Abstract
Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.
Highlights
Various pre- and partial orders have been the subject of extensive study both in mathematical statistics [24, 5, 32, 55, 2, 54, 21] and in information theory [49, 29, 20]
An example of paramount importance is that provided by the majorization preorder [34]: a probability distribution p1 is said to majorize another distribution p2, in formula p1 p2, whenever there exists a bistochastic1 transformation T such that T p1 = p2
In the limit α → ∞ the sandwiched quantum Renyi divergence converges to the so-called max-divergence [48, 22], i.e., Dmax(ρ σ) := inf λ ∈ R : ρ ≤ 2λσ. Both non-commutative families of Renyi divergences introduced above satisfy many desirable properties: they are monotonically increasing in α, they satisfiy the data-processing inequality, and in the limit α → 1 they both converge to the relative entropy
Summary
Various pre- and partial orders have been the subject of extensive study both in mathematical statistics [24, 5, 32, 55, 2, 54, 21] and in information theory [49, 29, 20]. As a consequence of Blackwell’s equivalence theorem [5], the more general case of dichotomies is completely characterized by a finite set of simple inequalities, which directly reduce to those of Hardy, Littlewood and Polya if q1 and q2 are both taken to be uniform In this more general scenario, a relative Lorenz curve can be associated to each dichotomy, and the preorder visualized [47]. This allows us to derive our main results, Theorems 3.4 and 3.6, which together show that the relative entropy fully characterizes when pairwise transformations are possible asymptotically.
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