Fundamental work costs of preparation and erasure in the presence of quantum side information

TL;DR

The paper develops a resource-theoretic framework for quantum thermodynamics in the presence of quantum side information to quantify work costs for preparation and erasure under Maxwellian demon scenarios. It introduces two minimal operational assumptions that define conditionally uniformity-covariant free operations and derives exact one-shot work costs in terms of $H_{\min}^{\downarrow,\varepsilon}$, $H_{\max}^{\uparrow,\varepsilon}$, and generalized mutual informations $I_{\max}^{\uparrow,\varepsilon}$, $I_{\min}^{\downarrow,\varepsilon}$ for trivial and nontrivial Hamiltonians, respectively. In the asymptotic limit these costs are governed by the generalized Umegaki mutual information $I(\rho_{AB}||\gamma_A)$ (and by the conditional von Neumann entropy in the trivial-H case), establishing macroscopic reversibility and a macroscopic second law via the conditional Helmholtz free energy $F(A|B)_\rho$. The work also proves the maximality of the conditional max-entropy within the axiomatic framework and connects these insights to Lieb–Yngvason-style thermodynamics and existing resource theories, thereby enriching the interface between thermodynamics and quantum information theory.

Abstract

The thought experiment of Maxwell's demon highlights the effect of side information in thermodynamics. In this paper, we present an axiomatic treatment of a quantum Maxwell's demon, by introducing a resource-theoretic framework of quantum thermodynamics in the presence of quantum side information. Under minimal operational assumptions that capture the demon's behaviour, we derive the one-shot work costs of preparing, as well as erasing, a thermodynamic system whose coupling with the demon's mind is described by a bipartite quantum state. With trivial Hamiltonians, these work costs are precisely captured by the smoothed conditional min- and max-entropies, respectively, thus providing operational interpretations for these one-shot information-theoretic quantities in microscopic thermodynamics. An immediate, information-theoretic implication of our results is an affirmative proof of the conjectured maximality of the conditional max-entropy among all axiomatically plausible conditional entropies, complementing the recently established minimality of the conditional min-entropy. We then generalize our main results to the setting with nontrivial Hamiltonians, wherein the work costs of preparation and erasure are captured by a generalized type of mutual information. Finally, we present a macroscopic second law of thermodynamics in the presence of quantum side information, in terms of a conditional version of the Helmholtz free energy. Our results extend the conceptual connection between thermodynamics and quantum information theory by refining the axiomatic common ground between the two theories and revealing fundamental insights of each theory in light of the other.

Figures (3)

• Figure 1: The operational setting of this paper is an abstraction of Maxwell's demon in the following way. Alice is in charge of a thermodynamic system, such as the particles in a chamber (red), through which she can invest or extract work. Bob is in charge of an informational system, such as the demon's mind (blue), which stores side information and is assumed to be only subject to the laws of quantum mechanics. Due to the possibility of the demon possessing knowledge about the particles, the state of the entire system is described by a bipartite quantum state shared by Alice and Bob. The demon's strategy for exploiting his knowledge and manipulating the particles is executed by a joint operation performed by Alice and Bob.
• Figure 2: In the resource theory of conditional nonuniformity, a free operation $\mathcal{N}_{AB\to A'B'}$ is composed of the following steps. First, Bob applies a channel $\mathcal{E}_{B\to B'B"}$ (in blue) and sends Alice a system $B"$. Then Alice manipulates her system jointly with $B"$ by applying a channel $\mathcal{F}_{AB"\to A'}$ (in red), in a way such that her system, should it originally be in informational equilibrium and uncorrelated with $B"$, remains in equilibrium afterwards. We also call the free operations conditionally uniformity-covariant operations due to Proposition \ref{['prop:free-nonuniformity']}.
• Figure 3: The one-shot work cost $W^\varepsilon(\rho_{AB};\sigma_{A'B'})$ of converting Alice's system in the presence of Bob's side information is calculated as follows. Initially, Alice's and Bob's systems are in the state $\rho_{AB}$, and Alice has access to a battery system $A_0$ in a pure state representing her initial work storage. Supposedly without doing work, they perform a conditionally uniformity-covariant operation $\mathcal{N}_{A_0AB\to A_1A'B'}$ such that the state $\tau_{A_1A'B'}$ after the operation is $\varepsilon$-close to the target state $\sigma_{A'B'}$ in tensor product with a pure state of a battery system $A_1$ representing Alice's final work storage. According to Landauer's principle, Alice has to invest $(1/\beta_\textnormal{b})(\log_2\lvert A_0\rvert-\log_2\lvert A_1\rvert)$ or extract $(1/\beta_\textnormal{b})(\log_2\lvert A_1\rvert-\log_2\lvert A_0\rvert)$ work to "recharge" or "discharge" her battery afterwards, returning her work storage to the initial level. This completes the the desired conversion from $A$ to $A'$ without causing significant change to the environment (specifically, the battery), and the work cost of the whole process boils down to the amount of work involved in Alice's recharging or discharging her battery.

Theorems & Definitions (50)

• Proposition 1
• Definition 1: gour2024InevitabilityKnowingLess
• Proposition 2
• proof
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Proposition 3
• proof