Measurements profoundly impact quantum systems and can be used to create novel states of matter out of equilibrium. We investigate the multipartite entanglement structure that emerges in hybrid quantum circuits involving unitaries and measurements. We describe how a balance between measurements and unitary evolution can lead to multipartite entanglement spreading to distances far greater than what is found in nonmonitored systems, thus evading the usual fate of entanglement. We introduce a graphical representation based on spanning graphs that allows to infer the evolution of genuine multipartite entanglement for general subregions. We exemplify our findings on hybrid random Haar circuits that realize a 1D measurement-induced dynamical phase transition, where we find genuine three-party entanglement at all separations. At criticality, our data are consistent with power-law decay with a tripartite exponent strictly larger than the one of the bipartite logarithmic negativity. The four-party case is also explored. Finally, we discuss how our approach can provide fundamental insights regarding entanglement dynamics for a wide class of quantum circuits and architectures.
Measurements profoundly impact quantum systems and can be used to create novel states of matter out of equilibrium. We investigate the multipartite entanglement structure that emerges in hybrid quantum circuits involving unitaries and measurements. We describe how a balance between measurements and unitary evolution can lead to multipartite entanglement spreading to distances far greater than what is found in nonmonitored systems, thus evading the usual fate of entanglement. We introduce a graphical representation based on spanning graphs that allows to infer the evolution of genuine multipartite entanglement for general subregions. We exemplify our findings on hybrid random Haar circuits that realize a 1D measurement-induced dynamical phase transition, where we find genuine three-party entanglement at all separations. At criticality, our data are consistent with power-law decay with a tripartite exponent strictly larger than the one of the bipartite logarithmic negativity. The four-party case is also explored. Finally, we discuss how our approach can provide fundamental insights regarding entanglement dynamics for a wide class of quantum circuits and architectures.
We study multiparty entanglement near measurement induced phase transitions (MIPTs), which arise in ensembles of local quantum circuits built with unitaries and measurements. In contrast to equilibrium quantum critical transitions, where entanglement is short-ranged, MIPTs possess long-range k-party genuine multiparty entanglement (GME) characterized by an infinite hierarchy of entanglement exponents for k ≥ 2. First, we represent the average spread of entanglement with "entanglement clusters", and use them to conjecture general exponent relations: 1) classical dominance, 2) monotonicity, 3) subadditivity. We then introduce measure-weighted graphs to construct such clusters in general circuits. Second, we obtain the exact entanglement exponents for a 1d MIPT in a measurement-only circuit that maps to percolation by exploiting non-unitary conformal field theory. The exponents, which we numerically verify, obey the inequalities. We also extend the construction to a 2d MIPT that maps to classical 3d percolation, and numerically find the first entanglement exponents. Our results provide a firm ground to understand the multiparty entanglement of MIPTs, and more general ensembles of quantum circuits.
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 d spatial dimensions and classical statistical mechanical models in d+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 d-dimensional percolation and (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.
The competition between scrambling unitary evolution and projective measurements leads to a phase transition in the dynamics of quantum entanglement. Here, we demonstrate that the nature of this transition is fundamentally altered by the presence of long-range, power-law interactions. For sufficiently weak power laws, the measurement-induced transition is described by conformal field theory, analogous to short-range-interacting hybrid circuits. However, beyond a critical power law, we demonstrate that long-range interactions give rise to a continuum of nonconformal universality classes, with continuously varying critical exponents. We numerically determine the phase diagram for a one-dimensional, long-range-interacting hybrid circuit model as a function of the power-law exponent and the measurement rate. Finally, by using an analytic mapping to a long-range quantum Ising model, we provide a theoretical understanding for the critical power law.
Measurement-induced phase transitions (MIPT) give rise to novel dynamical states of quantum matter realized by balancing unitary evolution and measurements. We present large-scale numerical simulations of a trapped-ion native MIPT, argued to belong to the universality class described by the Haar non-unitary conformal field theory. First, through a finite-size analysis we obtained the critical measurement rate, and correlation length exponent, which falls close to the percolation value. Second, by leveraging a monotone computable via semi-definite programming, we uncover robust algebraic decay of genuine multiparty entanglement (GME) versus separation for 2, 3, and 4 parties. The corresponding critical exponents are lower-bounded by those of the multiparty mutual information, which we determine up to 4 parties, and conjecture to be (k+2) for k parties. Additionally, we derive lower bounds for both GME and multiparty mutual information.
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 the structure of multipartite entanglement in monitored quantum circuits exhibiting measurement-induced phase transitions (MIPTs). Using a one-dimensional Clifford circuit subject to local measurements with a probability 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→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.