Convolution algebra of superoperators and nonseparability witnesses for quantum operations

Abstract

We define a product between quantum superoperators which is preserved under the Choi-Jamiołkowski-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.

Tristochastic operations and products of quantum states

“…However, we provide here a construction of quantum product of arbitrary density matrices of the same size, which results in a legitimate density matrix. A construction of convolution of quantum states, called twirled product, was recently proposed by Aniello [13][14][15], while other techniques were used to generate a composition of bi-partite density matrices [16] and convolution of quantum superoperators [17]. However, we believe that the extension of those products, with associativity possibly replaced by other properties, might result in new useful applications, especially in the areas where the associativity of convolution is not crucial.…”