Postselected communication over quantum channels

TL;DR

This work defines and fully characterises postselected communication over quantum channels, where the decoder may output an inconclusive result and the error is conditioned on conclusive outcomes. The authors prove a tight, single-letter characterization of both pEA and pNA capacities in terms of the channel's projective mutual information $I_\Omega(\mathcal{N})$, with achievability via a postselected teleportation-based scheme and converses via postselected hypothesis testing. In the asymptotic limit, all pEA and pNA capacities reduce to $I_\Omega(\mathcal{N})$ for classical communication and to $I_\Omega(\mathcal{N})/2$ for quantum communication, with strong converses. The results show that postselection greatly simplifies capacity calculations and reveal fundamental limits on communication even when powerful resources (entanglement or nonsignalling correlations) are available, including implications for postselected closed timelike curves.

Abstract

The single-letter characterisation of the entanglement-assisted capacity of a quantum channel is one of the seminal results of quantum information theory. In this paper, we consider a modified communication scenario in which the receiver is allowed an additional, `inconclusive' measurement outcome, and we employ an error metric given by the error probability in decoding the transmitted message conditioned on a conclusive measurement result. We call this setting postselected communication and the ensuing highest achievable rates the postselected capacities. Here, we provide a precise single-letter characterisation of postselected capacities in the setting of entanglement assistance as well as the more general nonsignalling assistance, establishing that they are both equal to the channel's projective mutual information -- a variant of mutual information based on the Hilbert projective metric. We do so by establishing bounds on the one-shot postselected capacities, with a lower bound that makes use of a postselected teleportation-based protocol and an upper bound in terms of the postselected hypothesis testing relative entropy. As such, we obtain fundamental limits on a channel's ability to communicate even when this strong resource of postselection is allowed, implying limitations on communication even when the receiver has access to postselected closed timelike curves.

Figures (5)

• Figure 1: Simulation of a subchannel ${\mathcal{N}}_{M\to{\widehat{M}}}'$ with a probabilistic supermap $\Theta_{(A\to B)\to(M\to{\widehat{M}})}$ over a channel ${\mathcal{N}}_{A\to B}$ [see Eq. \ref{['eq:transformation']}]. Note that $\Theta_{(A\to B)\to(M\to{\widehat{M}})}$ includes postselecting the conclusive outcome in the classical flag system.
• Figure 2: Postselected entanglement-assisted communication over a channel ${\mathcal{N}}_{A\to B}$ using a protocol $(\gamma_{A'B'},{\mathcal{E}}_{MA'\to A},{\mathcal{D}}_{BB'\to{\widehat{M}}})$ (see Definition \ref{['def:pEA']}). The red and green regions represent Alice's and Bob's operations, respectively. Note that the decoding operation ${\mathcal{D}}_{BB'\to{\widehat{M}}}$ includes postselecting the conclusive outcome in the classical flag system.
• Figure 3: Equivalence between a probabilistic supermap $\Theta_{(A\to B)\to(M\to{\widehat{M}})}$ (left) and the bipartite subchannel ${\widetilde{\Theta}}_{MB\to A{\widehat{M}}}$ (right). The systems $M$ and $A$ are held by Alice, and the systems $B$ and ${\widehat{M}}$ are held by Bob.
• Figure 4: The Alice-to-Bob-nonsignalling constraint on a bipartite subchannel ${\widetilde{\Theta}}_{MB\to A{\widehat{M}}}$ [see Eq. \ref{['eq:NA']}]. On coarse-graining Alice's output system $A$, no information can be transmitted from Alice's input system $M$ to Bob's output system ${\widehat{M}}$.
• Figure 5: Postselected teleportation-based coding over a channel ${\mathcal{N}}_{A\to B}$ (see Protocol \ref{['prot:teleportation']}). The red and green regions represent Alice's and Bob's operations, respectively.

Theorems & Definitions (31)

• Lemma 1: Characterisation of $D_{\textnormal{pH}}^\varepsilon$ regula_2022-4
• Definition 1: $(d_M,\varepsilon)$ protocols
• Definition 2: One-shot ${\mathscr{T}}$-assisted capacities
• Definition 3: Asymptotic ${\mathscr{T}}$-assisted capacities
• Remark 1: Note on related work
• Definition 4: pEA protocols
• Definition 5: pNA protocols
• Proposition 1: Characterisation of pNA protocols
• Remark 2: Equivalence of postselected protocols under rescaling
• Remark 3: pNA protocols need not be Bob-to-Alice nonsignalling