Randomized channel-state duality

TL;DR

This paper introduces a randomized approach to channel-state duality in quantum information, enabling efficient low-rank approximations and data compression for quantum channels and states, demonstrated on a spin system.

Contribution

It presents a novel randomized duality method that approximates quantum channels with fewer pure states, improving efficiency in quantum data processing.

Findings

Achieves accuracy of 1/N in representing quantum channels

Provides a low-rank approximation significantly smaller than the system dimension

Demonstrates applications on a chaotic 1D spin system

Abstract

Channel-state duality is a central result in quantum information science. It refers to the correspondence between a dynamical process (quantum channel) and a static quantum state in an enlarged Hilbert space. Since the corresponding dual state is generally mixed, it is described by a Hermitian matrix. In this article, we present a randomized channel-state duality. In other words, a quantum channel is represented by a collection of $N$ pure quantum states that are produced from a random source. The accuracy of this randomized duality relation is given by $1/N$, with regard to an appropriate distance measure. For large systems, $N$ is much smaller than the dimension of the exact dual matrix of the quantum channel. This provides a highly accurate low-rank approximation of any quantum channel, and, as a consequence of the duality relation, an efficient data compression scheme for mixed quantum states. We demonstrate these two immediate applications of the randomized channel-state duality with a chaotic $1$-dimensional spin system.