The Entropy Power Inequality states that the Shannon differential entropy of the convolution of two probability densities in R^n with given entropies is minimum when they are both Gaussian. This functional inequality is fundamental in information theory and was conjectured by C. Shannon himself, although he only proved that Gaussian densities are critical points; a full proof was later given by A. Stam, using an interpolation argument along the heat semigroup.
In this talk, we illustrate a quantum version of the Entropy Power Inequality, where probability densities on R^n are replaced with quantum states of the electromagnetic radiation, the Shannon entropy is replaced with the von Neumann entropy and the convolution operation with a natural ``beam-splitter'' operation. Moreover, we allow for the presence of a conditional ``external memory'', which would add no difficulty in the commutative scenario, but is rather delicate in the quantum scenario. The difficulty is due to the existence of entanglement, i.e. the existence of joint quantum states of the electromagnetic radiation and the memory where the two systems cannot be described independently, but must be necessarily described with a joint global state. Our proof is based on a quantum version of Stam's interpolation, where new ideas must be introduced to handle non-commutative features.
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