Holevo's theorem

Holevo's theorem is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a quantum state (accessible information). It was published by Alexander Holevo in 1973.

As for several concepts in quantum information theory, accessible information is best understood in terms of a 2-party communication. So we introduce two parties, Alice and Bob. Alice has a classical random variable X, which can take the values {1, 2, ..., n} with corresponding probabilities {p₁, p₂, ..., p_n}. Alice then prepares a quantum state, represented by the density matrix ρ_X chosen from a set {ρ₁, ρ₂, ... ρ_n}, and gives this state to Bob. Bob's goal is to find the value of X, and in order to do that, he performs a measurement on the state ρ_X, obtaining a classical outcome, which we denote with Y. In this context, the amount of accessible information, that is, the amount of information that Bob can get about the variable X, is the maximum value of the mutual information I(X : Y) between the random variables X and Y over all the possible measurements that Bob can do.

There is currently no known formula to compute the accessible information. There are however several upper bounds, the best-known of which is the Holevo bound, which is specified in the following theorem.

Let {ρ₁, ρ₂, ..., ρ_n} be a set of mixed states and let ρ_X be one of these states drawn according to the probability distribution P = {p₁, p₂, ..., p_n}.

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