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arXiv:2301.08255 · Physical review. B./Physical review. B

Entanglement in a one-dimensional critical state after measurements

Zhou Yang, Dan Mao, Chao‐Ming Jian

Published Oct 13, 2023• 47 citations
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Abstract

The entanglement entropy (EE) of the ground state of a one-dimensional Hamiltonian at criticality has a universal logarithmic scaling with a prefactor given by the central charge $c$ of the underlying $(1+1)$-dimensional conformal field theory. When the system is probed by measurements, the entanglement in the critical ground state is inevitably affected due to wave-function collapse. In this paper, we study the effect of weak measurements on the entanglement scaling in the ground state of the one-dimensional critical transverse-field 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 ${c}_{\mathrm{eff}}$. We derive the analytical expression of ${c}_{\mathrm{eff}}$ as a function of the measurement strength. With both numerical simulations and analytical studies, we show that for the EE averaged over all postmeasurement states based on their Born-rule probabilities, the effective central charge is 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 mean-field approximation to the Born-rule probabilities and lead to the same ${c}_{\mathrm{eff}}$ behavior. The effects of the measurements along the longitudinal spin axis and the postmeasurement correlation functions are also discussed.

DOI
10.1103/physrevb.108.165120

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The entanglement entropy (EE) of the ground state of a one-dimensional Hamiltonian at criticality has a universal logarithmic scaling with a prefactor given by the central charge ccc of the underlying (1+1)(1+1)(1+1)-dimensional conformal field theory. When the system is probed by measurements, the entanglement in the critical ground state is inevitably affected due to wave-function collapse. In this paper, we study the effect of weak measurements on the entanglement scaling in the ground state of the one-dimensional critical transverse-field 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 ceff{c}_{\mathrm{eff}}ceff​. We derive the analytical expression of ceff{c}_{\mathrm{eff}}ceff​ as a function of the measurement strength. With both numerical simulations and analytical studies, we show that for the EE averaged over all postmeasurement states based on their Born-rule probabilities, the effective central charge is 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 mean-field approximation to the Born-rule probabilities and lead to the same ceff{c}_{\mathrm{eff}}ceff​ behavior. The effects of the measurements along the longitudinal spin axis and the postmeasurement correlation functions are also discussed.
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)(1+0)(1+0)-dimensional defects in the (1+1)(1+1)(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 connections between the effects of measurements on many-body quantum states and of physical defects on low-energy equilibrium properties.
A bstract Critical systems host nontrivial entanglement structure that is generally sensitive to additional couplings. In the present work, we study the effect of weak measurements on the entanglement Hamiltonian of massless free fermions which are prepared in their critical ground state. While the power-law decaying correlation and logarithmic growing entanglement entropy have been observed as typical signatures of quantum criticality after the weak measurement, in this work we show that the conformal symmetry is lost and the entanglement Hamiltonian generally becomes gapped for arbitrary small measurement strength. To reveal this unconventional entanglement structure, we consider a field-theory description that allows us to establish an analytic mapping between the entanglement Hamiltonians before and after the weak measurements. From this mapping, we find that although the measurements lead to a significant modification of the entanglement spectrum, the real-space distribution of the eigenfunction of the kernel of entanglement Hamiltonian is unchanged, which is responsible for the coexistence of a gapped entanglement Hamiltonian and the logarithmic entanglement entropy. Moreover, as the mapping works for arbitrarily many disjoint intervals, the multi-interval entanglement entropy also exhibits the same scaling behavior as the critical ground state, and shares the same effective central charge with the single-interval case. We numerically demonstrate these field-theory results on a lattice model, where the entanglement Hamiltonian after weak measurements exhibits the typical signature of a finite gap and becomes long-ranged even in the single-interval case. This is distinct from the critical ground state, where the entanglement Hamiltonian for a single interval is gapless and local.
We study post-measurement ensembles of ground states of tricritical and critical 1D quantum Ising Hamiltonians subjected, respectively, to weak energy and spin measurements without post-selection. These measurements act as relevant perturbations about the unmeasured critical ground states. Using finite-size renormalization group (RG) crossover analyses, we characterize their universal properties through the entanglement effective central charge, effective Affleck-Ludwig boundary entropy, and signatures of multifractality from moments of measurement-averaged correlation functions. In both cases, we find evidence for "measurement-dominated" or "measurement-altered" fixed points governed by the underlying Born-rule randomness. For critical Ising, we find a direct RG flow to a projective-measurement fixed point with area-law entanglement, whereas for the tricritical Ising model, we find evidence for a weak-measurement fixed point with logarithmic entanglement. These results clarify the RG-flow structure of weakly measured multicritical Ising ground states and show how intrinsic measurement-induced randomness can generate complex and rich universal long-distance scaling behavior in the post-measurement ensembles, accessible to controlled analytical RG and numerical finite-size RG crossover analyses.
In this work, we numerically study the effect of weak measurement on deconfined quantum critical point(DQCP). Particularly, we consider the ground state of an one-dimensional spin 1/21/21/2 system with next-nearest-neighbour exchange interactions(KKK), which shows analogues phase transition to DQCP in the thermodynamic limit. This system is in the ferromagnetic phase below the critical exchange interaction KcK_cKc​ and in the valance bond solid phase above KcK_cKc​. The weak measurement is carried out by coupling a secondary ancilla system to the critical system via unitary interactions and later measuring the ancilla spins projectively. We numerically calculate entanglement entropy,correlation length, and order parameters of leading post-measurement states using uniform matrix product state representation of the quantum many-body state in the thermodynamic limit. We report asymmetric restructuring of entanglement of the post measurement states across the phase boundary under weak measurements. Especially, the trajectory (↓↓)\left(\downarrow \downarrow\right)(↓↓) describing a uniform measurement outcome given the all ancilla spins initiated in the same (↓)\left(\downarrow \right)(↓) state, shows anomalous entanglement when increasing the strength of weak measurement. The bipartite entanglement entropy strongly increases when K<KcK<K_cK<Kc​ whereas it weakly decreases when K>KcK>K_cK>Kc​. We argue with numerical evidences that observed asymmetry in entanglement would lead to a weak first order phase boundary in the thermodynamic limit. We also discuss important aspects in experimental observation of measurement induced effects linked to the strength of weak measurement and probability of post-measurement states.
Finding new critical states of matter is an important subject in modern many-body physics. Here we study the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory and show that it can lead to qualitatively new critical states. Depending on the Luttinger parameter KKK, the effect of measurement is irrelevant (relevant) at K>1K>1K>1 (K<1K<1K<1). We reveal that this causes an entanglement transition between two phases, one with logarithmic entanglement entropy for a subregion (K>1K>1K>1), and the other with algebraic entanglement entropy (K<1K<1K<1). At the critical point K=1K=1K=1, the measurement is marginal, and we find new critical states whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength. We also performed numerical density matrix renormalization group and fermionic Gaussian state simulations to support our results. We further discuss promising and feasible routes to experimentally realize new critical states in our work.
We study the statistics of measurement-induced entanglement (MIE) after partial measurement on a class of one-dimensional quantum critical states described by Tomonaga-Luttinger liquids at low energies. Using a replica trick to average over measurement outcomes in the charge basis and tools from conformal field theory (CFT), we derive closed-form expressions for the cumulants of MIE. We show that exact Born-averaging over microscopic measurement outcomes becomes equivalent at low energy to averaging over conformal boundary conditions weighted by their corresponding partition functions. Our results yield distinctive critical behavior across all cumulants in the regime where the unmeasured parts of the system are maximally separated. We also obtain the full distribution of MIE, finding that it is generically bimodal and exhibits fat-tails. We corroborate our analytical predictions by numerical calculations and find good agreement between them.