Joint quantum entropy

The joint quantum entropy is an entropy measure which attempts to generalize the classical joint entropy for quantum information theory. Intuitively, given two quantum states $\rho$ and $\sigma$, represented as density operators, the joint quantum entropy attempts to measure the total uncertainty or entropy of the joint system consisting of both states together. It is written $S(\rho ,\sigma )$ or $H(\rho ,\sigma )$, depending on the notation being used for the von Neumann entropy. Like other entropies, the joint quantum entropy is measured in bits.

In this article, we will use $S(\rho ,\sigma )$ for the joint quantum entropy.

Background

In information theory, for any classical random variable $X$, the classical Shannon entropy $H(X)$ is a measure of uncertain we are about the outcome of $X$. For example, if $X$ is a probability distribution concentrated at one point, the outcome of $X$ is certain and therefore its entropy $H(X)=0$. At the other extreme, if $X$ is the uniform probability distribution with $n$ possible values, intuitively one would expect $X$ is associated with the most uncertainty. Indeed such uniform probability distributions has maximum possible entropy $H(X)=\log _{2}(n)$.

In quantum information theory, the notion of entropy is extended from probability distributions to quantum states, or density matrices. For a state $\rho$, the von Neumann entropy is defined by

$-\operatorname {Tr} \rho \log \rho$

Applying the spectral theorem, or Borel functional calculus for infinite dimensional systems, we see that it generalizes the classical entropy. The physical meaning remains the same. A maximally mixed state, the quantum analog of the uniform probability distribution, has maximum von Neumann entropy. On the other hand, a pure state, or a rank one projection, will have zero von Neumann entropy. We write the von Neumann entropy $S(\rho )$ (or sometimes $H(\rho )$. See also von Neumann entropy).

Definition

Given two quantum states $\rho ^{A}$ and $\rho ^{B}$, if the joint system has a density operator $\rho ^{AB}$, the joint quantum entropy is then

$S(\rho ^{A},\rho ^{B})=S(\rho ^{AB})=-tr(\rho ^{AB}\log(\rho ^{AB}))$

Properties

The classical joint entropy is always at least equal to the entropy of each individual system. This is not the case for the joint quantum entropy. If the quantum state $\rho ^{AB}$ exhibits quantum entanglement, then the entropy of each subsystem may be larger than the joint entropy. This is equivalent to the fact that the conditional quantum entropy may be negative, while the classical conditional entropy may never be.

Consider a maximally entangled state such as a Bell state. If $\rho ^{AB}$ is a Bell state, say,

$\left|\Psi \right\rangle ={\frac {1}{\sqrt {2}}}\left(|00\rangle +|11\rangle \right),$

then the total system is a pure state, with entropy 0, while each individual subsystem is a maximally mixed state, with maximum von Neumann entropy $\log 2=1$. Thus the joint entropy of the combined system is less than that of subsystems. This is because for entangled states, definite states cannot be assigned to subsystems, resulting in positive entropy.

Relations to Other Entropy Measures

The joint quantum entropy $S(\rho ^{AB})$ can be used to define of the conditional quantum entropy:

$S(\rho ^{A}|\rho ^{B})\equiv S(\rho ^{A},\rho ^{B})-S(\rho ^{B})$

and the quantum mutual information:

$S(\rho ^{A}:\rho ^{B})\equiv S(\rho ^{A})+S(\rho ^{B})-S(\rho ^{A},\rho ^{B})$

These definitions parallel the use of the classical joint entropy to define the conditional entropy and mutual information.

Reference

Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information. Cambridge University Press, ISBN 0-521-63505-9.