Abstract
We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
Highlights
The relative entropy between two density operators ρ, σ, defined asS(ρ|σ ) = Tr[ρ(ln ρ − ln σ )], (1)is an asymptotic measure of their distinguishability
In the non-commutative setting, the relative entropy was later generalized to pairs ωψ, ωη of normal, positive functionals on a general von Neumann algebra by Araki [2–4] using relative modular Hamiltonians
By well-known theorems of the Stinespring-type, this includes in the case of finite-dimensional type I von Neumann algebras the familiar operations of a (i) a unitary time evolution of the density matrix, (ii) a von Neumann measurement followed by post-selection, (iii) forgetting part of the system
Summary
A attractive lower bound in the DPI has recently been given by Junge et al [29] for von Neumann algebras of type I. They consider a certain recovery channel α that is closely related to the Petz map We are motivated in particular by recent applications of the improved form of the DPI in high energy physics [14, 15], where such questions are natural in the algebraic approach to quantum field theory [20] In this context, the von Neumann algebras under consideration are of type III1 [11] and in many cases a distinguished reference state ωη exists, namely the vacuum. Our von Neumann algebras are assumed to be σ -finite, i.e., the existence of a faithful normal state is required
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