A complete hierarchy for the pure state marginal problem in quantum mechanics

Area of Science:

• Quantum Information Science
• Quantum Many-Body Physics

Background:

• Understanding the relationship between quantum systems and their subsystems is fundamental across scientific disciplines.
• The quantum marginal problem, central to quantum mechanics, investigates the existence of a global pure quantum state compatible with given marginal states.

Purpose of the Study:

• To establish a formal link between the quantum marginal problem and the quantum separability problem.
• To develop a computational framework for determining the existence of pure quantum states from given marginals.
• To explore applications of this framework in quantum entanglement and quantum error correction.

Main Methods:

• Establishing a correspondence between the quantum marginal problem and the quantum separability problem.
• Developing a hierarchy of semidefinite programs to solve the marginal problem.
• Applying the framework to analyze absolutely maximally entangled states and quantum error-correcting codes.

Main Results:

• A proven equivalence between the quantum marginal problem and the separability problem.
• A sequence of semidefinite programs capable of deciding the compatibility of marginals with a pure global state.
• Demonstrated that the existence of multiparticle absolutely maximally entangled states is equivalent to the separability of a specific two-party quantum state.
• Showcased that the existence of quantum codes with specified parameters can be framed as a marginal problem.

Conclusions:

• The developed semidefinite programming hierarchy provides a robust method for addressing the quantum marginal problem.
• The established connections offer new perspectives on quantum entanglement, particularly for absolutely maximally entangled states.
• The findings facilitate the design and analysis of quantum error-correcting codes by reinterpreting them within the marginal problem framework.