### 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.

## 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,y)}{p(x) p(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 analogy 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*with respect to system

^{AB}*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*

It can now be seen that the appropriate definition of quantum mutual information should be

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

Quantum mutual information can be interpreted the same way as in the classical case: it can be shown that

- I(\rho^{AB}) = S(\rho^{AB} \| \rho^A \otimes \rho^B)

where S(\cdot \| \cdot) denotes quantum relative entropy.