Entanglement of purification: from spin chains to holography (1709.07424v2)

Abstract: Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.

• The paper conjectures that entanglement of purification reduces the tensor network bond dimension needed for simulating thermal states in Ising spin chains.
• It demonstrates that in holographic models, Eₚ corresponds to the minimal entanglement wedge cross-section, linking quantum correlations to bulk geometry.
• The study reveals that in random stabilizer tensor networks, Eₚ aligns closely with half the mutual information, simplifying mixed-state entanglement characterization.

Entanglement of Purification: From Spin Chains to Holography

The concept of entanglement of purification (E_p) offers a refined measure of correlation in quantum systems characterized by mixed states. The paper "Entanglement of purification: from spin chains to holography" by Nguyen et al. explores E_p in three distinct quantum systems: the Ising spin chain, models relevant for the AdS/CFT correspondence, and random stabilizer tensor networks. The authors conjecture values for E_p in each model, supported by numerical and analytical methods, thereby illuminating its role in both theoretical and applied contexts, such as tensor network simulations and holography.

Model Systems and Analytical Conjectures

1. Ising Spin Chain: The paper leverages tensor network methods, particularly matrix product states (MPS), to explore the entanglement properties of thermal states. The authors conjecture that E_p can significantly reduce computational complexity in simulating such states. They demonstrate numerically that the bond dimension, a critical tensor network parameter, may be reduced to the square root of that required for thermofield double states.
2. Holographic Models (AdS/CFT): Within the context of holographic duality, where conformal field theories (CFTs) are mapped to gravitational theories in higher-dimensional anti-de Sitter (AdS) spaces, E_p is proposed to relate to the minimal cross-section of the entanglement wedge in the bulk geometry. This conjecture is significant, as it suggests that E_p not only measures correlation but also has a geometric interpretation in holography. The authors illustrate this with several geometric configurations and find agreement with expectations derived from the Ryu-Takayanagi formula.
3. Random Stabilizer Tensor Networks: These networks epitomize simplified quantum states that nevertheless obey an RT-like formula for entanglement entropy. The paper rigorously shows, using properties of stabilizer states, that in such networks, E_p aligns closely with half the mutual information—a fundamental lower bound—demonstrating a simplification not observed in the holographic context.

Properties and Computational Implications

The research rigorously validates several theoretical properties of E_p across these models:

• Inequality Bounds: E_p is shown to be bounded above by the entanglement entropy and below by half the mutual information. These bounds offer insights into its potential as a tool for characterizing total correlations in quantum systems.
• Monotonicity: The paper establishes that E_p is a monotonic quantity, decreasing upon tracing out subsystems, reinforcing its relevance in hierarchical system evaluations.
• Phase Sensitivity: Particularly in the holographic setup, E_p exhibits phase sensitivity, such as transitions where the dual geometry's connectedness shifts due to parameter changes, aligning with quantum phase transitions.

Future Directions and Applications

The implications of this work are manifold. The authors hint at advancing the practical use of E_p in improving tensor network algorithms for mixed states in strongly correlated quantum systems. This is especially pertinent given the potentially significant computational resources saved, which are crucial in large-scale simulations.

On the theoretical front, extending the understanding of E_p in holography may offer deeper insights into the very nature of quantum gravity and spaces. Given the intricate connection between geometry and quantum information, understanding E_p in this context could reveal novel pathways in exploring spacetime emergent properties in the AdS/CFT framework.

In summary, this research offers compelling conjectures and evidence that E_p serves as a functional bridge between quantum information theory and gravitational holography, providing a novel lens through which to view and solve problems in complex quantum systems. Such work pushes forward our conceptual and practical understandings in these domains, aligning with ongoing inquiries at the intersection of quantum computing and theoretical physics.