Nonlocality and entanglement in measured critical quantum Ising chains

TL;DR

This paper investigates how local measurements affect the entanglement and correlations in a critical quantum Ising chain, revealing that average effects leave the scaling unchanged while postselection induces continuous variation in key exponents.

Contribution

It analytically characterizes the impact of spatially dense measurements on critical Ising chains using conformal field theory and numerical methods, highlighting the role of defects as irrelevant or marginal perturbations.

Findings

Average correlations and entanglement scaling remain unchanged.

Postselection leads to continuously varying exponents.

Defects can be irrelevant or marginal depending on measurement conditions.

Abstract

We study the effects of measurements, performed with a finite density in space, on the ground state of the one-dimensional transverse-field Ising model at criticality. Local degrees of freedom in critical states exhibit long-range entanglement, and as a result, local measurements can have highly nonlocal effects. Our analytical investigation of correlations and entanglement in the ensemble of measured states is based on properties of the Ising conformal field theory (CFT), where measurements appear as (1+0)-dimensional defects in the (1+1)-dimensional Euclidean spacetime. So that we can verify our predictions using large-scale free-fermion numerics, we restrict ourselves to parity-symmetric measurements. To describe their averaged effects analytically we use a replica approach, and we show that the defect arising in the replica theory is an irrelevant perturbation to the Ising CFT. Strikingly, the asymptotic scalings of averaged correlations and entanglement entropy are therefore unchanged relative to the ground state. In contrast, the defect generated by postselecting on the most likely measurement outcomes is exactly marginal. We then find that the exponent governing postmeasurement order parameter correlations, as well as the ''effective central charge'' governing the scaling of entanglement entropy, vary continuously with the density of measurements in space. Our work establishes new connections between the effects of measurements on many-body quantum states and of physical defects on low-energy equilibrium properties.