In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy.
Motivation For simplicity, it will be assumed that all objects in the article are finite-dimensional. We first discuss the classical case. Suppose the probabilities of a finite sequence of events is given by the probability distribution P = {p1...pn}, but somehow we mistakenly assumed it to be Q = {q1...qn}. For instance, we can mistake an unfair coin for a fair one. According to this erroneous assumption, our uncertainty about the j-th event, or equivalently, the amount of information provided after observing the j-th event, is
− log q j . {\displaystyle \;-\log q_{j}.}
The (assumed) average uncertainty of all possible events is then
− ∑ j p j log q j . {\displaystyle \;-\sum _{j}p_{j}\log q_{j}.}
On the other hand, the Shannon entropy of the probability distribution p, defined by
− ∑ j p j log p j , {\displaystyle \;-\sum _{j}p_{j}\log p_{j},}
is the real amount of uncertainty before observation. Therefore the difference between these two quantities
− ∑ j p j log q j − ( − ∑ j p j log p j ) = ∑ j p j log p j − ∑ j p j log q j {\displaystyle \;-\sum _{j}p_{j}\log q_{j}-\left(-\sum _{j}p_{j}\log p_{j}\right)=\sum _{j}p_{j}\log p_{j}-\sum _{j}p_{j}\log q_{j}}
is a measure of the distinguishability of the two probability distributions p and q. This is precisely the classical relative entropy, or Kullback–Leibler divergence:
D K L ( P ‖ Q ) = ∑ j p j log p j q j . {\displaystyle D_{\mathrm {KL} }(P\|Q)=\sum _{j}p_{j}\log {\frac {p_{j}}{q_{j}}}\!.}
Note
In the definitions above, the convention that 0·log 0 = 0 is assumed, since lim x ↘ 0 x log ( x ) = 0 {\displaystyle \lim _{x\searrow 0}x\log(x)=0} . Intuitively, one would expect that an event of zero probability to contribute nothing towards entropy. The relative entropy is not a metric. For example, it is not symmetric. The uncertainty discrepancy in mistaking a fair coin to be unfair is not the same as the opposite situation.
Definition As with many other objects in quantum information theory, quantum relative entropy is defined by extending the classical definition from probability distributions to density matrices. Let ρ be a density matrix. The von Neumann entropy of ρ, which is the quantum mechanical analog of the Shannon entropy, is given by:510
S ( ρ ) = − Tr ρ log ρ . {\displaystyle S(\rho )=-\operatorname {Tr} \rho \log \rho .}
For two density matrices ρ and σ, the quantum relative entropy of ρ with respect to σ is defined by:511
S ( ρ ‖ σ ) = − Tr ρ log σ − S ( ρ ) = Tr ρ log ρ − Tr ρ log σ = Tr ρ ( log ρ − log σ ) . {\displaystyle S(\rho \|\sigma )=-\operatorname {Tr} \rho \log \sigma -S(\rho )=\operatorname {Tr} \rho \log \rho -\operatorname {Tr} \rho \log \sigma =\operatorname {Tr} \rho (\log \rho -\log \sigma ).}
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