# Entropy, entanglement, and area: analytical results for harmonic lattice systems.

@article{Plenio2005EntropyEA, title={Entropy, entanglement, and area: analytical results for harmonic lattice systems.}, author={Martin Bodo Plenio and Jens Eisert and J Dressig and Marcus Cramer}, journal={Physical review letters}, year={2005}, volume={94 6}, pages={ 060503 } }

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of…

## 117 Citations

### Entanglement-area law for general bosonic harmonic lattice systems (14 pages)

- Physics
- 2006

We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at…

### Entanglement versus gap for one-dimensional spin systems

- Computer Science, Mathematics
- 2010

A family of 1D local systems is constructed for which some intervals have entanglement entropy, which is polynomial in 1/Δ, whereas previously studied systems had the entropy of all intervals bounded by a constant time log(1/� Δ).

### Entanglement and alpha entropies for a massive Dirac field in two dimensions

- Physics
- 2005

We present some exact results on universal quantities derived from the local density matrix ρ, for a free massive Dirac field in two dimensions. We first find trρn in a novel fashion, which involves…

### Entanglement entropy of squeezed vacua on a lattice

- Physics
- 2015

We derive a formula for the entanglement entropy of squeezed states on a lattice in terms of the complex structure J. The analysis involves the identification of squeezed states with…

### A contour for the entanglement entropies in harmonic lattices

- Physics
- 2017

We construct a contour function for the entanglement entropies in generic harmonic lattices. In one spatial dimension, numerical analysis are performed by considering harmonic chains with either…

### The scaling of entanglement entropy in a honeycomb lattice on a torus

- Physics
- 2014

The entanglement entropy of a noninteracting fermionic system confined to a two-dimensional honeycomb lattice on a torus is calculated. We find that the entanglement entropy can characterize Lifshitz…

### Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices

- Physics, Mathematics
- 2005

We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in…

### Entanglement entropy of fermions in any dimension and the Widom conjecture.

- PhysicsPhysical review letters
- 2006

We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling L(d-1) for the harmonic lattice, for example. We also suggest and provide evidence in support…

### Divergence of entanglement entropy in quantum systems: Zero-modes

- Physics, Computer SciencePhysical Review D
- 2019

The free particle approximation of harmonic oscillator is developed, with which the properties of entropy divergence in continuous bi-partite quantum systems such as the coupled harmonic oscillators and the hydrogen atom are investigated.

### Bounds on the bipartite entanglement entropy for oscillator systems with or without disorder

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2019

A direct alternative proof of an area law for the entanglement entropy of the ground state of disordered oscillator systems is given and this area law in the disordered case is contrasted with divergent lower bounds on the entangling entropy of one-dimensional ordered oscillator chains.

## References

SHOWING 1-10 OF 46 REFERENCES

### Spatial structures and localization of vacuum entanglement in the linear harmonic chain

- Physics
- 2004

We study the structure of vacuum entanglement for two complimentary segments of a linear harmonic chain, applying the modewise decomposition of entangled Gaussian states discussed by Boteno and…

### Entanglement in quantum critical phenomena.

- PhysicsPhysical review letters
- 2003

The results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories.

### Entanglement frustration for Gaussian states on symmetric graphs.

- PhysicsPhysical review letters
- 2004

The maximal entanglement of formation that can be achieved in symmetric graphs like chains, 2D and 3D lattices, mean field models and the platonic solids is determined.

### Ground state entanglement in quantum spin chains

- PhysicsQuantum Inf. Comput.
- 2004

A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions and a possible connection be-tween majorization and renormalization group irreversibility emerges from the numerical analysis.

### Geometric entropy, area and strong subadditivity

- Physics, Computer Science
- 2003

This work shows that the area law follows in general from simple considerations based on quantum mechanics and relativity.

### Quantum-information approach to the Ising model: Entanglement in chains of qubits

- Physics
- 2004

Simple physical interactions between spin-1/2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite…

### Entropy growth of shift-invariant states on a quantum spin chain

- Computer Science, Mathematics
- 2003

The asymptotics of the total entropy is investigated in detail and a class of states derived from quasi-free states on a CAR algebra is investigated, characterized by a measurable subset of the unit interval.

### Generic Bell correlation between arbitrary local algebras in quantum field theory

- Mathematics
- 2000

We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum…