Beyond Complete Positivity

TL;DR

This paper broadens the understanding of quantum dynamical maps by showing that, under relaxed initial state assumptions, many non-completely positive maps can be physically realized, challenging traditional constraints in quantum information theory.

Contribution

It introduces a general framework for subsystem quantum maps that includes non-CP maps, relaxing the usual initial state assumptions and expanding the set of physically realizable dynamics.

Findings

Many non-CP maps can be physically realized as subsystem dynamical maps.

The framework demonstrates a trade-off between initial state restrictions and allowed joint unitaries.

A non-CP map violating quantum data processing inequality is explicitly constructed.

Abstract

We provide a general and consistent formulation for linear subsystem quantum dynamical maps, developed from a minimal set of postulates, primary among which is a relaxation of the usual, restrictive assumption of uncorrelated initial system-bath states. We describe the space of possibilities admitted by this formulation, namely that, far from being limited to only completely positive (CP) maps, essentially any $\mathbb{C}$-linear, Hermiticity-preserving, trace-preserving subsystem map can arise as a legitimate subsystem dynamical map from a joint unitary evolution of a system coupled to a bath. The price paid for this added generality is a trade-off between the set of admissible initial states and the allowed set of joint system-bath unitary evolutions. As an application we present a simple example of a non-CP map constructed as a subsystem dynamical map that violates some fundamental inequalities in quantum information theory, such as the quantum data processing inequality.