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arXiv:1512.03940 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results

M. A. Rajabpour

Published Jun 1, 2016• 37 citations
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Abstract

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.

DOI
10.1088/1742-5468/2016/06/063109

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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.
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.
The bipartite entanglement entropy of a segment of length lll in 1+11+11+1-dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\mathrm{ln}l+\ensuremath{\gamma}$, where ccc is the central charge of the CFT and $\ensuremath{\gamma}$ is a cutoff-dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have more bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy, but also have more multipartite entanglement content. In particular, we argue that since a system with a smaller smallest scaling dimension has a larger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in 1+11+11+1 dimensions. Furthermore, we also argue that the QFI decays under renormalization group flow between two unitary CFTs. Finally, we also comment on nonconformal but scale-invariant systems.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
We study bipartite post measurement entanglement entropy after selective measurements in quantum chains. We first study the quantity for the critical systems that can be described by conformal field theories. We find a connection between post measurement entanglement entropy and the Casimir energy of floating objects. Then we provide formulas for the post measurement entanglement entropy for open and finite temperature systems. We also comment on the Affleck-Ludwig boundary entropy in the context of the post measurement entanglement entropy. Finally, we also provide some formulas regarding modular hamiltonians and entanglement spectrum in the after measurement systems. After through discussion regarding CFT systems we also provide some predictions regarding massive field theories. We then discuss a generic method to calculate the post measurement entanglement entropy in the free fermion systems. Using the method we study the post measurement entanglement entropy in the XY spin chain. We check numerically the CFT and the massive field theory results in the transverse field Ising chain and the XX model. In particular, we study the post meaurement entanglement entropy in the infinite, periodic and open critical transverse field Ising chain and the critical XX model. The effect of the temperature and the gap is also discussed in these models.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.