The entropy power inequality for quantum systems

Abstract

When two independent analog signals, $X$ and $Y$ are added together giving $Z=X+Y$, the entropy of $Z$, $H(Z)$, is not a simple function of the entropies $H(X)$ and $H(Y)$, but rather depends on the details of $X$ and $Y$'s distributions.  Nevertheless, the entropy power inequality (EPI), which states that $e^{2H(Z)} \geq e^{2H(X)} + e^{2H(Y)}$, gives a very tight restriction on the entropy of $Z$. This inequality has found many applications in information theory and statistics.  The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this talk is to give an outline of the proof of two separate generalizations of the entropy power inequality to the quantum regime.  These inequalities provide strong new upper bounds for the classical capacity of quantum additive noise channels, including quantum analogues of the additive white Gaussian noise channels.   Our proofs are similar in spirit to standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. Specifically, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.   This is joint work with Graeme Smith.