-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.
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.
In the asymptotic scenario, relative entropy acts as a parent quantity for other measures besides being an important measure itself.
-relative entropy functions as a parent quantity for other measures in the one-shot scenario.
, consider the information processing task of hypothesis testing.
In hypothesis testing, we wish to devise a strategy to distinguish between two density operators
The probability that the strategy produces a correct guess on input
and the probability that it produces a wrong guess is given by
-relative entropy captures the minimum probability of error when the state is
Suppose the trace distance between two density operators
, inheriting this property from the trace distance.
This result and its proof can be found in Dupuis et al.[2] Upper bound: Trace distance can be written as This maximum is achieved when
is the orthogonal projector onto the positive eigenspace of
be the orthogonal projection onto the positive eigenspace of
, we can re-write this as Hence To derive this Pinsker-like inequality, observe that A fundamental property of von Neumann entropy is strong subadditivity.
denote the von Neumann entropy of the quantum state
be a quantum state on the tensor product Hilbert space
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:[3] for every CPTP map
denotes the relative entropy of the quantum states
-relative entropy also obeys monotonicity under quantum operations:[4] for any CPTP map
Since the adjoint of any CPTP map is also positive and unital, this is a valid POVM.
Not only is this interesting in itself, but it also gives us the following alternative method to prove the data processing inequality.
[2] By the quantum analogue of the Stein lemma,[5] where the minimum is taken over
Applying the data processing inequality to the states