Role of complementarity in superdense coding

TL;DR

This paper investigates how the quantum concept of complementarity influences the capacity of superdense coding, revealing that non-commuting encodings enable quantum advantage and deriving explicit formulas for capacity based on complementarity measures.

Contribution

It establishes a quantitative link between the complementarity of observables and superdense coding capacity, including formulas for noisy and ideal resources, advancing understanding of quantum communication advantages.

Findings

Complementarity is necessary for superdense coding advantage.

Derived explicit formulas for superdense coding capacity based on complementarity.

Extended results to noisy quantum resources like Werner states.

Abstract

The complementarity of two observables is often captured in uncertainty relations, which quantify an inevitable tradeoff in knowledge. Here we study complementarity in the context of an information processing task: we link the complementarity of two observables to their usefulness for superdense coding (SDC). In SDC, Alice sends two classical dits of information to Bob by sending a single qudit. However, we show that encoding with commuting unitaries prevents Alice from sending more than one dit per qudit, implying that complementarity is necessary for SDC to be advantagous over a classical strategy for information transmission. When Alice encodes with products of Pauli operators for the $X$ and $Z$ bases, we quantify the complementarity of these encodings in terms of the overlap of the $X$ and $Z$ basis elements. Our main result explicitly solves for the SDC capacity as a function of the complementarity, showing that the entropy of the overlap matrix gives the capacity, when the preshared state is maximally entangled. We generalise this equation to resources with symmetric noise such as a preshared Werner state. In the most general case of arbitrary noisy resources, we obtain an analogous lower bound on the SDC capacity. Our results shed light on the role of complementarity in determining the quantum advantage in SDC and also seem fundamentally interesting since they bear a striking resemblance to uncertainty relations.