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arXiv:1105.1211 · Physical Review A

Entanglement in bipartite pure states of an interacting boson gas obtained by local projective measurements

Francis N. C. Paraan, Javier Molina-Vilaplana, V. E. Korepin, Sougato Bose

Published Sep 20, 2011• 2 citations
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Abstract

We quantify the extractable entanglement of excited states of a Lieb-Liniger gas that are obtained from coarse-grained measurements on the ground state in which the boson number in one of two complementary contiguous partitions of the gas is determined. Numerically exact results obtained from the coordinate Bethe ansatz show that the von Neumann entropy of the resulting bipartite pure state increases monotonically with the strength of repulsive interactions and saturates to the impenetrable-boson limiting value. We also present evidence indicating that the largest amount of entanglement can be extracted from the most probable projected state having half the number of bosons in a given partition. Our study points to a fundamental difference between the nature of the entanglement in free-bosonic and free-fermionic systems, with the entanglement in the former being zero after projection, while that in the latter (corresponding to the impenetrable-boson limit) being nonzero.

DOI
10.1103/physreva.84.032330

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We quantify the extractable entanglement of excited states of a Lieb-Liniger gas that are obtained from coarse-grained measurements on the ground state in which the boson number in one of two complementary contiguous partitions of the gas is determined. Numerically exact results obtained from the coordinate Bethe ansatz show that the von Neumann entropy of the resulting bipartite pure state increases monotonically with the strength of repulsive interactions and saturates to the impenetrable-boson limiting value. We also present evidence indicating that the largest amount of entanglement can be extracted from the most probable projected state having half the number of bosons in a given partition. Our study points to a fundamental difference between the nature of the entanglement in free-bosonic and free-fermionic systems, with the entanglement in the former being zero after projection, while that in the latter (corresponding to the impenetrable-boson limit) being nonzero.
We study a way of establishing a pure entangled state between two segments of a 1D ring using impenetrable bosons. The two bosons are initially simply placed at two positions on the ring which is an unentangled state. After some time evolution, we project the segments to a pure state through coarse grained measurements which ascertain whether there is a particle present in a given segment without revealing any information about its position within the segment. Subject to finding a particle in each segment, we quantify the entanglement established between the segments through the von Neumann entropy of a subsystem. We also investigate whether this entanglement increases with the number of particles and compare the entanglement in dynamical and ground states.
The entanglement of the N-harmonium system (i.e., a completely-integrable model of N particles where both the confinement and the two-particle interaction are har-monic) is shown to be analytically determined in terms of N and the relative interaction strength. First, the entanglement of the N-boson system via the von Neumann entropy is computed by use of the corresponding natural occupation numbers. Below a critical value Nc of N it is found that, for positive values of the coupling constant the entan-glement grows when the number of particles is increasing; opposite situation occurs for N> Nc. Then the entanglement of the N-fermion system is investigated by means of the linear entropy, which is a linearized form of the von Neumann entropy of the one-body reduced density matrix. We have shown that the entanglement grows when the number of particles is increasing in the region of negative coupling. For positively increasing values of the coupling the entanglement as a function of N varies according to various regimes. Moreover, in the strong coupling regime the entropy of the system decreases when N is growing. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Finally, let us point out that the spatial and spin contributions to the total entanglement are of comparable size.
The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of NNN particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of NNN and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number NcN_cNc​ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for N>NcN > N_cN>Nc​. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when NNN is growing. For fixed NNN, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.
We consider the entanglement between two spatial subregions in the Lieb-Liniger model of bosons in one spatial dimension interacting via a contact interaction. Using ground-state path integral quantum Monte Carlo we numerically compute the R\'enyi entropy of the reduced density matrix of the subsystem as a measure of entanglement. Our numerical algorithm is based on a replica method previously introduced by the authors, which we extend to efficiently study the entanglement of spatial subsystems of itinerant bosons. We confirm a logarithmic scaling of the R\'enyi entropy with subsystem size that is expected from conformal field theory, and compute the nonuniversal subleading constant for interaction strengths ranging over two orders of magnitude. In the strongly interacting limit, we find agreement with the known free fermion result.
Abstract Entanglement entropy under a particle bipartition generates complementary information to mode entanglement because it is sensitive to interactions and particle statistics at leading order and does not depend on any externally imposed length scale. In this paper, we investigate the particle entanglement entropy in a system of N interacting spinless lattice fermions in one spatial dimension by combining bosonization techniques with exact and approximate numerical methods. We introduce a general scaling form for the fermionic particle entanglement entropy captured by a shape function that enters as an extensive interaction-induced correction to a known free fermion result. A general asymptotic expansion in the total number of particles demonstrates that its form is robust for different values of the Rényi index and highlights how quantum correlations are encoded in the n-particle density matrix of a pure many-body quantum state.
The entanglement of a pure bipartite state is uniquely measured by the von-Neumann entropy of its reduced density matrices. Though it cannot specify all the non-local characteristics of pure entangled states. It was proven that for every possible value of entanglement of a bipartite system, there exists an infinite number of equally entangled pure states, not comparable(satisfies Nielsen’s criteria) to each other. In this work, we investigate other correlation measures of pure bipartite states that are able to differentiate the quantum correlations of the states with entropy of entanglement. In Schmidt rank 3, we consider the whole set of states having same entanglement and compare how minutely such states can be distinguished by other correlation measures. Then for different values of entanglement we compare the sets of states belonging to the same entanglement and also investigate the graphs of different correlation measures. We extend our search to Schmidt rank 4 and 5 also.