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arXiv:1501.07831 · Physical Review B

Post-measurement bipartite entanglement entropy in conformal field theories

M. A. Rajabpour

Published Aug 5, 2015• 38 citations
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Abstract

We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $(1+1)$-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-$1/2 XX$ 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.

DOI
10.1103/physrevb.92.075108

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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.
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 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.
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.
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.
We study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that, for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge c=1/2c=1/2c=1/2 and 1.
The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a 2D conformal field theory~(CFT), the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum, where one of the boundary conditions arise due to the `entanglement cut'. The latter has been argued to be conformal and has been numerically found to be the `free' boundary condition for Ising, Potts and free boson theories. For these models, the `free' boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary minimal models of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the entanglement entropy from the CFT partition functions. These are compared with numerical results.