Conditional channel entropy sets fundamental limits on thermodynamic quantum information processing

Abstract

The thermodynamic resourcefulness of quantum channels primarily depends on their underlying causal structure and their ability to generate quantum correlations. We quantify this interplay within the resource theory of athermality for bipartite quantum channels in the presence of a side channel acting as memory, referred to as the resource theory of conditional athermality. For channels with trivial output Hamiltonians, we characterize the optimal one-shot rates for distilling the identity gate from a given channel, as well as the cost of simulating the channel using the identity gate, under conditional Gibbs-preserving superchannels. We show that these rates have a direct trade-off relation with the conditional channel entropies, attributing operational significance to signaling in quantum processes. Furthermore, we establish an equipartition property for the conditional channel min-entropy for classes of channels that are either tele-covariant or no-signaling from the non-conditioning input to the conditioning output. As a consequence, we demonstrate asymptotic reversibility of the resource theory for these channels. The asymptotic conditional athermality capacity of a tele-covariant channel is half the superdense coding capacity of its Choi state. Our work establishes the conditional channel entropy as a primitive information-theoretic concept for quantum processes, elucidating its potential for wider applications in quantum information science.

Figures (3)

• Figure 1: Summary of the signaling properties of channels and the bounds on the conditional channel min-entropy $S_{\infty}[A|B]_{\mathcal{N}}$ for different classes of bipartite channels $\mathcal{N}_{A'B'\to AB}$. For simplicity, we have assumed that $|A'|=|B'|=|B|=|A|$. Note that the causal structure of the channel manifests in its position on this line: the more signaling from $A'$ to $B$, the further to the right it lies. Swap-like unitary channels sit at the right extremal point of $-3\log |A|$, while the controlled unitary channels $\mathcal{C}_\mathcal{U}$ are lower bounded by $-2\log|A|$, indicating a clear gap between the signaling ability of swap-like and controlled unitary channels. $\widehat{\mathcal{C}}_\mathcal{U}$ denotes the controlled unitary with orthogonal unitary operators. The conditional min-entropy of channels $\mathcal{N}$ in the sets ${\rm C{\text{-}}SEP{\text{-}}P}$, ${\rm C{\text{-}}PPT{\text{-}}P}$, and ${\rm S}_{A'\not\to B}$ are all lower bounded by $-\log|A|$. See Section \ref{['sec:signaling']} for related results in generality.
• Figure 2: Plot showing the conditional min-entropy $S_{\infty}[A|B]_{\mathcal{N}}$ along with the upper and lower bounds $S_{\infty}(AR_A|BR_B)_{\Phi^\mathcal{N}}-1$ and $S^\downarrow_{\infty}(AR_A|BR_B)_{\Phi^\mathcal{N}}-1$ for two-qubit channels parametrized as $\mathcal{N}_p:=p\mathcal{R}^{\pi}+(1-p)\mathcal{N}$ for $p\in[0,1]$, where $\mathcal{N}$ is chosen to be $\mathrm{CNOT}$, $\mathrm{SWAP}$, $\mathrm{id}$. We see that the conditional channel min-entropy coincides with the upper bound. For the case of a noisy two-qubit swap channel, both the upper and lower bounds coincide for all values of $p$.
• Figure 3: Plot showing the conditional min-entropy $S_{\infty}[A|B]_{\mathcal{N}}$ along with the upper and lower bounds $S{\infty}(AR_A|BR_B)_{\Phi^\mathcal{N}}-1$ and $S_{\infty}(AR_A|BR_B)_{\Phi^\mathcal{N}}-1$ for two-qubit channels parametrized as $\mathcal{N}_p:=p\mathcal{R}^{\pi}+(1-p)\mathcal{N}$ for $p\in[0,1]$, where $\mathcal{N}$ is a numerically generated random channel using Quantinf package Cub10. There is a clear separation between conditional min-entropy and its bounds in \ref{['thm:min-ent-bounds']}.

Theorems & Definitions (79)

• Definition 1: One-shot conditional athermality distillation and formation
• Definition 2: Conditional athermality distillation and formation capacity
• Theorem 1: Single-shot yield and cost
• Corollary 1
• proof
• Proposition 1: One-shot yield-cost trade-off
• Lemma 1
• proof
• Proposition 2
• proof