Entanglement, Entropy and Modular Hamiltonians in Conformal Field Theories.

Research demonstrates two-point correlation functions of modular Hamiltonians share properties with von Neumann entropy and entanglement capacity. Calculations for spherical subregions in conformal field theories confirm equivalence to the stress-tensor conformal block, even with imaginary time separation, offering insights into generic systems and holographic duals.

The behaviour of entanglement, a fundamental property of quantum systems where particles become linked regardless of distance, continues to reveal subtle connections to broader areas of physics, including gravity and information theory. Recent research delves into the statistical relationships between ‘modular Hamiltonians’, operators describing the energy associated with quantum entanglement, and their ‘two-point correlation functions’, which quantify how these energies fluctuate across different parts of a system. Mathew W. Bub and Allic Sivaramakrishnan, both from the Walter Burke Institute for Theoretical Physics at the California Institute of Technology, present a detailed analysis of these correlation functions, demonstrating their connection to established concepts like von Neumann entropy – a measure of entanglement – and the capacity of a quantum channel. Their work, titled ‘Correlation functions of von Neumann entropy’, focuses specifically on conformal field theories, a class of quantum field theories exhibiting scale invariance, and provides analytical results for spherical subregions, offering insights relevant to systems with holographic duals – theoretical frameworks linking quantum gravity in a given spacetime to a quantum field theory on its boundary.

Recent investigations explore a connection between quantum entanglement and the fundamental structure of spacetime, utilising concepts from quantum field theory and general relativity. Researchers are examining how entanglement, quantified through the use of modular Hamiltonians, relates to the dynamics of spacetime itself. Modular Hamiltonians, a tool originating in the mathematical study of von Neumann algebras, provide a means to characterise the entanglement structure within a quantum system, moving beyond simple bipartite entanglement measures.

The research focuses on analysing two-point correlation functions, which measure the statistical relationship between different points in space, and their behaviour when considering imaginary time separation. Imaginary time, a mathematical construct where time is treated as an imaginary number, allows physicists to explore connections between seemingly disparate physical scenarios, often simplifying calculations and revealing hidden symmetries. This technique is frequently employed when transitioning between Minkowski space (the spacetime of special relativity) and Euclidean space, a space without a notion of time.

Specifically, the investigations centre on how these correlation functions behave when the time separation between the measured points is imaginary. This approach suggests an exploration of geometries beyond classical descriptions, potentially linking entanglement to concepts such as wormholes or other non-classical spacetime structures. The underlying premise is that the degree of entanglement between regions of space may be directly related to the geometric connection between those regions, potentially offering a pathway to understanding the emergence of spacetime from quantum mechanical principles. The research aims to establish whether the entanglement structure, as described by modular Hamiltonians and reflected in the two-point correlation functions, can provide insights into the dynamic behaviour of spacetime itself.