Zero-Error Accessible Information of a Quantum Source

Elloá B. Guedes, Francisco de Assis, Rex A. C. Medeiros

Jop Briët, Harry Buhrman, Monique Laurent +2 more

We study a problem from zero-error information theory—a topic well-known for its rich connections to combinatorics [1,8,10–12,14]—in a setting where a sender and receiver may use quantum entanglement, one of the most striking features of quantum mechanics. The problem that we consider is the classical source-channel coding problem , where Alice and Bob are each given an input from a random source and get access to a noisy channel through which Alice can send messages to Bob. Their goal is to minimize the average number of channel uses per source input while allowing Bob to learn Alice’s inputs. Here we show that entanglement can allow for an unbounded decrease in the asymptotic rate of classical source-channel codes. We also consider the source problem , the case where Alice can send messages to Bob without noise. We prove a lower bound on the rate of source codes with entanglement in terms of a variant of the Lovász theta number [10,13], a graph parameter given by a semidefinite program.

Rex A. C. Medeiros, Francisco de Assis

Jianxin Chen, Toby S. Cubitt, Aram W. Harrow +1 more

The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly, so that they can be decoded with zero probability of error. The study of zero-error capacities dates right back to Shannon and the early days of information theory. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel. We further strengthen this result to show that there exist pairs of channels, neither of which have any zero-error classical capacity (as before), yet for which access to one copy of the joint channel even allows far more delicate quantum information to be transmitted perfectly. This subsumes the first result, and also implies that the quantum zero-error capacity can be superactivated. But it is strictly stronger than either of these. Indeed, this is the strongest conceivable form of superactivation, and nothing similar is possible for standard Shannon capacities of quantum channels or for zero-error capacities of classical channels.

Zahra Baghali Khanian, Andreas Winter

We consider a generalization of the quantum state redistribution task, where pure multipartite states from an ensemble source are distributed among an encoder, a decoder and a reference system. The encoder, Alice, has access to two quantum systems: system A which she compresses and sends to the decoder, Bob, and the side information system C which she wants to keep at her site. Bob has access to quantum side information in a system B, wants to decode the compressed information in such a way to preserve the correlations with the reference system on average. As figures of merit, we consider both block error (which is the usual one in source coding) and per-copy error (which is more akin to rate-distortion theory), and find the optimal compression rate for the second criterion, and achievable and converse bounds for the first. The latter almost match in general, up to an asymptotic error and an unbounded auxiliary system; for so-called irreducible sources they are provably the same. Full paper forthcoming [1].

Elloá B. Guedes, Francisco de Assis, Rex A. C. Medeiros