Abstract
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 $p_c$, and is characterized by volume-law entanglement in the steady-state and `ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ 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 R\'{e}nyi 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 R\'{e}nyi 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 $1+1$D. 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.
Highlights
When left unobserved, quantum systems tend to evolve toward states of higher entanglement [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]
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
Do such measurements collapse the many-body wave function into something close to a product state with area-law entanglement, or can volume-law entanglement survive?. We answer this question for the simplest type of measurement, which is a projective measurement of a discrete degree of freedom. We show that both types of dynamics can occur, leading to volume-law or area-law entanglement, depending on the value of the measurement rate p
Summary
Quantum systems tend to evolve toward states of higher entanglement [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. At a nonzero rate p per degree of freedom Do such measurements collapse the many-body wave function into something close to a product state with area-law entanglement, or can volume-law entanglement survive?. The specific models we study all have discrete time dynamics While this discretization is important in order for the dynamics of S0 to be well defined [21] (i.e., for the construction of the toy model), we do not expect that it will affect the existence or universality class of the generic transition that is manifest in physically meaningful quantities (such as the von Neumann entanglement entropy S1 or the mutual information between separated spins). The transition we introduce is in the entanglement structure of pure states
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