Witnessing random unitary and projective quantum channels: Complementarity between separable and maximally entangled states

TL;DR

This paper develops a method to witness and distinguish between different quantum channels and states, revealing a fundamental complementarity between separable and maximally entangled states through geometric and algebraic insights.

Contribution

It introduces a novel approach to compute witnesses for quantum channels and states, highlighting their entanglement properties and geometric structures using the Choi-Jamiolkowski isomorphism and a new Schmidt decomposition.

Findings

Mixtures of unitary evolutions map to maximally entangled states

Separable states originate from quantum-state reduction maps

The complementarity is characterized by a new Schmidt decomposition

Abstract

Modern applications in quantum computation and quantum communication require the precise characterization of quantum states and quantum channels. In practice, this means that one has to determine the quantum capacity of a physical system in terms of measurable quantities. Witnesses, if properly constructed, succeed in performing this task. We derive a method that is capable to compute witnesses for identifying deterministic evolutions and measurement-induced collapse processes. At the same time, applying the Choi-Jamiolkowski isomorphism, it uncovers the entanglement characteristics of bipartite quantum states. Remarkably, a statistical mixture of unitary evolutions is mapped onto mixtures of maximally entangled states, and classical separable states originate from genuine quantum-state reduction maps. Based on our treatment we are able to witness these opposing attributes at once and, furthermore, obtain an insight into their different geometric structures. The complementarity is further underpinned by formulating a complementary Schmidt decomposition of a state in terms of maximally entangled states and discrete Fourier-transformed Schmidt coefficients.