Multifold behavior of the information transmission by the quantum 3-switch

TL;DR

This paper explores how information transmission via three quantum channels in superposition of causal orders behaves under noise, revealing multiple behaviors and classifying quantum switch matrices to understand quantum control of causal orders.

Contribution

It introduces a classification of quantum switch matrices based on their invariants and provides analytical expressions for Holevo information in key cases, advancing understanding of quantum causal order control.

Findings

Transmission exhibits three distinct behaviors with varying noise levels.

Classification of quantum switch matrices by characteristic polynomials.

Analytical expressions for Holevo information in representative cases.

Abstract

We uncover new behaviors of the transmission of information by three quantum channels in superposition of causal orders subject to some level of noise. We find that the transmission can exhibit three different behaviors as the level of noise is varied. This multifold behavior can be explained by the different equivalence classes of quantum switch matrices related to specific combinations of causal orders. We classify these matrices using their characteristic polynomials and matrix invariants, and we calculate analytical expressions for the Holevo information in three representative cases. Our results are a step forward to understand and harness quantum control of causal orders with different levels of noise. We also study the Holevo information as function of a continuous order parameter and analyse transitions at integer values.