arXiv:2012.00040

Entanglement negativity at the critical point of measurement-driven transition

Bowen Shi, Xin Dai, Yuan-Ming Lu

Published Nov 30, 2020• 19 citations

Abstract

We study the entanglement behavior of a random unitary circuit punctuated by projective measurements at the measurement-driven phase transition in one spatial dimension. We numerically study the logarithmic entanglement negativity of two disjoint intervals and find that it scales as a power of the cross-ratio. We investigate two systems: (1) Clifford circuits with projective measurements, and (2) Haar random local unitary circuit with projective measurements. Remarkably, we identify a power-law behavior of entanglement negativity at the critical point. Previous results of entanglement entropy and mutual information point to an emergent conformal invariance of the measurement-driven transition. Our result suggests that the critical behavior of the measurement-driven transition is distinct from the ground state behavior of any \emph{unitary} conformal field theory.

DOI: 10.48550/arxiv.2012.00040

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arXiv:2012.00040

Entanglement negativity at the critical point of measurement-driven transition

Bowen Shi, Xin Dai, Yuan-Ming Lu

Published Nov 30, 2020• 19 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2012.00040

Related papers

Measurement-induced criticality in ( 2 + 1 ) -dimensional hybrid quantum circuits

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Measurement-Induced Phase Transitions: Universal Entanglement Scaling at Quantum Criticality — E8 Intelligence Research

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FINDING: Measurement-induced phase transitions (MIPT) in quantum systems reveal a sharp entanglement phase boundary driven by the competition between unitary evolution and projective measurements, with critical points often characterized by universal exponents and effective central charges. | MATH: The MIPT is typically studied via the entanglement entropy $S = -\text{Tr}(\rho_A \ln \rho_A)$ scaling as $S \sim \frac{c}{6} \ln L$ at criticality (for 1+1D conformal field theory), where $c$ is the effective central charge. The critical measurement rate $p_c$ separates volume-law ( $S \propto L$ ) from area-law ( $S \propto \text{const}$ ) phases. For Clifford circuits, the transition is in the percolation universality class with $c = 0$ (or $c = \ln 2$ for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate $\gamma$ , the effective Hamiltonian is $H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Measurement-induced phase transitions (MIPT) in quantum systems reveal a sharp entanglement phase boundary driven by the competition between unitary evolution and projective measurements, with critical points often characterized by universal exponents and effective central charges. | MATH: The MIPT is typically studied via the entanglement entropy \(S = -\text{Tr}(\rho_A \ln \rho_A)$ scaling as $S \sim \frac{c}{6} \ln L$ at criticality (for 1+1D conformal field theory), where $c$ is the effective central charge. The critical measurement rate $p_c$ separates volume-law ($S \propto L$) from area-law ($S \propto \text{const}$) phases. For Clifford circuits, the transition is in the percolation universality class with $c = 0$ (or $c = \ln 2$ for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate $\gamma$, the effective Hamiltonian is \(H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Critical properties of the measurement-induced transition in random quantum circuits

Aidan Zabalo, Michael J. Gullans, Justin H. Wilson +3 more

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate ${p}_{c}=0.17(1)$ . We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from those for stabilizer circuits or percolation, but we cannot definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies ${S}_{n}$ depends strongly on the R\'enyi index $n$ ; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate ${p}_{c}=0.17(1)$. We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from those for stabilizer circuits or percolation, but we cannot definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies ${S}_{n}$ depends strongly on the R\'enyi index $n$; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.

arXiv:1911.00008Physical review. B./Physical review. BFeb 20, 2020349 citations

L=48

for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.

U(1)

symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.

S = -\text{Tr}(\rho_A \ln \rho_A)

scaling as

S \sim \frac{c}{6} \ln L

at criticality (for 1+1D conformal field theory), where

c

is the effective central charge. The critical measurement rate

p_c

separates volume-law (

S \propto L

) from area-law (

S \propto \text{const}

) phases. For Clifford circuits, the transition is in the percolation universality class with

c = 0

(or

c = \ln 2

for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate

\gamma

, the effective Hamiltonian is \(H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

p

, we show numerically that the entanglement depth, corresponding to the size of the largest cluster of entangled qubits scales as a power law with system size on both sides of the transition. The power law exponent is 1 in the entangling phase and continuously decreases to 0 as

p \to 1

in the disentangling phase. In addition, we find that the spatial support of the largest cluster exhibits an approximate fractal geometry with a tunable fractal dimension controlled by the measurement rate. We argue that this structure arises from a competition between unitary-driven coagulation of entangled clusters and measurement-induced fragmentation, giving rise to a fractal steady state reminiscent of classical coagulation-fragmentation models. Away from the MIPT critical point, the fractal dimension matches the entanglement depth power law exponent. These results show that multipartite entanglement structure provides a fresh perspective on the emergent quantum correlations in monitored quantum circuits and noisy quantum dynamics.

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in

1+1

dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate

{p}_{c}=0.17(1)

. We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from those for stabilizer circuits or percolation, but we cannot definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies

{S}_{n}

depends strongly on the R\'enyi index

n

; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.