Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy

TL;DR

This work provides an operational thermodynamic interpretation of the dynamical min-entropy by showing that the one-shot adversarial erasure cost of a quantum channel is (approximately) proportional to the negative of its dynamical min-entropy at fixed temperature. Through a quantum channel decoupling theorem, the authors connect a channel's ability to decouple its output from a reference to its erasure cost, with negative entropy enabling potential work extraction. The results are developed within both thermodynamic and resource-theoretic frameworks, highlighting a deep link between decoupling, min-entropy, and the limits of one-shot quantum computation. These insights offer a quantitative tool for assessing the energetic costs of resetting and reusing quantum hardware and suggest connections to quantum capacity and secure information processing.

Abstract

The erasure of information is fundamentally an irreversible logical operation, carrying profound consequences for the energetics of computation and information processing. We investigate the thermodynamic costs associated with erasing (and preparing) quantum processes. Specifically, we analyze an arbitrary bipartite unitary gate acting on logical and ancillary input-output systems, where the ancillary input is always initialized in the ground state. We focus on the adversarial erasure cost of the reduced dynamics -- that is, the minimal thermodynamic work cost to erase the logical output of the gate for any logical input, assuming full access to the ancilla but no access to any purifying reference of the logical input state. We determine that this adversarial erasure cost is directly proportional to the negative min-entropy of the reduced dynamics, thereby giving the dynamical min-entropy a clear operational meaning. The dynamical min-entropy can take positive and negative values, depending on the underlying quantum dynamics. The negative value of the erasure cost implies that the extraction of thermodynamic work is possible instead of its consumption during the process. A key foundation of this result is the quantum process decoupling theorem, which quantitatively relates the decoupling ability of a process with its min-entropy. This insight bridges thermodynamics, information theory, and the fundamental limits of quantum computation.

Figures (3)

• Figure 1: Pictorial representation of the main problem: We consider a bipartite unitary gate (channel) $\mathcal{U}_{A'E'\to AE}$ where $E',E$ are ancilla ($\op{0}_{E'}$ being the initial state). $R$ is a reference that purifies the logical input $A'$. The Hamiltonian of the logical output $A$ is trivial. The goals are to determine: (a) the optimal erasure cost of $A$ of the quantum process $\mathcal{U}$ when the eraser has access to $E$ but not $R$, (b) the optimal preparation cost of $A$ when the preparer has access to $R$ but not ancilla.
• Figure 2: The picture illustrates the mechanism of the decoupling theorem for quantum channels. Decoupling theorem provides a fundamental limit on the ability of a channel $\mathcal{N}_{A'\to A}$ post-processed with $\mathcal{T}_{A\to B}\circ\mathcal{U}_A$, where $\mathcal{U}_A$ is a Haar-random unitary channel, to decouple the output $B$ from its reference $R$. See Theorem \ref{['thm:de-ch']} for the formal, precise statement.
• Figure 3: We plot the numerical values for the negative of the dynamical min-entropy $-S_{\min}[\mathcal{N}_p]$ of a $p$-parametrized quantum channel $\mathcal{N}_p$ vs the parameter $p\in[0,1]$, for three families of qubit channels: depolarizing channels $\mathcal{E}_p$\ref{['eq:polch']}, first-kind dephasing channels $\mathcal{D}^{(1)}_p$\ref{['eq:dep1']}, and second-kind dephasing channels $\mathcal{D}^{(2)}_p$\ref{['eq:dep2']}.

Theorems & Definitions (23)

• Lemma 1
• Theorem 1: Decoupling theorem for processes
• Proposition 1
• Lemma 2
• Theorem 2
• Lemma 3
• Proposition 2
• Lemma 4: cf. Proposition 8 of SPSD25
• proof
• Lemma