Universal work extraction in quantum thermodynamics

Abstract

Evaluating the maximum amount of work extractable from a nanoscale quantum system is one of the central problems in quantum thermodynamics. Previous works identified the free energy of the input state as the optimal rate of extractable work under the crucial assumption: experimenters know the description of the given quantum state, which restricts the applicability to significantly limited settings. Here, we show that this optimal extractable work can be achieved without knowing the input states at all, removing the aforementioned fundamental operational restrictions. We achieve this by presenting a universal work extraction protocol, whose description does not depend on input states but nevertheless extracts work quantified by the free energy of the unknown input state. Remarkably, our result partially encompasses the case of infinite-dimensional systems, for which optimal extractable work has not been known even for the standard state-aware setting. Our results clarify that, in spite of the crucial difference between the state-aware and state-agnostic scenarios in accomplishing information-theoretic tasks, whether we are in possession of information on the given state does not influence the optimal performance of the asymptotic work extraction.

Figures (9)

• Figure 1: Schematic figure of the universal work extraction protocol. The work extraction protocol discussed in the previous results (left) is tailored according to the information of the initial state, and the optimal extractable work rate is shown to be characterized by the Helmholtz free energy. The universal work extraction protocol introduced in our result (right) is independent of the input state but achieves the same extractable rate as the state-dependent protocol in the asymptotic limit.
• Figure 2: Overview of the universal work extraction protocol for finite-dimensional systems. First, we apply the channel called Schur pinching to obtain the state diagonalized with a specific energy eigenbasis that also respects the permutation symmetry. After that, we apply a thermal operation that simulates the type measurement on a sublinear number of subsystems and the work extraction protocol conditioned on the measurement outcomes. Since the projector corresponding to the measurement is the projector onto the energy eigenspace, we can realize the same action solely by a thermal operation.
• Figure 3: The structure of the Hamiltonian with the Schur basis. Due to the Schur-Weyl duality, the Hamiltonian $H^{\times k}$ of $k$ systems is block-diagonalized as above. Since all $H_\lambda$ are Hermitian, we can find an orthonormal basis of the whole Hilbert space that diagonalizes $H_\lambda$ in each block.
• Figure 4: The matrix representation of $\rho^{\otimes k}$. The blocks indicate each direct sum element $\mathcal{W}_\lambda\otimes \mathcal{U}_\lambda$. Each block consists of $n_\lambda\times n_\lambda$ blocks $(\rho_\lambda)_{ij}I_{m_\lambda}$, where $m_\lambda=\dim \mathcal{U}_\lambda$ and $n_\lambda=\dim\mathcal{W}_\lambda$.
• Figure S.1: Overview of the universal work extraction protocol for finite-dimensional systems. First, we apply the channel called Schur pinching to obtain the state diagonalized with a specific energy eigenbasis that also respects the permutation symmetry. After that, we apply an appropriate work extraction protocol, which is conditioned by the type measurement. Since the projector corresponding to the measurement is the projector onto the energy eigenspace, we can realize the same action solely by a thermal operation.

Theorems & Definitions (22)

• Definition 1
• Theorem 2
• Theorem 3
• Lemma S.1
• proof
• Lemma S.2
• proof
• Lemma S.3: Bluhm_general_continuityBluhm_Continuity_of
• proof
• Definition S.4