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

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

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 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^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

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.

Open-source code (SOMIM and SeCQC)

SOMIM: There is an open-source program code called SOMIM (Search for Optimal Measurements by an Iterative Method), which calculates the maximal mutual information (accessible information). For a given set of statistical operators, SOMIM finds the POVMs that maximize the accessed information, and thus determines the accessible information and one or all of the POVMs that retrieve it. The maximization procedure is a steepest-ascent method that follows the gradient in the POVM space, and also uses conjugate gradients for speed-up.

The complete set of files including the codes and manual can be found at the SOMIM website: http://www.quantumlah.org/publications/software/SOMIM/.

SeCQC: Another open-source program code called SeCQC (Search for the classical Capacity of Quantum Channels). Given a quantum channel, SeCQC finds the statistical operators and POVM outcomes that maximize the accessible information, and thus determines the classical capacity of the quantum channel.

The complete set of files including the codes and manual can be found at the SeCQC website: http://www.quantumlah.org/publications/software/SeCQC/.