# Measurement-Induced Phase Transitions in the Dynamics of Entanglement

## 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 $1+1\mathrm{D}$, 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 $1+1\mathrm{D}$. 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.

- Authors:

- Publication Date:

- Research Org.:
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES); Gordon and Betty Moore Foundation (GBMF)

- OSTI Identifier:
- 1543370

- Alternate Identifier(s):
- OSTI ID: 1610523

- Grant/Contract Number:
- SC0001088

- Resource Type:
- Published Article

- Journal Name:
- Physical Review. X

- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 9 Journal Issue: 3; Journal ID: ISSN 2160-3308

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Physics; Condensed Matter Physics; Quantum Physics; Statistical Physics

### Citation Formats

```
Skinner, Brian, Ruhman, Jonathan, and Nahum, Adam. Measurement-Induced Phase Transitions in the Dynamics of Entanglement. United States: N. p., 2019.
Web. doi:10.1103/PhysRevX.9.031009.
```

```
Skinner, Brian, Ruhman, Jonathan, & Nahum, Adam. Measurement-Induced Phase Transitions in the Dynamics of Entanglement. United States. doi:10.1103/PhysRevX.9.031009.
```

```
Skinner, Brian, Ruhman, Jonathan, and Nahum, Adam. Mon .
"Measurement-Induced Phase Transitions in the Dynamics of Entanglement". United States. doi:10.1103/PhysRevX.9.031009.
```

```
@article{osti_1543370,
```

title = {Measurement-Induced Phase Transitions in the Dynamics of Entanglement},

author = {Skinner, Brian and Ruhman, Jonathan and Nahum, Adam},

abstractNote = {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.},

doi = {10.1103/PhysRevX.9.031009},

journal = {Physical Review. X},

number = 3,

volume = 9,

place = {United States},

year = {2019},

month = {7}

}

DOI: 10.1103/PhysRevX.9.031009

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