#jsDisabledContent { display:none; } My Account |  Register |  Help

# Quantum mutual information

Article Id: WHEBN0006011769
Reproduction Date:

 Title: Quantum mutual information Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 ρAB 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 TrB is partial trace with respect to system B. This is the reduced state of ρAB on system A. The reduced von Neumann entropy of ρAB with respect to system A is

\;S(\rho^A).

S(ρB) 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.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.