Uniqueness of Purifications Is Equivalent to Haag Duality

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Algebraic Quantum Field Theory

Background:

• The uniqueness of purifications of quantum states is a fundamental concept in quantum information theory.
• Purifications are analyzed using von Neumann algebras, M_A and M_B, acting on a Hilbert space H.
• Haag duality, M_A = M_B', is a key condition in quantum field theory.

Purpose of the Study:

• To establish the equivalence between the uniqueness of purifications and Haag duality for commuting von Neumann algebras.
• To investigate the conditions under which the uniqueness of purifications can fail, particularly in systems with infinite degrees of freedom.
• To demonstrate the failure of uniqueness of purifications and Haag duality in specific condensed matter models.

Main Methods:

• Modeling quantum systems A and B using commuting von Neumann algebras M_A and M_B.
• Applying the concept of Haag duality (M_A = M_B') to analyze purification uniqueness.
• Investigating systems with infinite degrees of freedom and their algebraic properties.
• Analyzing ground state sectors of topologically ordered models on infinite lattices.

Main Results:

• Uniqueness of purifications is shown to be equivalent to Haag duality (M_A = M_B') for commuting von Neumann algebras.
• The uniqueness of purifications can fail in systems with infinitely many degrees of freedom.
• Failure of uniqueness and Haag duality is demonstrated in ground state sectors of 2D topologically ordered models at RG fixed points.

Conclusions:

• The study provides a direct link between purification uniqueness and Haag duality in quantum systems.
• Infinite degrees of freedom introduce complexities that can lead to the breakdown of purification uniqueness.
• The findings have implications for understanding quantum states in topologically ordered phases of matter.