The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.
The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.
Understanding thermalization in quantum many-body systems is among the most enduring problems in modern physics. A particularly interesting question concerns the role played by quantum mechanics in this process, i.e., whether thermalization in quantum many-body systems is fundamentally different from that in classical many-body systems and, if so, which of its features are genuinely quantum. Here, we study this question in minimally structured many-body systems that are only constrained to have local interactions, i.e., local random circuits. In particular, we introduce random permutation circuits (RPCs), which are circuits comprising gates that locally permute basis states, as a counterpart to random unitary circuits (RUCs), a standard toy model for generic quantum dynamics. RPCs represent a model for generic microscopic classical reversible dynamics but, interestingly, can be interpreted both as classical or as quantum dynamics. We show that, upon averaging over all circuit realizations, RPCs permit the analytical computation of several key quantities such as out-of-time-order correlators (OTOCs) and entanglement entropies. In the classical setting, we obtain similar exact results relating (quantum) purity to (classical) growth of mutual information and (quantum) OTOCs to (classical) decorrelators. We, thus, discover a series of exact relations, connecting quantities in RUC and (quantum or classical) RPCs. Our results indicate that, despite the fundamental differences between quantum and classical systems, their many-body dynamics exhibits remarkably similar behaviors.
We investigate the subsystem entanglement asymmetry in random quantum automaton ensembles, which are generated by permuting the basis states in the Hilbert space and applying global phase shifts. We compute the ensemble average of the U(1) subsystem asymmetry in different connectivity geometries, showing that the late-time limit of the ensemble associated to a 2-local circuit geometry coincides with the all-to-all ensemble average. By focusing on different subsystem sizes, we demonstrate that, similarly to Haar-random circuits, the system locally symmetrizes. However, in sharp contrast to the Haar-random setting, the scale at which symmetrization happens depends on the initial state, a phenomenon we associate with the interplay of conservation of the participation entropy and the uniform exploration of charge sectors. Additionally, we connect the growth of the subsystem asymmetry to the subsystem coherence and show that their growth is characterized by the same symmetrization scale.
The efficient generation of random quantum states is a long-standing challenge, motivated by their diverse applications in quantum information processing tasks. In this work, we identify entanglement as the key resource that enables local random unitaries to generate global random states by effectively gluing randomness across the system. Specifically, we demonstrate that approximate random states can be produced from an entangled state ∣ψ⟩ through the application of local random unitaries. We show that the resulting ensemble forms an approximate state design with an error saturating as Θ(e−N2(ψ)), where N2(ψ) is the second Rényi entanglement entropy of ∣ψ⟩. Furthermore, we prove that this tight bound also applies to the second Rényi entropy of coherence when the ensemble is constructed using coherence-free operations. These results imply that, when restricted to resource-free gates, the quality of the generated random states is determined entirely by the resource content of the initial state. Notably, we find that among all α-Rényi entropeis, the second Rényi entropy yields the tightest bounds. Consequently, these second Rényi entropies can be interpreted as the maximal capacities for generating randomness using resource-free operations. Finally, moving beyond approximate state designs, we utilize this entanglement-assisted gluing mechanism to present a novel method for generating pseudorandom states in multipartite systems from a locally entangled state via pseudorandom unitaries in each of parties.
Measurement-induced entanglement phase transitions, caused by the competition between entangling unitary dynamics and disentangling projective measurements, have been studied in various random circuit models in recent years. In this paper, we study the dynamics of averaged purity for a simple N-qudit Brownian circuit model with all-to-all random interaction and measurements. In the large-N limit, our model is mapped to a one-dimensional quantum chain in the semi-classical limit, which allows us to analytically study critical behaviors and various other properties of the model. We show that there are two phases distinguished by the behavior of the total system entropy in the long time. In addition, the two phases also have distinct subsystem entropy behavior. The low measurement rate phase has a first-derivative discontinuity in the behavior of second Renyi entropy versus subsystem size, similar to the "Page curve" of a random state, while the other phase has a smooth entropy curve.
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.
The notion of typicality in statistical mechanics is essential to characterize a macroscopic system. An overwhelming majority of the pure state looks almost identical if we neglect macroscopic non-local correlations, suggesting that thermal equilibrium is the collection of the typical properties. Quantum entanglement, which characterizes a non-local correlation, also has a typical behavior in equilibrium systems. However, it remains elusive whether there is a typical behavior of entanglement in dynamical non-equilibrium systems. To investigate the typicality, we consider a situation where a system in a pure state starts to share entanglement with its environment system due to the interaction between them. Assuming the initial state is randomly chosen from an ensemble of pure states, a criteria for the typicality of the Rényi entropies is presented. In addition, it is analytically proven that the second Rényi entropy has a typical behavior in two cases. The first one is an energy dissipation process in a multiple-qubit system which is initially in a random pure state in an energy shell. Since the typical behavior is qualitatively the same as the prediction of the Page curve conjecture, it gives the first proof of the Page curve conjecture in a dynamical process. In the second case, the typicality is proven for any dynamics described by a multiple-product of a single-qudit channel when the system is initially in a pure state randomly chosen from the whole Hilbert space. This result shows that entanglement typicality is not a specific feature of energy dissipating processes.