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Entanglement Induced Fluctuations of Cold Bosons

Brustein, R, Yarom, A

Published Jan 9, 2005• 3 citations
OpenAlex

Abstract

We show that due to entanglement, quantum fluctuations may differ significantly from standard statistical fluctuations. We calculate quantum fluctuations of the particle number and of the energy in a sub-volume of a system of bosons in a pure state, and briefly discuss the possibility of measuring them. We explain how area-scaling of the energy fluctuations may be relevant to black hole physics.

Related papers

Entanglement induced fluctuations of cold bosons

Ram Brustein, Amos Yarom

We show that due to entanglement, quantum fluctuations may differ significantly from statistical fluctuations. We calculate quantum fluctuations of the particle number and of the energy in a sub-volume of a system of bosons in a pure state, and briefly discuss the possibility of measuring them. We find that energy fluctuations have a non-extensive nature.

arXiv:hep-th/0501058•Journal of Statistical Mechanics Theory and Experiment•Jul 29, 2008•3 citations

Colloquium : Area laws for the entanglement entropy

Jens Eisert, M. Cramer, Martin Bodo Plenio

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

arXiv:0808.3773•Reviews of Modern Physics•Feb 4, 2010•2,867 citations

Entanglement of Coupled Harmonic Oscillators

H. Snijder

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 $N$ with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out $N-l$ 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 $N\\rightarrow\\infty$, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large $N$, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by $S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421$ in the discrete case and $S=\\frac{0.3744}{3} \\ln{l} + 0.6421$ in the infinite case.

•Jan 1, 2019•0 citations

Black holes and quantumness on macroscopic scales

Daniel Flassig, Alexander Pritzel, Nico Wintergerst

It has recently been suggested that black holes may be described as condensates of weakly interacting gravitons at a critical point, exhibiting strong quantum effects. In this paper, we study a model system of attractive bosons in one spatial dimension which is known to undergo a quantum phase transition. We demonstrate explicitly that indeed quantum effects are important at the critical point, even if the number of particles is macroscopic. Most prominently, we evaluate the entropy of entanglement between different momentum modes and observe it to become maximal at the critical point. Furthermore, we explicitly see that the leading entanglement is between long-wavelength modes and is hence a feature independent of ultraviolet physics. If applicable to black holes, our findings substantiate the conjectured breakdown of semiclassical physics even for large black holes. This can resolve long-standing mysteries, such as the information paradox and the no-hair theorem.

arXiv:1212.3344•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 2, 2013•50 citations

Boson-fermion universality of mesoscopic entanglement fluctuations in free systems

Cunzhong Lou, Chushun Tian, Zhixing Zou +2 more

Entanglement fluctuations associated with the Schrödinger evolution of wave functions offer a unique perspective on various fundamental issues ranging from quantum thermalization to state preparation in quantum devices. Very recently, a subset of the present authors showed that, in a class of free-fermion lattice models and interacting spin chains, entanglement dynamics enters into a new regime at long time, with entanglement probes displaying persistent temporal fluctuations, whose statistics falls into the seemingly disparate paradigm of mesoscopic fluctuations in condensed-matter physics. This motivates us to revisit here entanglement dynamics of a canonical bosonic model in many-body physics, i.e., a coupled harmonic oscillator chain. We find that when the system is driven out of equilibrium, the long-time entanglement dynamics exhibits strictly the same statistical behaviors as those of free-fermion models. Specifically, irrespective of entanglement probes and microscopic parameters, the statistical distribution of entanglement fluctuations is flanked by asymmetric tails: sub-Gaussian for upward fluctuations and sub-Gamma for downward; moreover, the variance exhibits a crossover from the scaling ∼ 1 / L to ∼ L A 3 / L 2 as the subsystem size L A increases ( L the total system size). This insensitivity to the particle statistics, dubbed boson-fermion universality, is contrary to the common wisdom that statistical phenomena of a many-body nature depend strongly on particle statistics. Together with our previous work, the present study indicates rich fluctuation phenomena in long-time entanglement dynamics that await in-depth exploration.

arXiv:2411.14687•Physical Review Research•May 30, 2025•1 citation

Entanglement entropy of an acoustic black hole

P.C. Van der Graaf, Henk T. C. Stoof

We introduce a method to numerically compute the entanglement entropy of an acoustic black hole. It is shown that the entanglement entropy of sufficiently large subregions scales linearly with size and thus shows a volume law instead of an area law. The origin of this scaling can be traced back to the non-separable long-distance correlations due to the production of phonon pairs at the horizon. The system is shown to be locally thermal, such that the part of the entanglement entropy scaling with volume is well approximated by the thermal entropy of the outgoing Hawking radiation.

arXiv:2605.30540•May 28, 2026•0 citations
We show that due to entanglement, quantum fluctuations may differ significantly from standard statistical fluctuations. We calculate quantum fluctuations of the particle number and of the energy in a sub-volume of a system of bosons in a pure state, and briefly discuss the possibility of measuring them. We explain how area-scaling of the energy fluctuations may be relevant to black hole physics.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
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 recently been suggested that black holes may be described as condensates of weakly interacting gravitons at a critical point, exhibiting strong quantum effects. In this paper, we study a model system of attractive bosons in one spatial dimension which is known to undergo a quantum phase transition. We demonstrate explicitly that indeed quantum effects are important at the critical point, even if the number of particles is macroscopic. Most prominently, we evaluate the entropy of entanglement between different momentum modes and observe it to become maximal at the critical point. Furthermore, we explicitly see that the leading entanglement is between long-wavelength modes and is hence a feature independent of ultraviolet physics. If applicable to black holes, our findings substantiate the conjectured breakdown of semiclassical physics even for large black holes. This can resolve long-standing mysteries, such as the information paradox and the no-hair theorem.
Entanglement fluctuations associated with the Schrödinger evolution of wave functions offer a unique perspective on various fundamental issues ranging from quantum thermalization to state preparation in quantum devices. Very recently, a subset of the present authors showed that, in a class of free-fermion lattice models and interacting spin chains, entanglement dynamics enters into a new regime at long time, with entanglement probes displaying persistent temporal fluctuations, whose statistics falls into the seemingly disparate paradigm of mesoscopic fluctuations in condensed-matter physics. This motivates us to revisit here entanglement dynamics of a canonical bosonic model in many-body physics, i.e., a coupled harmonic oscillator chain. We find that when the system is driven out of equilibrium, the long-time entanglement dynamics exhibits strictly the same statistical behaviors as those of free-fermion models. Specifically, irrespective of entanglement probes and microscopic parameters, the statistical distribution of entanglement fluctuations is flanked by asymmetric tails: sub-Gaussian for upward fluctuations and sub-Gamma for downward; moreover, the variance exhibits a crossover from the scaling ∼ 1 / L to ∼ L A 3 / L 2 as the subsystem size L A increases ( L the total system size). This insensitivity to the particle statistics, dubbed boson-fermion universality, is contrary to the common wisdom that statistical phenomena of a many-body nature depend strongly on particle statistics. Together with our previous work, the present study indicates rich fluctuation phenomena in long-time entanglement dynamics that await in-depth exploration.
We introduce a method to numerically compute the entanglement entropy of an acoustic black hole. It is shown that the entanglement entropy of sufficiently large subregions scales linearly with size and thus shows a volume law instead of an area law. The origin of this scaling can be traced back to the non-separable long-distance correlations due to the production of phonon pairs at the horizon. The system is shown to be locally thermal, such that the part of the entanglement entropy scaling with volume is well approximated by the thermal entropy of the outgoing Hawking radiation.