Abstract
Hybrid evolution protocols, composed of unitary dynamics and repeated, weak or projective measurements, give rise to new, intriguing quantum phenomena, including entanglement phase transitions and unconventional conformal invariance. Defying the complications imposed by the non-linear and stochastic nature of the measurement process, we introduce a scenario of measurement-induced many body evolution, which possesses an exact analytical solution: bosonic Gaussian measurements. The evolution features a competition between the continuous observation of linear boson operators and a free Hamiltonian, and it is characterized by a unique and exactly solvable covariance matrix. Within this framework, we then consider an elementary model for quantum criticality, the free boson conformal field theory, and investigate in which way criticality is modified under measurements. Depending on the measurement protocol, we distinguish three fundamental scenarios (a) enriched quantum criticality, characterized by a logarithmic entanglement growth with a floating prefactor, or the loss of criticality, indicated by an entanglement growth with either (b) an area-law or (c) a volume-law. For each scenario, we discuss the impact of imperfect measurements, which reduce the purity of the wavefunction and are equivalent to Markovian decoherence, and present a set of observables, e.g., real-space correlations, the relaxation time, and the entanglement structure, to classify the measurement-induced dynamics for both pure and mixed states. Finally, we present an experimental tomography scheme, which grants access to the density operator of the system by using the continuous measurement record only.
Highlights
The reason for this surprising result is rooted in the fact that (i) Gaussians states can be described by a polynomial number of parameters that can be numerically integrated for rather large system sizes, and that (ii) due to the deterministic dynamics of the covariance matrix we do not have the problem of postselection
We have demonstrated that bosonic Gaussian measurements represent a rather unique class of systems, whose conditioned dynamics under continuous measurements is exactly solvable
Starting from the unitary free boson conformal field theory (CFT), we examined three different types of continuous measurements, and found that the corresponding stationary states cover the three paradigmatic cases of area law, volume law, or a logarithmic scaling of the entanglement
Summary
Quantum critical behavior, emerging from the competition between a set of non-commuting operators, is a paradigmatic and fascinating phenomenon in many-body quantum systems [1–4]. When the time-evolution is constrained by certain symmetries, e.g., for Gaussian circuits or a free Hamiltonian, the measurement-induced transition has been associated with a quantum phase transition [34, 39–41, 43, 44, 62] This enables the notion of quantum critical behavior, and emergent conformal invariance, in a nonequilibrium wave function, generated by the hybrid evolution protocol [51–53, 61]. We start from an paradigmatic model for quantum criticality, the free boson conformal field theory (CFT) [12], and examine how continuous measurements may either enrich or destroy its quantum critical behavior To this end, we introduce an elementary, and exactly solvable setup for the measurement-induced many-body evolution of bosons, which we term Gaussian measurements. In a corresponding experiment it is not necessary to perform additional measurements or decoding operations on the system [71,72], but rather one processes the measurement outcomes that are available from the measurements itself [73–76]
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