Approximation of the Nearest Classical-Classical State to a Quantum State

TL;DR

This paper introduces a gradient-based method on Stiefel manifolds to approximate the nearest classical-classical state to a given quantum state, facilitating quantification of quantum correlations.

Contribution

It proposes a novel gradient descent approach using Frobenius norm for approximating quantum states, ensuring decomposition into tensor products and applicability in real-world scenarios.

Findings

Objective value decreases along the flow

Method guarantees quantum state decomposition

Numerical results confirm real-world applicability

Abstract

The capacity of quantum computation exceeds that of classical computers. A revolutionary step in computation is driven by quantumness or quantum correlations, which are permanent in entanglements but often in separable states; therefore, quantifying the quantumness of a state in a quantum system is an important task. The exact quantification of quantumness is an NP-hard problem; thus, we consider alternative approaches to approximate it. In this paper, we take the Frobenius norm to establish an objective function and propose a gradient-driven descent flow on Stiefel manifolds to determine the quantity. We show that the objective value decreases along the flow by proofs and numerical results. Besides, the method guarantees the ability to decompose quantum states into tensor products of certain structures and maintain basic quantum assumptions. Finally, the numerical results eventually confirm the applicability of our method in real-world settings.