We consider the universal aspects of two problems: (i) the singular value structure of a product M t = m t m t − 1 … m 1 of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices m i , representing time steps in the evolution, but the m i are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.
We consider the universal aspects of two problems: (i) the singular value structure of a product M t = m t m t − 1 … m 1 of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices m i , representing time steps in the evolution, but the m i are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.
We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; ii) weak continuous-time monitoring that induces a Dyson brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus's operator space, focusing in particular on the unitary and orthogonal cases, respectively β=2 and β=1, with β the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any β, thanks to the mapping of the Fokker-Planck evolution of eigenvalues onto the Calogero-Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
Quantum circuits under sufficiently weak monitoring purify on a timescale TP exponentially long in the system size. This slowness underlies a universal purification dynamics, whose quantitative description has so far required the replica trick, with a delicate analytic continuation. We show that monitored Clifford circuits on L qudits of prime dimension q bypass this construction entirely: in the scaling limit at fixed x=t/TP(L), purification reduces to the Markovian decay of the density-matrix rank, an exactly solvable death process descending from infinity. We compute the full scaling functions in compact form: all Rényi entropies collapse onto a universal curve ⟨S(x)⟩. Exact stabilizer simulations at q=2,3,5 confirm the predictions, with no fitting parameter for the global model and TP as the only fitted scale for local brick-wall circuits. Also, the replica problem amounts to a tilted version of the same Markov process, in agreement with exact computations from the Clifford commutant. Finally, the quantization of the rank leaves two hallmarks that distinguish Clifford dynamics from generic monitored circuits: the entropy fluctuations saturate at short scaled times x→0 to an O(1) variance, instead of vanishing, and observables develop a temporal modulation periodic in logqx, which cannot be captured by the replica approach.
Abstract We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement-induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: (i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; (ii) weak continuous-time monitoring that induces a Dyson Brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus’s operator space, focusing in particular on the unitary and orthogonal cases, respectively β = 2 and β = 1 , with β the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any β , thanks to the mapping of the Fokker–Planck evolution of eigenvalues onto the Calogero–Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.
We address entanglement transitions in monitored random quantum circuits of non-interacting fermions, in particular, the question of whether Born-rule and forced measurements yield the same universality class. For a generic circuit with no symmetry other than fermion parity, acting on a one-dimensional Majorana chain, we numerically obtain several critical exponents, providing clear evidence that the two transitions with Born-rule and forced measurements are in different universality classes. We provide a theoretical understanding for our numerical results by identifying the underlying statistical mechanics model which follows from the general correspondence, established in Jian et al., Phys. Rev. B 106, 134206, between non-unitary circuits of non-interacting fermions and the ten-fold Altland-Zirnbauer (AZ) symmetry classes. The AZ class is the same for Born-rule and forced measurements of the circuits. For the circuit under consideration (in AZ class DIII), the statistical mechanics model describing the transition is the principal chiral non-linear sigma model whose field variable is an SO(n) matrix in the replica limits n→0 and n→1 for forced and Born-rule measurements, respectively. The former is in an Anderson localization universality class while we show that the latter is in a novel universality class beyond Anderson localization. Both entanglement transitions are driven by proliferation of Z2 topological defects. The different replica limits account for the difference in the universality classes. Furthermore, we provide numerical and symmetry-based arguments that the entanglement transition in the previously-studied monitored circuit of Majorana fermions based on the loop model with crossings, a highly fine-tuned circuit, belongs to a universality class different from both transitions in the generic circuits discussed in this paper.
We study an exactly solvable model of monitored dynamics in a system of N spin-1/2 particles with pairwise all-to-all noisy interactions, where each spin is continuously weakly measured along a random direction. Using the replica trick to incorporate the Born-rule weighting of measurement outcomes, we obtain an exact large-N description of purification and of the statistics of local observables. We find that the nature of the phase transition strongly depends on the number n of replicas: non-perturbative logarithmic corrections appear in the physically relevant n→1 limit and destroy the purifying phase present at finite integer n. As a consequence, the purification time of an initially mixed state is always exponentially long in the system size, even at arbitrarily large measurement rate.