Quantum mutual information

In quantum information theory, quantum mutual information, or von Neumann mutual information, after John von Neumann, is a measure of correlation between subsystems of quantum state. It is the quantum mechanical analog of Shannon mutual information.

For simplicity, it will be assumed that all objects in the article are finite-dimensional.

The definition of quantum mutual entropy is motivated by the classical case. For a probability distribution of two variables p(x, y), the two marginal distributions are

The classical mutual information I(X, Y) is defined by

where S(q) denotes the Shannon entropy of the probability distribution q.

One can calculate directly

So the mutual information is

But this is precisely the relative entropy between p(x, y) and p(x)p(y). In other words, if we assume the two variables x and y to be uncorrelated, mutual information is the discrepancy in uncertainty resulting from this (possibly erroneous) assumption.

It follows from the property of relative entropy that I(X,Y) ≥ 0 and equality holds if and only if p(x, y) = p(x)p(y).

The quantum mechanical counterpart of classical probability distributions are density matrices.

Consider a composite quantum system whose state space is the tensor product

Let ρ^AB be a density matrix acting on H. The von Neumann entropy of ρ, which is the quantum mechanical analogy of the Shannon entropy, is given by

For a probability distribution p(x,y), the marginal distributions are obtained by integrating away the variables x or y. The corresponding operation for density matrices is the partial trace. So one can assign to ρ a state on the subsystem A by

where Tr_B is partial trace with respect to system B. This is the reduced state of ρ^AB on system A. The reduced von Neumann entropy of ρ^AB with respect to system A is

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