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arXiv:1503.07771

Fate of the area-law after partial measurement in quantum field theories

M. A. Rajabpour

Published Mar 26, 2015• 7 citations
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Abstract

We calculate numerically the Rényi bipartite entanglement entropy of the ground state of Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators in $d=1,2$ and $3$ dimensions. We show that after partial measurement as far as the two parts are connected the area-law is valid in generic dimensions except in $d=1$ conformal field theories. When due to the measured region the two parts are disconnected, the entanglement entropy decreases exponentially with respect to the minimum distance between the regions. In the massless case the decay is power-law with a universal dimension-dependent exponent.

DOI
10.48550/arxiv.1503.07771

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We calculate numerically the Rényi bipartite entanglement entropy of the ground state of Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators in d=1,2d=1,2d=1,2 and 333 dimensions. We show that after partial measurement as far as the two parts are connected the area-law is valid in generic dimensions except in d=1d=1d=1 conformal field theories. When due to the measured region the two parts are disconnected, the entanglement entropy decreases exponentially with respect to the minimum distance between the regions. In the massless case the decay is power-law with a universal dimension-dependent exponent.
Abstract We calculate analytically the Rényi bipartite entanglement entropy S α of the ground state of 1 + 1 dimensional conformal field theories (CFT) after performing a projective measurement in part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region A separates the two parts B and B ¯ , the entanglement entropy between B and B ¯ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank α of the Rényi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein–Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.
We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in (1+1)(1+1)(1+1)-dimensional conformal field theories with periodic and open boundary conditions. After defining the setup we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-1/2XX1/2 XX1/2XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.
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.
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 most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions---corresponding to the Klein-Gordon case---upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 (2005). These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.
An upper bound of the relative entanglement entropy of thermal states at an inverse temperature βββ of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance ddd in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times ln⁡∣tanh⁡(πd/2β)∣\ln | \tanh (πd/ 2 β) |ln∣tanh(πd/2β)∣ which signifies power law decay for asymptotic ddd where the exponent depends on β<∞β< \inftyβ<∞.
We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz free scalar theory. We present simple intuitive arguments based on “non-local” effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the z → ∞ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in support of this behavior. In (2 + 1)-dimensions we also study some shape dependent aspects of entanglement. We argue that in the massless limit corner contributions are no more additive for large enough dynamical exponent due to non-local effects of Lifshitz theories. We also comment on possible holographic duals of such theories based on the sign of tripartite information.