Monotonicity of quantum relative entropy and recoverability

Abstract

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

Recoverability in quantum information theory

“…This answers an open question discussed in [34]. In particular, we can say that there is a map R P,t σ ,N that perfectly recovers σ from N (σ ), while having a performance limited by (3.10) when recovering ρ from N (ρ).…”

“…A breakthrough result from [18] established an inequality of the form in corollary 5.1, with however essentially nothing being known about the input and output unitaries. The methods of Fawzi & Renner [18] were generalized in [34] to produce an inequality of the form in (3.10), again with essentially nothing known about the input and output unitaries. Meanwhile, operational proofs for physically meaningful lower bounds on conditional mutual information have appeared as well [51,52], the latter in part based on the notion of fidelity of recovery [22].…”

“…The ideas in this section follow a line of thought developed in [34]. Let E ≡ {p X (x), ρ x AB } be an ensemble of bipartite quantum states with expectationρ AB ≡ x p X (x)ρ x AB .…”

Thermodynamic Implementations of Quantum Processes

“…We proceed to construct a universal conditional typical projector based on ideas from Schur-Weyl duality. The construction presented here is similar to, and inspired by, techniques put forward in earlier work [22,[24][25][26]47,48].…”

Universal recoverability in quantum information

“…The main purpose of this paper is to present several further refinements of the entropy inequality in (1), recently reported in [8] and [9] (see also the earlier contributions on this topic in [10], [11] and the more recent one in [12]). One of the refinements can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum channel is relatively small, then it is possible to perform a recovery channel, such that we can perfectly recover one state while approximately recovering the other.…”