Two Types of Trilocality of Probability and Correlation Tensors

Area of Science:

• Quantum information theory
• Foundations of quantum mechanics
• Tensor network theory

Background:

• Trilocality is a key concept in quantum information theory, extending notions of nonlocality to three-party systems.
• Hidden variable models offer a framework to understand the underlying reality of quantum correlations.
• Triangle networks are fundamental structures for studying multipartite quantum correlations.

Purpose of the Study:

• To define and analyze continuous (C-triLHVM) and discrete (D-triLHVM) trilocal hidden variable models.
• To investigate the properties of C-trilocal and D-trilocal probability tensors (PTs) and correlation tensors (CTs) in a triangle network.
• To establish conditions for the realizability and characterization of these trilocal tensors.

Main Methods:

• Development of continuous and discrete trilocal hidden variable models.
• Analysis of probability and correlation tensors within a triangle network structure.
• Application of concepts from quantum measurement theory, including local POVMs.
• Investigation of convex combinations and tensor properties.

Main Results:

• A PT (or CT) is D-trilocal if and only if it can be realized by shared separable states and local POVMs in a triangle network.
• A CT is C-trilocal (or D-trilocal) if it's a convex combination of product deterministic CTs with a C-trilocal (or D-trilocal) PT.
• Properties such as path-connectedness and partial star-convexity of the sets of C-trilocal and D-trilocal PTs/CTs were proven.

Conclusions:

• The study provides a rigorous mathematical framework for understanding trilocality in quantum systems.
• The findings offer new insights into the structure and realizability of multipartite quantum correlations.
• The characterization of C-trilocal and D-trilocal tensors contributes to the broader understanding of quantum nonlocality and hidden variable theories.