Quantum mutual information

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

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Motivation

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

$p(x) = \sum_{y} p(x,y)\; , \; p(y) = \sum_{x} p(x,y).$

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

$\;I(X,Y) = S(p(x)) + S(p(y)) - S(p(x,y))$

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

One can calculate directly

$\; S(p(x)) + S(p(y))$

$\; = \sum_x p_x \log p(x) + \sum_y p_y \log p(y)$

$\; = \sum_x \; ( \sum_{y'} p(x,y') \log \sum_{y'} p(x,y') ) + \sum_y ( \sum_{x'} p(x',y) \log \sum_{x'} p(x',y))$

$\; = \sum_{x,y} p(x,y) (\log \sum_{y'} p(x,y') + \log \sum_{x'} p(x',y))$

$\; = \sum_{x,y} p(x,y) \log p(x) p(y) .$

So the mutual information is

$I(X,Y) = \sum_{x,y} p(x,y) \log \frac{p(x) p(y)}{p(x,y)}.$

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).

Definition

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

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

$H = H_A \otimes H_B.$

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

$S(\rho^{AB}) = - \operatorname{Tr} \rho^{AB} \log \rho^{AB}.$

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

$\rho^A = \operatorname{Tr}_B \; \rho^{AB}$

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

$\;S(\rho^A).$

S(ρ^B) is defined in the same way.

Technical Note: In mathematical language, passing from the classical to quantum setting can be described as follows. The algebra of observables of a physical system is a C*-algebra and states are unital linear functionals on the algebra. Classical systems are described by commutative C*-algebras, therefore classical states are probability measures. Quantum mechanical systems have non-commutative observable algebras. In concrete considerations, quantum states are density operators. If the probability measure μ is a state on classical composite system consisting of two subsystem A and B, we project μ onto the system A to obtain the reduced state. As stated above, the quantum analog of this is the partial trace operation, which can be viewed as projection onto a tensor component. End of note

We can see now the appropriate definition of quantum mutual information should be

$\; I(\rho^{AB}) = S(\rho^A) + S(\rho^B) - S(\rho^{AB}).$