Measurement-induced phase transitions have largely been explored for projective or continuous measurements of Hermitian observables, assuming perfect detection without information loss. Yet such transitions also arise in more general settings, including quantum-jump processes with non-Hermitian jump operators, and under inefficient detection. A theoretical framework for treating these broader scenarios has been missing. Here we develop a comprehensive replica Keldysh field theory for general quantum-jump processes in both bosonic and fermionic systems. Our formalism provides a unified description of pure-state quantum trajectories under efficient detection and mixed-state dynamics emerging from inefficient monitoring, with deterministic Lindbladian evolution appearing as a limiting case. It thus establishes a direct connection between phase transitions in nonequilibrium steady states of driven open quantum matter and in measurement-induced dynamics. As an application, we study imbalanced and inefficient fermion counting in a one-dimensional lattice system: monitored gain and loss of fermions occurring at different rates, with a fraction of gain and loss jumps undetected. For imbalanced but efficient counting, we recover the qualitative picture of the balanced case: entanglement obeys an area law for any nonzero jump rate, with an extended quantum-critical regime emerging between two parametrically separated length scales. Inefficient detection introduces a finite correlation length beyond which entanglement, as quantified by the fermionic logarithmic negativity, obeys an area law, while the subsystem entropy shows volume-law scaling. Numerical simulations support our analytical findings. Our results offer a general and versatile theoretical foundation for studying measurement-induced phenomena across a wide class of monitored and open quantum systems.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
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 connections between the effects of measurements on many-body quantum states and of physical defects on low-energy equilibrium properties.
We analyze the effect of decoherence, modeled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "g function" in conformal field theory, allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the conformal field theory. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.
This thesis studies the effects of continuous measurements of local occupation numbers on a one-dimensional free fermionic system with periodic boundary conditions and a fixed number of particles. The specific model that is studied here, is a phase-dependent quantum state diffusion of the free fermionic system. The system is studied by both analytical and numerical means.\\ Analytically, this thesis uses the framework of replica Keldysh field theory in order to obtain a field theoretical description, in terms of a functional integral formulation, that describes the conditional dynamics of the quantum system, which in turn is described by a stochastic Lindblad master equation. From the replica Keldysh field theory, a Gaussian theory that incorporates linear fluctuations around the saddle point of the replica Keldysh action is derived, yielding analytical insights into the behavior of density-density correlations and the entanglement entropy of the system at different system sizes. In addition to this, the effects of non-linear fluctuations around the Gaussian saddle point, in particular the effects of massless Goldstone modes that are described by a non-linear sigma model, are taken into account. The resulting RG flow equation of the NLSM coupling shows that the system enters a strong-measurement phase at exponentially large system sizes
l∗, resulting in the absence of a measurement-induced phase transition. A one-loop renormalization group analysis leads to a logarithmic correction of the Gaussian theory at large system sizes. Due to this correction, the Fourier space density-density correlation function is expected to display local maxima at momenta
qC,max that scale as
qC,max∼γ2, where
γ is the measurement rate. \\ As for the entanglement entropy, the Gaussian theory predicts an unbounded logarithmic scaling of the entanglement entropy with the subsystem size
ℓ, resulting in a constant scale-dependent effective central charge
cℓ∼const. at large system sizes
ℓ→∞. However, due to the RG correction to the Gaussian theory a local maximum of the central charge at intermediate length scales is formed, where the position
lc,max is expected to scale as
lc,max∼γ−3/2.\\ The analytical predictions from the replica Keldysh field theory are backed up by numerical data of a system with
L=1000 lattice sites and
N=L/2 particles, obtained by means of a quantum trajectory simulation for different unraveling phases
φ∈[0,π/2] and a variety of different measurement rates
γ. The numerical data clearly shows the predicted scalings
qC,max∼γ2 and
lc,max∼γ−3/2 for unraveling phases
φ=π/2, however modifying the scaling of the central charge maximum to
lc,max∼γ−1 for smaller values of
γ. The modified scaling can be explained by an emergent local maximum in the Gaussian prediction for the central charge for the corresponding values of
φ. Conclusively, the thesis comes to the conclusion that there is no measurement-induced phase transition in the studied system. However, for the unraveling
φ=π/2 the system is always found to be in the rare-measurement phase, characterized by volume-law entanglement.
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
ceff. We derive the analytical expression of
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 behavior. The effects of the measurements along the longitudinal spin axis and the postmeasurement correlation functions are also discussed.