Universal thermodynamic implementation of a process with a variable work cost

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.

💡 Research Summary

This paper investigates the fundamental thermodynamic work cost required to implement quantum processes, focusing on the case of time‑covariant quantum channels applied to many independent and identically distributed (i.i.d.) copies. Building on the resource theory of thermal operations, the authors revisit the notion of thermodynamic capacity—a quantity that gives the worst‑case work per copy needed to implement a channel when a deterministic, state‑independent work budget is imposed. While previous works constructed universal implementations that consume a fixed amount of work equal to this capacity, recent studies have shown that work extraction can be performed universally with a variable work cost that adapts to the actual input state.

The central contribution of this work is a protocol that combines thermal operations with a conditional erasure sub‑routine to achieve a universal, variable‑work implementation of any time‑covariant channel E. The protocol proceeds in three stages. First, the Stinespring dilation of E^{⊗n} is enacted via an energy‑conserving unitary; this step requires no work because the unitary commutes with the total Hamiltonian. Second, the environment registers generated by the dilation are conditionally erased using the system copies X^{n} as a memory. The erasure cost is precisely β⁻¹