Thermodynamic criteria for signaling in quantum channels

TL;DR

This work develops a resource-theoretic framework linking the signalling capacity of quantum channels to thermodynamic athermality by introducing robust measures $R_T$ (ather-mality), $R_S$ (signalling), and the joint $R(\Lambda||\Gamma)$ relative to the completely thermalising channel $\Gamma$. It identifies three thermodynamic tasks—generation, preservation, and transmission of athermality—and proves precise relations showing that signalling is bounded below by athermality transmission and above by athermality preservation, with corresponding operational interpretations via Gibbs-preserving dilations. The authors demonstrate these bounds concretely in the quantum switch setup, revealing a trade-off between signaling and thermodynamic resources that constrains the switch’s ability to generate and transmit athermality. Collectively, the results provide a unified framework to assess quantum channels under thermal constraints and offer practical tools for evaluating signaling and thermodynamic capabilities in quantum communication devices.

Abstract

Signaling quantum channels are fundamental to quantum communication, enabling the transfer of information from input to output states. In contrast, thermalisation erases information about the initial state. This raises a crucial question: How does the thermalising tendency of a quantum channel constrain its signaling power and vice versa? In this work, we address this question by considering three thermodynamic tasks associated with a quantum channel: the generation, preservation, and transmission of athermality. We provide faithful measures for athermality generation and athermality preservation of quantum channels, and prove that their difference quantifies athermality transmission. Analysing these thermodynamic tasks, we find that the signaling ability of a quantum channel is upper-bounded by its athermality preservation and lower-bounded by its athermality transmission, thereby establishing a fundamental relationship between signaling and thermodynamic properties of channels for quantum communication. We demonstrate this interplay for the example of the quantum switch, revealing an explicit trade-off between the signaling ability and athermality of the quantum channels it can implement.

Figures (10)

• Figure 1: Geometric interpretation of robustness measure. The distance between $X$ and the free state $Z$ corresponds to the factor $s/(1+s)$. Minimising $s$ finds the closest free state $Z^*$ to $X$, and hence, quantifies the distance between $X$ and the free set $\mathcal{F}$.
• Figure 2: Main results. (a) Three thermodynamic tasks associated with a quantum channel $\Lambda\in\mathcal{O}(A'\rightarrow B)$ correspond to three quantifiable resources: Task I (athermality generation) -- how much athermality can $\Lambda$ generate from a thermal input? Task II (athermality preservation) -- how much input athermality does $\Lambda$ preserve if $A'$ is part of a pure, locally thermal state? Task III (athermality transmission) -- how much of the initial global athermality from correlations between $A$ and $A'$ does $\Lambda$ transfer? Athermality is denoted by charged batteries and empty batteries indicate thermal states. (b) The signalling ability of the quantum channel $\Lambda$ is bounded by its thermodynamic capacities (Theorem \ref{['thm:R_S_bounds']}).
• Figure 3: GPO dilation. A general channel $\Lambda$ can be dilated by a channel $G\in\mathrm{GPO}(CA\rightarrow B)$ with some state $\rho_C$.
• Figure 4: Resource measures $R_\mathrm{T}$, $R_\mathrm{S}$ and $R$ of random qubit channels.$20000$ qubit channels in $\mathcal{O}(A\rightarrow B)$ with $d_A=d_B=2$ and $\gamma_A=\gamma_B = 0.75\mathinner{|{0}\rangle}\!\mathinner{\langle{0}|}+0.25\mathinner{|{1}\rangle}\!\mathinner{\langle{1}|}$ are randomly generated. Channels with $1/R_\mathrm{T} \le\!\!(>) R$ are marked by black (blue) edgecolor. The set GPO lies in $[0, d_A^2-1]$ on the line of $R_\mathrm{T}=0$ while the set NSO lies in $[0, 1/g_{\min}^{(B)}-1]$ on the line of $R_\mathrm{S}=0$. The completely thermalsing channel $\Gamma$ is at the origin. The highest $R_\mathrm{T}$ and $R_\mathrm{S}$ are achieved by the most energetic channels (sending $\gamma_A$ to $\mathinner{|{1}\rangle}\!\mathinner{\langle{1}|}_B$) and unitary channels, respectively. The symbol ✕ denotes that there is no intersection between the most energetic channels and unitary channels.
• Figure 5: Quantum switch. A quantum switch $\mathdutchcal{S}$ with a control qubit $\rho_C$ and two GPOs as input can be considered as a channel simulation of $\Lambda_{\mathdutchcal{S}}$.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma A.1
• proof
• Theorem A.1: The most athermal state
• proof