Generalized relative entropy (-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. -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, -relative entropy functions as a parent quantity for other measures in the one-shot scenario.
To motivate the definition of the -relative entropy , consider the information processing task of hypothesis testing. In hypothesis testing, we wish to devise a strategy to distinguish between two density operators and . A strategy is a POVM with elements and . The probability that the strategy produces a correct guess on input is given by and the probability that it produces a wrong guess is given by . -relative entropy captures the minimum probability of error when the state is , given that the success probability for is at least .
For , the -relative entropy between two quantum states and is defined as
From the definition, it is clear that . This inequality is saturated if and only if , as shown below.
To derive this Pinsker-like inequality, observe that
Alternative proof of the Data Processing inequality
A fundamental property of von Neumann entropy is strong subadditivity. Let denote the von Neumann entropy of the quantum state , and let be a quantum state on the tensor product Hilbert space. Strong subadditivity states that
where refer to the reduced density matrices on the spaces indicated by the subscripts. When re-written in terms of mutual information, this inequality has an intuitive interpretation; it states that the information content in a system cannot increase by the action of a local quantum operation on that system. In this form, it is better known as the data processing inequality, and is equivalent to the monotonicity of relative entropy under quantum operations:
for every CPTP map, where denotes the relative entropy of the quantum states .
It is readily seen that -relative entropy also obeys monotonicity under quantum operations:
for any CPTP map . To see this, suppose we have a POVM to distinguish between and such that . We construct a new POVM to distinguish between and . Since the adjoint of any CPTP map is also positive and unital, this is a valid POVM. Note that , where is the POVM that achieves . Not only is this interesting in itself, but it also gives us the following alternative method to prove the data processing inequality.