Convolution algebra of superoperators and nonseparability witnesses for quantum operations

TL;DR

This paper introduces a convolution product for quantum superoperators that preserves channel-state duality and enables the construction of nonseparability witnesses for quantum operations.

Contribution

It defines a new algebraic structure on superoperators and demonstrates its use in identifying nonseparability of quantum channels.

Findings

The convolution product is preserved under channel-state isomorphism.

Nonseparability witnesses can be constructed within the superoperator space.

The algebraic framework simplifies the detection of nonseparable quantum operations.

Abstract

We define a product between quantum superoperators which is preserved under the Choi-Jamio{\l}kowski-Kraus-Sudarshan channel-state isomorphism. We then identify the product as the convolution on the space of superoperators, with respect to which the channel-state duality is also an algebra isomorphism. We find that any witness operator for detecting nonseparability of quantum operations on separated parties can be written entirely within the space of superoperators with the help of the convolution product.