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arXiv:2509.15634 · Physica Scripta

Measurement-driven transitions between area law phases

Hui Yu, Jiangping Hu

Published May 8, 2025• 3 citations
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Abstract

Abstract In recent years, quantum circuits consisting of unitary gates and projective measurements have become valuable tools for stimulating or preparing quantum many-body states with non-trivial properties. Here, we introduce and examine a measurement-only circuit (the projective quantum Ising model with three-spin interactions) that involves three non-commuting projective measurements. This model features three distinct phases, separated by two critical lines. We utilize two entanglement measures (topological entanglement entropy and mutual information) to identify the phase boundaries and derive various critical exponents through scaling analysis. We establish a relationship between our model and a two-dimensional statistical model (bond percolation) within certain limits. We hope that our results will shed light on further studies using other measurement-only models.

DOI
10.1088/1402-4896/add656

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Abstract In recent years, quantum circuits consisting of unitary gates and projective measurements have become valuable tools for stimulating or preparing quantum many-body states with non-trivial properties. Here, we introduce and examine a measurement-only circuit (the projective quantum Ising model with three-spin interactions) that involves three non-commuting projective measurements. This model features three distinct phases, separated by two critical lines. We utilize two entanglement measures (topological entanglement entropy and mutual information) to identify the phase boundaries and derive various critical exponents through scaling analysis. We establish a relationship between our model and a two-dimensional statistical model (bond percolation) within certain limits. We hope that our results will shed light on further studies using other measurement-only models.
A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to L=48L=48L=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.
We study a free fermion model where two sets of non-commuting non-projective measurements stabilize area-law entanglement scaling phases of distinct topological order. We show the presence of a topological phase transition that is of a different universality class than that observed in stroboscopic projective circuits. In the presence of unitary dynamics, the two topologically distinct phases are separated by a region with sub-volume scaling of the entanglement entropy. We find that this entanglement transition is well identified by a combination of the bipartite entanglement entropy and the topological entanglement entropy. We further show that the phase diagram is qualitatively captured by an analytically tractable non-Hermitian model obtained via post-selecting the measurement outcome. Finally we introduce a partial-post-selection continuous mapping, that uniquely associates topological indices of the non-Hermitian Hamiltonian to the distinct phases of the stochastic measurement-induced 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.
Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate.This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements.We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone.This finding motivates us to introduce measurement-only models, in which the "scrambling" and "unscrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes.These models represent a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits.We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models.We find that the principle driving the EPTs in these models is frustration, or mutual incompatibility, of the measurements.Surprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local (≳3-body) operators.We identify a class of exceptions to this behavior ("bipartite ensembles") which cannot sustain an entangling phase but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points.Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light cone, despite the nonlocality inherent to quantum measurements.
Quantum circuits provide an emerging controllable platform to realize novel dynamical non-equilibrium phases including topologically ordered states. The Kitaev model has become a cornerstone of quantum magnetism due to its quantum spin liquid ground state and rich phase diagram. The Kitaev model has also been treated in the monitored circuit setting, giving rise to topological area-law and critical-law entanglement entropy phases. In this article, we study the evolution of its phase diagram under the addition of new terms, motivated by their effects in the Kitaev model. We find that a single-qubit term, analogous to a magnetic field, leads to a trivial state in the high field limit, but with an additional intermediate volume-law phase. A three-qubit operator that commutes with the flux operators has the opposite effect: it stabilizes the critical-law phase against the short ranged area-law entanglement. We also employ a four-qubit plaquette commuting operator that simultaneously measures two opposite identical-type bonds on a plaquette. This generates a distinct volume-law phase and preserves the plaquette fluxes and associated topological order, yielding extensive entanglement while coexisting with the topological memory characteristic of the area-law phase. We quantitatively locate phase boundaries using stabilizer (Clifford) simulations together with tripartite mutual information and entanglement entropy measures. Our results highlight the rich phase diagram accessible from the measurement-only Kitaev model as well as suggesting rules relating the newly added operators to the phases they promote.