Properties of Conjugate Channels with Applications to Additivity and Multiplicativity

TL;DR

This paper investigates the properties of conjugate quantum channels, demonstrating their equivalence in key measures like entropy and norms, and shows how these properties relate to additivity and multiplicativity conjectures, with applications to well-known channel types.

Contribution

It introduces methods to construct conjugate channels, proves their key properties, and reduces complex conjectures to simpler cases, providing explicit formulas for common channels.

Findings

Conjugate channels share the same minimal output entropy and maximal output p-norm as original channels.

Additivity and multiplicativity conjectures hold for channels if and only if they hold for their conjugates.

Explicit formulas for conjugates of well-known channels, including entanglement-breaking and depolarizing channels.

Abstract

Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.