Generalized relative entropy ( ε {\displaystyle \varepsilon } -relative entropy) is a measure of dissimilarity between two quantum states. It is a "one-shot" analogue of quantum relative entropy and shares many properties of the latter quantity. In the study of quantum information theory, we typically assume that information processing tasks are repeated multiple times, independently. The corresponding information-theoretic notions are therefore defined in the asymptotic limit. The quintessential entropy measure, von Neumann entropy, is one such notion. In contrast, the study of one-shot quantum information theory is concerned with information processing when a task is conducted only once. New entropic measures emerge in this scenario, as traditional notions cease to give a precise characterization of resource requirements. ε {\displaystyle \varepsilon } -relative entropy is one such particularly interesting measure. In the asymptotic scenario, relative entropy acts as a parent quantity for other measures besides being an important measure itself. Similarly, ε {\displaystyle \varepsilon } -relative entropy functions as a parent quantity for other measures in the one-shot scenario.
Definition To motivate the definition of the ε {\displaystyle \varepsilon } -relative entropy D ε ( ρ ‖ σ ) {\displaystyle D^{\varepsilon }(\rho \|\sigma )} , consider the information processing task of hypothesis testing. In hypothesis testing, we wish to devise a strategy to distinguish between two density operators ρ {\displaystyle \rho } and σ {\displaystyle \sigma } . A strategy is a POVM with elements Q {\displaystyle Q} and I − Q {\displaystyle I-Q} . The probability that the strategy produces a correct guess on input ρ {\displaystyle \rho } is given by Tr ( ρ Q ) {\displaystyle \operatorname {Tr} (\rho Q)} and the probability that it produces a wrong guess is given by Tr ( σ Q ) {\displaystyle \operatorname {Tr} (\sigma Q)} . ε {\displaystyle \varepsilon } -relative entropy captures the minimum probability of error when the state is σ {\displaystyle \sigma } , given that the success probability for ρ {\displaystyle \rho } is at least ε {\displaystyle \varepsilon } . For ε ∈ ( 0 , 1 ) {\displaystyle \varepsilon \in (0,1)} , the ε {\displaystyle \varepsilon } -relative entropy between two quantum states ρ {\displaystyle \rho } and σ {\displaystyle \sigma } is defined as
D ε ( ρ ‖ σ ) = − log 1 ε min { ⟨ Q , σ ⟩ | 0 ≤ Q ≤ I and ⟨ Q , ρ ⟩ ≥ ε } . {\displaystyle D^{\varepsilon }(\rho \|\sigma )=-\log {\frac {1}{\varepsilon }}\min \left\{\left\langle Q,\sigma \right\rangle |0\leq Q\leq I{\text{ and }}\left\langle Q,\rho \right\rangle \geq \varepsilon \right\}~.}
From the definition, it is clear that D ε ( ρ ‖ σ ) ≥ 0 {\displaystyle D^{\varepsilon }(\rho \|\sigma )\geq 0} . This inequality is saturated if and only if ρ = σ {\displaystyle \rho =\sigma } , as shown below.
Relationship to the trace distance Suppose the trace distance between two density operators ρ {\displaystyle \rho } and σ {\displaystyle \sigma } is
‖ ρ − σ ‖ 1 = δ . {\displaystyle {\left\|\rho -\sigma \right\|}_{1}=\delta ~.}
For 0 < ε < 1 {\displaystyle 0<\varepsilon <1} , it holds that
log ε ε − ( 1 − ε ) δ ≤ D ε ( ρ ‖ σ ) ≤ log ε ε − δ . {\displaystyle \log {\frac {\varepsilon }{\varepsilon -(1-\varepsilon )\delta }}\quad \leq \quad D^{\varepsilon }(\rho \|\sigma )\quad \leq \quad \log {\frac {\varepsilon }{\varepsilon -\delta }}~.}
… excerpt ends here. Continue reading the full article.
