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 $X$, we define an entropy $H(X)$. The entropy attempts to measure how uncertain we are about the outcome of $X$. If $X$ always has the same value, we know it exactly, and it has an entropy $H(X)=0$. Conversely, if $X$ can take any of $n$ different values, and each value has equal probability of occurring, we know the least possible about it, and it has entropy $H(X)=\log _{2}(n)$.

Similarly, in quantum information theory, for any quantum state $\rho$, the von Neumann entropy attempts to measure our uncertainty about the state. A perfectly mixed state will have the highest possible von Neumann entropy, while a pure state will have a von Neumann entropy of 0. We write the von Neumann entropy $S(\rho )$ (or sometimes $H(\rho )$; see 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. This is particularly striking for fully entangled states such as the Bell states. If $\rho ^{AB}$ is a Bell state, such as $\left|\Psi \right\rangle \equiv {\frac {1}{\sqrt {2}}}\left(|00\rangle +|11\rangle \right)$, then the total system is a pure state, with entropy 0, while each individual subsystem is a perfectly mixed state, with entropy 1 bit.

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 mutual quantum 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.