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arXiv:2206.05781

Entanglement in harmonic systems at coherent states

Dimitrios Katsinis, Georgios Pastras

Published Jun 12, 2022• 0 citations
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Abstract

It is well-known that entanglement entropy in field theory at its ground state is dominated by an area law term, presenting a similarity to the entropy of black holes. It is interesting to investigate whether this similarity can be extended by showing that gravitational dynamics emerges from the first law of entanglement thermodynamics. Answering this question requires the specification of the modular Hamiltonian. Motivated by the above, we study entanglement in the toy model of harmonic systems lying at any classicalmost state, i.e. any coherent state. We specify explicitly the reduced density matrix and its time-evolution, as well as the modular Hamiltonian. Interestingly, the time evolution is unitary and we specify the effective Hamiltonian which generates it. Our results provide the tools to investigate the similarity between gravity and entanglement in discretized free scalar field theory in the framework of (Phys Rev Lett 71:666, 1993).

DOI
10.48550/arxiv.2206.05781

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It is well-known that entanglement entropy in field theory at its ground state is dominated by an area law term, presenting a similarity to the entropy of black holes. It is interesting to investigate whether this similarity can be extended by showing that gravitational dynamics emerges from the first law of entanglement thermodynamics. Answering this question requires the specification of the modular Hamiltonian. Motivated by the above, we study entanglement in the toy model of harmonic systems lying at any classicalmost state, i.e. any coherent state. We specify explicitly the reduced density matrix and its time-evolution, as well as the modular Hamiltonian. Interestingly, the time evolution is unitary and we specify the effective Hamiltonian which generates it. Our results provide the tools to investigate the similarity between gravity and entanglement in discretized free scalar field theory in the framework of (Phys Rev Lett 71:666, 1993).
A bstract The entanglement entropy of a free scalar field in its ground state is dominated by an area law term. It is noteworthy, however, that the study of entanglement in scalar field theory has not advanced far beyond the ground state. In this paper, we extend the study of entanglement of harmonic systems, which include free scalar field theory as a continuum limit, to the case of the most general Gaussian states, namely the squeezed states. We find the eigenstates and the spectrum of the reduced density matrix and we calculate the entanglement entropy. We show that our method is equivalent to the correlation matrix method. Finally, we apply our method to free scalar field theory in 1+1 dimensions and show that, for very squeezed states, the entanglement entropy is dominated by a volume term, unlike the ground-state case. Even though the state of the system is time-dependent in a non-trivial manner, this volume term is time-independent. We expect this behaviour to hold in higher dimensions as well, as it emerges in a large-squeezing expansion of the entanglement entropy for a general harmonic system.
The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size NNN with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out N−lN-lN−l oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for NrightarrowinftyN\\rightarrow\\inftyNrightarrowinfty, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large NNN, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by S=frac0.27763lnbigg(fracNpisinfracpilNbigg)+0.6421S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421S=frac0.27763lnbigg(fracNpisinfracpilNbigg)+0.6421 in the discrete case and S=frac0.37443lnl+0.6421S=\\frac{0.3744}{3} \\ln{l} + 0.6421S=frac0.37443lnl+0.6421 in the infinite case.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.
We reexamine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between the inside and outside of the horizon. Motivated by the observation that certain modes of gravitational fluctuations in a black-hole background behave as scalar fields, we compute the entanglement entropy of such a field, by tracing over its degrees of freedom inside a sphere. We show that while this entropy is proportional to the area of the sphere when the field is in its ground state, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. The area law is thus recovered for large areas. Further, we identify the location of the degrees of freedom that give rise to the above entropy.
We investigate the dynamics of the ground state entanglement entropy for a discretized scalar field propagating within the Oppenheimer-Snyder collapse metric. Starting from a well-controlled initial configuration, we follow the system as it evolves toward the formation of a horizon and, eventually, a singularity. Our approach employs an Ermakov-like equation to determine the time-dependent ground state of the field and calculates the resulting entanglement entropy by tracing out the degrees of freedom inside a spherical region within the matter sphere. We find that the entanglement entropy exhibits nontrivial scaling and time dependence during collapse. Close to the horizon, the entropy can deviate from the simple area law, reflecting the rapid changes in geometry and field configuration. Although the model is idealized, these results provide insights into the generation and scaling of entanglement in the presence of realistic, dynamically evolving gravitational fields.
We study a proposed statistical explanation for the Bekenstein-Hawking entropy of a black hole in which entropy arises quantum-mechanically as a result of entanglement. Arguments for the identification of black hole entropy with entanglement entropy are reviewed in the framework of quantum field theory, emphasizing the role of renormalization and the need for a physical short-distance cutoff. \n \nOur main novel contribution is a calculation of entanglement entropy in loop quantum gravity. The kinematical Hilbert space and spin network states are introduced, and the entanglement entropy of these states is calculated using methods from quantum information theory. The entanglement entropy is compared with the density of states previously computed for isolated horizons in loop quantum gravity, and the two are found to agree up to a topological term. \n \nWe investigate a conjecture due to Sorkin that the entanglement entropy must be a monotonically increasing function of time under the assumption of causality. For a system described by a finite-dimensional Hilbert space, the conjecture is found to be trivial, and for a system described by an infinite-dimensional Hilbert space a counterexample is provided. \n \nFor quantum states with Euclidean symmetry, the area scaling of the entanglement entropy is shown to be equivalent to the strong additivity condition on the entropy. The strong additivity condition is naturally interpreted in information-theoretic terms as a continuous analog of the Markov property for a classical random variable. We explicitly construct states of a quantum field theory on the one-dimensional real line in which the area law is exactly satisfied.