On the notion of quantum copulas

TL;DR

This paper introduces a quantum analogue of copulas, establishing a unique copula for any quantum state of a composite system that preserves entanglement properties and simplifies the state space analysis.

Contribution

It presents the first quantum version of copulas, showing their existence, uniqueness, and relation to entanglement, with a method based on quantum channels and positive maps.

Findings

Existence and uniqueness of quantum copulas for any state

Quantum copulas preserve entanglement or separability

Dimension reduction of quantum states via copulas

Abstract

Working with multivariate probability distributions Sklar introduced the notion of copula in 1959, which turned out to be a key concept to understand the structure of distributions of composite systems. Roughly speaking Sklar proved that a joint distribution can be represented with its marginals and a copula. The main goal of this paper is to present a quantum analogue of the notion of copula. Our main theorem states that for any state of a composite quantum system there exists a unique copula such that they are connected to each other by invertible matrices, moreover, they are both separable or both entangled. So considering copulas instead of states is a separability preserving transformation and efficiently decreases the dimension of the state space of composite systems. The method how we prove these results draws attention to the fact that theorem for states can be achieved by considering the states as quantum channels and using theorems for channels and different kind of positive maps.