the output system faithfully reproduces that of the desired process to be implemented. As an immediate consequence of our results, we recover recent results for optimal work extraction from i.i.d. states up to the error scaling and implementation specifics, and propose an optimal preparation protocol for time-covariant i.i.d. states.

Figures (5)

• Figure 1: Thermal operations with a battery can be used to implement an operation on the system $S$ while consuming work; they involve a global energy-conserving unitary $U_{SWB}$ operating on $S$, a battery $W$ with a family of charge states $'{\tau_W^{(E)}}_E$, and a heat bath $B$. The quantum channel induced on $S$ by the thermal operation is revealed in the correlations between $S$ and $R$ in the output state $\rho'_{SR}$, for entangled inputs $\rho_{SR}$. The battery's output charge state is guaranteed to be at least $E'$ if its state passes a hypothetical test represented by a POVM effect $\Pi_W^{\geqslant E'}$; the battery's output state can then also be assumed uncorrelated with the system by explicitly thermalizing it within $\Pi_W^{\geqslant E'}$'s support. The bath is eventually traced out. Multiple heat baths can be combined into a single large heat bath. Similarly, multiple batteries can equivalently be combined into a single one, because the battery states are reversibly transformable with thermal operations; therefore, a sequence of thermal operations operating on different batteries is equivalent to a single operation acting on a single battery using the sum of the work costs. Pure time-covariant ancillary systems can be borrowed at the cost of transforming a suitable bath system from its thermal state to the desired state; such an ancilla may be used, for instance, to perform for free another thermal operation conditionally on a measurement outcome.
• Figure 2: Types of thermodynamic implementations of an i.i.d. quantum process $\mathcal{E}$. An implementation tailored for $\rho_{XR}$ fails if any other state is used as an input. A universal implementation produces the correct output for all input states, using a deterministic amount of work per copy. A variable-work universal implementation produces the correct output (up to a dephasing) for all input states, using an amount of work that varies depending on the actual input. A semiuniversal implementation for work $w$ behaves like a universal implementation, but it only produces the correct output if the work required to implement the process for the given input state does not exceed $w$.
• Figure 3: Implementation of a universal, variable-work conditional reset operation of systems $E^n$ conditioned on $X^n$ (Main Result; version for conditional reset). The systems $E^n$ and $X^n$ start in some joint pure state $\ket\rho_{E^nX^nR}$ with a reference system $R$. The reset operation should reset each copy of $E$ to the fixed state $\kappa_E$ while preserving the reduced state $\rho_{X^nR}$. Here we suppose $\kappa_E$ is the thermal state on $E$; a similar argument holds if $\kappa_E$ is any energy eigenstate. (If $X^n$ is trivial, this task is equivalent to work extraction.) Our variable-work implementation $\mathcal{R}_{E^nX^nW}$ consumes an amount of work $w$, using a battery $W$, which is asymptotically optimal for each i.i.d. input state $\rho_{EX}^{\otimes n}$. The output state $\tilde{\rho}_{X^nR}$ on $X^nR$ differs from $\rho_{X^nR}$ by a dephasing operation due to the fact that the process interacts differently with the environment for different amounts of consumed work (which is revealed in the battery's output state). If $\rho_{E^nX^n}$ is time-covariant and permutation-invariant, we can show that, in fact, $\tilde{\rho}_{X^n} = \rho_{X^n}$.
• Figure 4: Variable-work implementation of any time-covariant i.i.d. process $\mathcal{E}^{\otimes n}$ (Main Result; version for time-covariant processes). Given any i.i.d. input $\sigma_{XR}^{\otimes n}$, the implementation outputs $\widetilde{\mathcal{E}}_{X^n}(\sigma_{XR}^{\otimes n})$, a work-cost-dephased version of $\mathcal{E}^{\otimes n}(\sigma_{XR}^{\otimes n})$, and uses the asymptotically optimal work per copy $W[\mathcal{E},\sigma]$. (For any time-covariant $\sigma_X^{\otimes n}$, the work-cost-dephasing is irrelevant without $R$, i.e., $\widetilde{\mathcal{E}}_{X^n}(\sigma_{X}^{\otimes n}) = \mathcal{E}^{\otimes n}(\sigma_{X}^{\otimes n})$.) Our implementation borrows fresh heat baths, resets them to a suitable pure state, carries out the unitary Stinespring dilation of $\mathcal{E}^{\otimes n}$ using these ancillary systems, and invokes our variable-work conditional reset operation to reset the ancillas to their thermal state before they are traced out. Accounting the work cost of each step yields the asymptotically optimal amount of work per copy $W[\mathcal{E},\sigma]$ when the input state is $\ket\sigma_{XR}^{\otimes n}$. All ancillas and heat baths can be merged into a single heat bath, and all work expenditure steps can be merged into a single battery use; the overall process is one large thermal operation acting on a suitably large and accurate battery $W$, the systems $X^n$ and a possibly very large bath $B$.
• Figure 5: A variable-work cost implementation of a process necessarily induces decoherence, because the implementation interacts differently with the environment when different amounts of work are required. In general, this dephasing can be significant in the presence of a reference system (not depicted). For time-covariant and permutation-invariant inputs, the dephasing has no effect if we ignore the reference system. a. Example variable-work implementation of a process that aligns spins along some input-dependent axis (for illustration purposes). If the spins are already aligned, the process costs no work. b. The same process might consume a lot of work on a different input state. c. A superposition of different types of input states might decohere into a mixture because the process interacts differently with the heat bath for those input state, which is reflected in the different battery output state. Note the work-cost-dephasing is significantly subtler than a dephasing of the input or output state in the energy basis: For instance, the identity process suffers no work-cost-dephasing, and is implemented accurately by our construction for arbitrary input states, because all input states require the same amount of work (zero work).

Theorems & Definitions (20)

• theorem 1: Main Result; Conditional Erasure
• theorem 2: Main Result; Quantum Process
• proposition 1
• proposition 2
• theorem 3
• proof : **thm:UniversalTheoremCovariantInputSet
• theorem 4
• proof : **thm:UniversalTheoremGpmInputSet
• lemma 1
• proof : *thm:UnivCondErasureForEachIidInput