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arXiv:2103.06873 · Physical Review X

Fractal, Logarithmic, and Volume-Law Entangled Nonthermal Steady States via Spacetime Duality

Matteo Ippoliti, Tibor Rakovszky, Vedika Khemani

Published Mar 9, 2022• 112 citations
arXivHTMLPDF

Abstract

The extension of many-body quantum dynamics to the nonunitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions.We show how a duality transformation between space and time, on one hand, and unitarity and nonunitarity, on the other, can be used to realize steady-state phases of nonunitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size-from logarithmic to extensive to fractal.We show how these outcomes in nonunitary circuits (that are "spacetime dual" to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets "radiated away" from one edge of the system.In spacetime duals of chaotic unitary circuits, this mapping allows us to analytically derive a nonthermal volume-law entangled phase with a universal logarithmic correction to the entropy, previously observed in unitary-measurement dynamics.Notably, we also find robust steady-state phases with fractal entanglement scaling, SðlÞ ∼ l α with tunable 0 < α < 1 for subsystems of size l in one dimension.We present an experimental protocol for preparing these novel steady states with only a vanishing density of postselected measurements via a type of "teleportation" between spacelike and timelike slices of quantum circuits.

DOI
10.1103/physrevx.12.011045

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The extension of many-body quantum dynamics to the nonunitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions.We show how a duality transformation between space and time, on one hand, and unitarity and nonunitarity, on the other, can be used to realize steady-state phases of nonunitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size-from logarithmic to extensive to fractal.We show how these outcomes in nonunitary circuits (that are "spacetime dual" to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets "radiated away" from one edge of the system.In spacetime duals of chaotic unitary circuits, this mapping allows us to analytically derive a nonthermal volume-law entangled phase with a universal logarithmic correction to the entropy, previously observed in unitary-measurement dynamics.Notably, we also find robust steady-state phases with fractal entanglement scaling, SðlÞ ∼ l α with tunable 0 < α < 1 for subsystems of size l in one dimension.We present an experimental protocol for preparing these novel steady states with only a vanishing density of postselected measurements via a type of "teleportation" between spacelike and timelike slices of quantum circuits.
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries.Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to nonunitary evolution, e.g., via projective measurements.Here we employ the idea of spacetime rotation of a circuit to explore the relation between systems that fall into these two classes.In particular, by spacetime rotating unitary Floquet circuits that display a localization transition, we construct nonunitary circuits that display a rich variety of entanglement scaling and phase transitions.One outcome of our approach is a nonunitary circuit for free fermions in one dimension that exhibits an entanglement transition from logarithmic scaling to volume-law scaling.This transition is accompanied by a "purification transition" analogous to that seen in hybrid projective-unitary circuits.We follow a similar strategy to construct a nonunitary two-dimensional (2D) Clifford circuit that shows a transition from area to volume-law entanglement scaling.Similarly, we spacetime rotate a 1D spin chain that hosts many-body localization to obtain a nonunitary circuit that exhibits an entanglement transition.Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition then the correlator is expected to be singular in its nonunitary counterpart as well.
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to non-unitary evolution, e.g., via projective measurements. Here we employ the idea of space-time rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by space-time rotating unitary Floquet circuits that display a localization transition, we construct non-unitary circuits that display a rich variety of entanglement scaling and phase transitions. One outcome of our approach is a non-unitary circuit for free fermions in 1d that exhibits an entanglement transition from logarithmic scaling to volume-law scaling. This transition is accompanied by a 'purification transition' analogous to that seen in hybrid projective-unitary circuits. We follow a similar strategy to construct a non-unitary 2d Clifford circuit that shows a transition from area to volume-law entanglement scaling. Similarly, we space-time rotate a 1d spin chain that hosts many-body localization to obtain a non-unitary circuit that exhibits an entanglement transition. Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition, then the correlator is expected to be singular in its non-unitary counterpart as well.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.
We study measurement-induced phase transitions in quantum circuits consisting of kicked Ising models with postselected weak measurements, whose dynamics can be mapped onto a classical dynamical system. For a periodic (Floquet) non-unitary evolution, such circuits are exactly tractable and admit volume-to-area law transitions. We show that breaking time-translation symmetry down to a quasiperiodic (Fibonacci) time evolution leads to the emergence of a critical phase with tunable effective central charge and with a fractal origin. Furthermore, for some classes of random non-unitary circuits, we demonstrate the robustness of the volume-to-area law phase transition for arbitrary random realizations, thanks to the emergent compactness of the classical map encoding the circuit's dynamics.
Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes hybrid quantum dynamics, consisting of unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume- and area-law entanglement. The role of dimension in the nature of these transitions is an open problem. There is a dimensional correspondence between measurement-induced transitions in non-unitary quantum circuits in ddd spatial dimensions and classical statistical mechanical models in d+1d+1d+1 dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with ddd-dimensional percolation and (d+1)(d+1)(d+1)-dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.
© 2019 authors. Published by the American Physical Society. We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate p for each degree of freedom, we show that the system has two dynamical phases: "entangling" and "disentangling." The former occurs for p smaller than a critical rate pc and is characterized by volume-law entanglement in the steady state and "ballistic" entanglement growth after a quench. By contrast, for p>pc the system can sustain only area-law entanglement. At p=pc the steady state is scale invariant, and in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains and show that the phenomenology of the two phases is similar to that of the toy model but with distinct "quantum" critical exponents, which we calculate numerically in 1+1D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.