Recovery of the second law in fully quantum thermodynamics

TL;DR

This work resolves the long-standing question of whether the second law can be recovered for fully quantum states under thermal operations without external coherence assistance. It proves that, provided the coherence structure satisfies $\mathcal{C}(\rho')\subseteq\mathcal{C}(\rho)$, a coherent quantum state $\rho$ can be converted to $\rho'$ by a thermal operation with a correlated catalyst if and only if the nonequilibrium Helmholtz free energy obeys $F(\rho)\ge F(\rho')$, rendering the theory effectively reversible. The authors develop a constructive two-part proof: first an asymptotic marginal transformation using ladder-system technology and catalytic coherence, then a single-shot correlated-catalytic reduction, thereby closing the gap between classical and quantum coherence treatments. A key consequence is the collapse of thermal operations and Gibbs-preserving transformations in the correlated-catalytic regime, showing that no external coherence sources are needed. The results provide a rigorous, operational recovery of the second law in the small-quantum regime and offer practical insights for reversible quantum thermodynamics and resource theories.

Abstract

Quantum thermodynamics investigates how robust the second law of thermodynamics serves as the unique fundamental law in the small quantum world. To tackle this problem, the quantum coherence constitutes a major difficulty of investigations, which provides severe constraints hindering the recovery of a single thermodynamic potential. Here we solve this long-standing problem of quantum information theory by revealing that the state convertibility under thermal operations is fully characterized by the second law of thermodynamics. Specifically, we prove that whether a quantum state with quantum coherence is convertible to another by a thermal operation with a correlated catalyst is completely determined by the free energy ordering. Unlike previous attempts, our setting does not resort to any additional external coherent assist, providing a faithful operational characterization of thermodynamic state transformation.

Figures (7)

• Figure 1: Thermal operation with and without a correlated catalyst.a-b.(a) In a thermal operation, we employ an auxiliary systems; a thermal environment in its Gibbs state $\tau_{\mathop{\mathrm{Gibbs}}\nolimits , E}$. We apply an energy-conserving unitary operation on them. The final reduced state of the system is $\rho'$. (b) In a thermal operation with a correlated-catalyst, we employ a further external auxiliary system; a catalyst in state $c$, which is arbitrary but should return to its original state. We apply an energy-conserving unitary operation on the system and two external systems. The final reduced state of the system is $\rho'$ and that of the catalyst returns to $c$.
• Figure 2: Overview of the protocol. We first divide $n$ copies of the initial state $\rho$ into $\mu\nu$ copies and $o(n)$ copies. Using the dephasing channel, we remove off-diagonal elements of each $\mu$ copies of $\rho$ and obtain energy-diagonal state $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}$ on $S^{\otimes \mu}$. We then convert $\nu$ copies of $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}$ into another energy-diagonal state $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}'$ with the same number of copies. The state $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}'$ has the same spectrum as $\rho'^{\otimes \mu}$. Separately from this process, we distill state $\eta$ with broad coherence from $o(n)$ copies of $\rho$. We finally recover $\rho'^{\otimes \mu}$ from $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}'$ by an energy-conserving unitary operation by using $\eta$ as catalytic coherence repeatedly.
• Figure 3: Random walks modeling the energy distribution of the coherent resource state. If the free energy ordering $F(\rho)\geq F(\rho')$ is satisfied, the expected energy of the coherent resource state increases, corresponding to the random walk with moving upward on average. In this case, the probability of hitting the ground energy can be arbitrarily suppressed. On the other hand, if the free energy ordering is violated, i.e., $F(\rho)<F(\rho')$, then the expected energy of the coherent resource state decreases by this process, resulting in hitting the ground energy with high probability and causing undesirable error.
• Figure 4: Overview of the asymptotic marginal conversion protocol (same as Fig. \ref{['f:schematic']} in the main article). We transform $n$ copies of $\rho$ into $n'=n-o(n)$ copies of $\rho'$ in the sense of the asymptotic marginal conversion. As its preliminary treatment, we first divide $n$ copies of the initial state $\rho$ into $n'=\mu\nu$ copies and $o(n)$ copies. In step (1) we use the dephasing channel and remove all the off-diagonal elements of each $\mu$ copies of $\rho$. The state becomes an energy-diagonal state $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}$ on $S^{\otimes \mu}$. Through this process, the decrease of the free energy density is arbitrarily suppressed. In step (2), we convert $\nu$ copies of $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}$ into $\nu$ copies of another energy-diagonal state $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}'$, which has the same spectrum as $\rho'^{\otimes \mu}$. The existence of this conversion is confirmed by the second law for classical states. In step (3), we distill state $\eta$ with broad coherence from $o(n)$ copies of $\rho$. In step (4), using $\eta$ as catalytic coherence repeatedly, we recover $\rho'^{\otimes \mu}$ from $\rho_{\mathop{\mathrm{cl}}\nolimits,\mu}'$ by an energy-conserving unitary operation. The accuracy of this implementation is confirmed by the random-walk argument.
• Figure 5: Sequential implementation of the coherent unitary $U_\mu$ with a coherent resource state $\eta$. This approximately produces a state whose marginal on each subsystem $S$ is close to $\rho'$ with potential correlation with other subsystems and catalytic system.

Theorems & Definitions (25)

• Theorem
• Definition S.1: Thermal operations Sagawa2021asymptotic-reversibility
• Definition S.2: Covariant operations
• Definition S.3: Correlated-catalytic transformation
• Definition S.4: Coherent modes MS14
• Lemma S.5: Monotonicity of coherent modes MS14
• Definition S.6: Resonant coherent modes ST23
• Theorem S.7
• Definition S.8
• Lemma S.9