Generalized relative entropy

Generalized relative entropy (formula_1-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. formula_1-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, formula_1-relative entropy functions as a parent quantity for other measures in the one-shot scenario.
Definition.
To motivate the definition of the formula_1-relative entropy formula_5, consider the information processing task of hypothesis testing. In hypothesis testing, we wish to devise a strategy to distinguish between two density operators formula_6 and formula_7. A strategy is a POVM with elements formula_8 and formula_9. The probability that the strategy produces a correct guess on input formula_6 is given by formula_11 and the probability that it produces a wrong guess is given by formula_12. formula_1-relative entropy captures the minimum probability of error when the state is formula_7, given that the success probability for formula_6 is at least formula_1.
For formula_17, the formula_1-relative entropy between two quantum statesformula_6 and formula_7 is defined as
From the definition, it is clear that formula_22. This inequality is saturated if and only if formula_23, as shown below.
Relationship to the trace distance.
Suppose the trace distance between two density operators formula_6 and formula_7 is
For formula_27, it holds that
In particular, this implies the following analogue of the Pinsker inequality
Furthermore, the proposition implies that for any formula_17, formula_31 if and only if formula_23, inheriting this property from the trace distance. This result and its proof can be found in Dupuis et al.
Proof of inequality a).
"Upper bound": Trace distance can be written as
This maximum is achieved when formula_8 is the orthogonal projector onto the positive eigenspace of formula_35. For any POVM element formula_8 we have
so that if formula_38, we have
From the definition of the formula_1-relative entropy, we get
"Lower bound": Let formula_8 be the orthogonal projection onto the positive eigenspace of formula_35, and let formula_44 be the following convex combination of formula_45 and formula_8:
where formula_48
This means
and thus 
Moreover,
Using formula_49, our choice of formula_8, and finally the definition of formula_54, we can re-write this as
Hence
Proof of inequality b).
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 formula_59 denote the von Neumann entropy of the quantum state formula_7, and let formula_61 be a quantum state on the tensor product Hilbert space formula_62. Strong subadditivity states that
where formula_64 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 formula_66, where formula_67 denotes the relative entropy of the quantum states formula_68.
It is readily seen that formula_1-relative entropy also obeys monotonicity under quantum operations:
for any CPTP map formula_66.
To see this, suppose we have a POVM formula_72 to distinguish between formula_73 and formula_74 such that formula_75. We construct a new POVM formula_76 to distinguish between formula_6 and formula_7. Since the adjoint of any CPTP map is also positive and unital, this is a valid POVM. Note that formula_79, where formula_80 is the POVM that achieves formula_5.
Not only is this interesting in itself, but it also gives us the following alternative method to prove the data processing inequality.
By the quantum analogue of the Stein lemma,
where the minimum is taken over formula_85 such that formula_86
Applying the data processing inequality to the states formula_87 and formula_88 with the CPTP map formula_89, we get
Dividing by formula_91 on either side and taking the limit as formula_92, we get the desired result.