Private Quantum Channels, Conditional Expectations, and Trace Vectors

TL;DR

This paper explores the connection between private quantum channels, conditional expectations, and trace vectors, providing new geometric characterizations and insights into private states in quantum information theory.

Contribution

It introduces a geometric characterization of single qubit private quantum channels using trace vectors and links trace vectors to private states of channels that are conditional expectations.

Findings

Trace vectors provide a complete description of private states for certain channels.

A new geometric framework for understanding single qubit private quantum channels.

Examples illustrating the theoretical connections between concepts.

Abstract

Private quantum channels are the quantum analogue of the classical one-time pad. Conditional expectations and trace vectors are notions that have been part of operator algebra theory for several decades. We show that the theory of conditional expectations and trace vectors is intimately related to that of private quantum channels. Specifically we give a new geometric characterization of single qubit private quantum channels that relies on trace vectors. We further show that trace vectors completely describe the private states for quantum channels that are themselves conditional expectations. We also discuss several examples.