Spectral Properties of Tensor Products of Channels

TL;DR

This paper studies the spectral properties of tensor products of quantum channels, characterizing their fixed points and eigenoperators, and provides bounds and criteria for channel factorizability with applications in quantum error correction.

Contribution

It offers a complete description of the spectral structure of tensor product channels and introduces bounds on the multiplicative index, aiding in understanding channel factorization.

Findings

The multiplicative domain of a tensor product splits into the tensor product of individual domains.

Full characterization of fixed points and peripheral eigenoperators of tensor product channels.

Provided criteria and examples for when channels cannot be decomposed into tensor products.

Abstract

We investigate spectral properties of the tensor products of two quantum channels defined on matrix algebras. This leads to the important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ the multiplicative domain of $\mathcal{E}_1\otimes\mathcal{E}_2$ splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper $^*$-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the 'multiplicative index' of a unital channel which was recently introduced. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which can not be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum error correction.