Recoverability in quantum information theory

Abstract

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra and the recently introduced Rényi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.

Measures and applications of quantum correlations

“…Several of the measures introduced extend straightforwardly to multipartite systems, in particular the geometric ones (by defining the set of fully classical states as the states diagonal with respect to a product of orthonormal bases for each subsystem [24,93], and any set of partially classical-quantum states with respect to a selection of subsystems treated as classical, as detailed in [28]), the measurement induced geometric and informational ones (again depending on a selection of subsystems on which local measurements are applied), the entanglement activation ones [35,36,28], and the recoverability ones [316]. As previously pointed out, it will be an interesting future direction to develop consistent generalisations of the unitary response and coherence based (in the particular case of asymmetry measures) approaches to define faithful two-sided and more general multipartite measures of QCs, especially in view of the operational merits that these types of measures exhibit in the one-sided case for bipartite systems.…”

Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle

“…The Rényi entropy is widely used in communication and coding theory, quantum information theory, signal processing, data mining and many other areas [4,5]. Stationary states which maximize Rényi entropy have already been well investigated.…”

“…Extension of the Rényi entropy for the continuous case can be defined as H(X, β) = 1 1 − β log p β (x) dx , β = 1, β > 0, (1.3) where X is an absolutely continuous random variable with probability density function (PDF) p. The Rényi entropy is often taken as a typical measure of complexity to describe dynamical systems in physics, engineering and information theory [4]. The Rényi entropy applications are overviewed in [5].…”

Approximate Quantum Markov Chains

“…More precisely, it was shown [28,53,85,142,145,153,175] that for any state ρ ABC there exists a recovery map R B→BC such that …”

“…Beigi [16] and Dupuis [48] used variations of the Riesz-Thorin theorem based on Hadamard's three line theorem to show properties of the minimal Rényi relative entropy and conditional Rényi entropy, respectively. Wilde [175] first used complex interpolation theory to prove remainder terms for the monotonicity of quantum relative entropy. Extensions and further applications of this approach are discussed by Dupuis and Wilde [49].…”