Entanglement in onedimensional critical state after measurements
Abstract
The entanglement entropy (EE) of the ground state of a onedimensional Hamiltonian at criticality has a universal logarithmic scaling with a prefactor given by the central charge $c$ of the underlying 1+1d conformal field theory. When the system is probed by measurements, the entanglement in the critical ground state is inevitably affected due to wavefunction collapse. In this paper, we study the effect of weak measurements on the entanglement scaling in the ground state of the onedimensional critical transversefield Ising model. For the measurements of the spins along their transverse spin axis, we identify interesting postmeasurement states associated with spatially uniform measurement outcomes. The EE in these states still satisfies the logarithmic scaling but with an alternative prefactor given by the effective central charge $c_{\text{eff}}$. We derive the analytical expression of $c_{\text{eff}}$ as a function of the measurement strength. Using numerical simulations, we show that for the EE averaged over all postmeasurement states based on their Bornrule probabilities, the numerically extracted effective central charge appears to be independent of the measurement strength, contrary to the usual expectation that local and nonoverlapping measurements reduce the entanglement in the system. We also examine the behavior of the average EE under (biased) forced measurements where the measurement outcomes are sampled with a predetermined probability distribution without intersite correlations. In particular, we find an optimal probability distribution that can serve as a meanfield approximation to the Bornrule probabilities and lead to the same $c_{\text{eff}}$ behavior. The effects of the measurements along the longitudinal spin axis and the postmeasurement correlation functions are also discussed.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.08255
 arXiv:
 arXiv:2301.08255
 Bibcode:
 2023arXiv230108255Y
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 12+5 pages, 8+2 figures