[Paper Review] Multipartite classical states and detecting quantum discord

Lin Chen, Eric Chitambar+2 more|arXiv (Cornell University)|May 24, 2010

Quantum Information and Cryptography3 citations

TL;DR

This paper introduces and characterizes multipartite pseudo-classical and classical states, providing polynomial-time decidable necessary and sufficient conditions for membership. It shows these states form a measure-zero set in state space and proposes a physical criterion for detecting non-classicality based on commutativity of reduced states after local POVMs.

ABSTRACT

We study various types of multipartite separable states in terms of their inherent classical features. For the two important classes of pseudo-classical, introduced here, and classical states, we provide necessary and sufficient conditions for deciding membership in both which can be checked in polynomial running time. Geometrically, the volume of these states in multipartite state space is found to be measure zero. We also provide a physical criterion for detecting non-classical states based on the commutivity of reduced states following local POVMs performed on individual subsystems.

Motivation & Objective

• To define and analyze multipartite separable states with inherent classical features, particularly pseudo-classical and classical states.
• To establish necessary and sufficient conditions for identifying such states that can be verified efficiently in polynomial time.
• To investigate the geometric measure of these classical states within the full multipartite quantum state space.
• To develop a physical, operational criterion for detecting non-classical correlations (quantum discord) in multipartite systems.

Proposed method

• Introduce the concept of pseudo-classical states as a new class of multipartite separable states with specific classical correlations.
• Define classical states as those admitting a local classical model, and derive conditions for membership using local measurements.
• Use local positive operator-valued measures (POVMs) to probe the reduced states of subsystems and assess their commutativity.
• Formulate a criterion based on the commutativity of post-measurement reduced states following local POVMs to detect non-classicality.
• Analyze the volume of classical and pseudo-classical states in the full state space using geometric and measure-theoretic methods.
• Apply polynomial-time algorithms to test membership in the defined classes of classical and pseudo-classical states.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a multipartite quantum state to be classified as pseudo-classical or classical?
• RQ2 How can one efficiently determine membership in the set of classical or pseudo-classical states using computational methods?
• RQ3 What is the geometric measure of the set of classical and pseudo-classical states within the full multipartite quantum state space?
• RQ4 Can a physical, operational criterion be derived to detect non-classical correlations (quantum discord) in multipartite systems?
• RQ5 How does the commutativity of reduced states after local POVMs relate to the presence or absence of quantum discord?

Key findings

• The paper establishes polynomial-time decidable necessary and sufficient conditions for membership in both pseudo-classical and classical states.
• The set of classical and pseudo-classical states has measure zero in the full multipartite quantum state space, indicating they are rare in a geometric sense.
• A physical criterion for detecting non-classical states is proposed, based on the non-commutativity of reduced density matrices after local POVMs.
• The proposed detection criterion provides a practical, operational method to identify quantum discord in multipartite systems without full state tomography.
• The analysis reveals that classical and pseudo-classical states are structurally constrained, with their existence tied to specific algebraic and geometric properties of the state space.