The dynamics of monitored systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling phases, tuned by the measurement rate. When the dynamics obeys a continuous symmetry, the entangling phase further splits into a fuzzy phase and a sharp phase based on the scaling of fluctuations of the symmetry charge. While the sharpening transition for Abelian symmetries is well understood analytically, no such understanding exists for the non- Abelian case. In this work, building on a recent analytical solution of the MIPT on tree-like circuit architectures (where qubits are repatedly added or removed from the system in a recursive pattern), we study entanglement and sharpening transitions in monitored dynamical quantum trees obeying U(1) and SU(2) symmetries. The recursive structure of tree tensor networks enables powerful analytical and numerical methods to determine the phase diagrams in both cases. In the U(1) case, we analytically derive a Fisher-KPP-like differential equation that allows us to locate the critical point and identify its properties. We find that the entanglement/purification and sharpening transitions generically occur at distinct measurement rates. In the SU(2) case, we find that the fuzzy phase is generic, and a sharp phase is possible only in the limit of maximal measurement rate. In this limit, we analytically solve the boundaries separating the fuzzy and sharp phases, and find them to be in agreement with exact numerical simulations.
The dynamics of monitored systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling phases, tuned by the measurement rate. When the dynamics obeys a continuous symmetry, the entangling phase further splits into a fuzzy phase and a sharp phase based on the scaling of fluctuations of the symmetry charge. While the sharpening transition for Abelian symmetries is well understood analytically, no such understanding exists for the non- Abelian case. In this work, building on a recent analytical solution of the MIPT on tree-like circuit architectures (where qubits are repatedly added or removed from the system in a recursive pattern), we study entanglement and sharpening transitions in monitored dynamical quantum trees obeying U(1) and SU(2) symmetries. The recursive structure of tree tensor networks enables powerful analytical and numerical methods to determine the phase diagrams in both cases. In the U(1) case, we analytically derive a Fisher-KPP-like differential equation that allows us to locate the critical point and identify its properties. We find that the entanglement/purification and sharpening transitions generically occur at distinct measurement rates. In the SU(2) case, we find that the fuzzy phase is generic, and a sharp phase is possible only in the limit of maximal measurement rate. In this limit, we analytically solve the boundaries separating the fuzzy and sharp phases, and find them to be in agreement with exact numerical simulations.
Monitored quantum circuits exhibit entanglement transitions at certain measurement rates. Such a transition separates phases characterized by how much information an observer can learn from the measurement outcomes. We study SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping onto an effective statistical-mechanics model. Due to the symmetry's non-Abelian nature, measuring qubit pairs allows for nontrivial entanglement scaling even in the measurement-only limit. We find a transition between a volume-law entangled phase and a critical phase whose diffusive purification dynamics emerge from the non-Abelian symmetry. Additionally, we identify a ``spin-sharpening transition.'' Across the transition, the rate at which measurements reveal information about the total spin quantum number changes parametrically with system size.
Monitored quantum circuits (MRCs) exhibit a measurement-induced phase transition between area-law and volume-law entanglement scaling. MRCs with a conserved charge additionally exhibit two distinct volume-law entangled phases that cannot be characterized by equilibrium notions of symmetry-breaking or topological order, but rather by the nonequilibrium dynamics and steady-state distribution of charge fluctuations. These include a charge-fuzzy phase in which charge information is rapidly scrambled leading to slowly decaying spatial fluctuations of charge in the steady state, and a charge-sharp phase in which measurements collapse quantum fluctuations of charge without destroying the volume-law entanglement of neutral degrees of freedom. By taking a continuous-time, weak-measurement limit, we construct a controlled replica field theory description of these phases and their intervening charge-sharpening transition in one spatial dimension. We find that the charge fuzzy phase is a critical phase with continuously evolving critical exponents that terminates in a modified Kosterlitz-Thouless transition to the short-range correlated charge-sharp phase. We numerically corroborate these scaling predictions also hold for discrete-time projective-measurement circuit models using large-scale matrix-product state simulations, and discuss generalizations to higher dimensions.
Monitored many-body systems fall broadly into two dynamical phases, "entangling" or "disentangling," separated by a transition as a function of the rate at which measurements are made on the system.Producing an analytical theory of this measurement-induced transition is an outstanding challenge.Recent work made progress in the context of tree tensor networks, which can be related to all-to-all quantum circuit dynamics with forced (postselected) measurement outcomes.So far, however, there are no exact solutions for dynamics of spin-1/2 degrees of freedom (qubits) with "real" measurements, whose outcome probabilities are sampled according to the Born rule.Here we define dynamical processes for qubits, with real measurements, that have a treelike space-time interaction graph, either collapsing or expanding the system as a function of time.The former case yields an exactly solvable measurement transition.We explore these processes analytically and numerically, exploiting the recursive structure of the tree.We compare the case of "real" measurements with the case of "forced" measurements.Both cases show a transition at a nontrivial value of the measurement strength, with the real measurement case exhibiting a smaller entangling phase.Both exhibit exponential scaling of the entanglement near the transition, but they differ in the value of a critical exponent.An intriguing difference between the two cases is that the real measurement case lies at the boundary between two distinct types of critical scaling.On the basis of our results, we propose a protocol for realizing a measurement phase transition experimentally via an expansion process.
Monitored many-body quantum systems can undergo sharp learnability transitions characterized by how much information can be learned by the observer. When the dynamics conserves a non-Abelian charge, such as an SU(2) spin, understanding how the observer learns the total charge remains an outstanding problem. Unlike the Abelian case, where charge measurements on distinct sites commute, the SU(2)-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of 1+1d monitored quantum dynamics with SU(2) symmetry, and show that it can be described by an effective replicated loop model comprised of a replica-pairing field and a diffusive (z=2) background sector that carries the SU(2) charge and remains gapless throughout the phase diagram. Our theory predicts that the "spin-sharpening'' and entanglement transitions coincide as a single transition. Ordering of the pairing field produces volume-law entanglement and hides the background sector from measurements, leading to a learning time of t∼L3 for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time t∼L2. Our analysis is controlled by a large-loop-fugacity expansion.
We study critical properties of the entanglement and charge-sharpening measurement-induced phase transitions in a nonunitary quantum circuit evolving with a U(1) conserved charge. Our numerical estimation of the critical properties of the entanglement transition at finite system sizes appears distinct from the generic nonconserving case and percolation. We provide two possible interpretations of this observation: (a) these two transitions occur at different measurement rates in the thermodynamic limit, but at finite system sizes their critical fans overlap and the critical exponents we probed here show a combination of both criticalities. Nonetheless, the multifractal properties of the entanglement transition remain distinct from the generic case without any symmetry, indicating a unique universality class due to the U(1) symmetry. (b) These two transitions occur at the same measurement rate at any lengthscale. Within this interpretation, our estimation of all the critical exponents is sharply different from the nonconserving case, again confirming the presence of a new universality class due to U(1) symmetry. We compute entanglement critical exponents and correlation functions via various ancilla measures, we use a transfer matrix for multifractality, and we compute correlators associated with charge sharpening to explain these findings. Through these correlators, we also find evidence consistent with the charge-sharpening transition being of the Berezinskii-Kosterlitz-Thouless type (including the predicted ``jump'' in stiffness), which simultaneously argues for a broad critical fan for this transition. As a result, attempts to measure critical properties in this finite-size system will see anomalously large exponents predicted by our numerical analysis.
We develop a novel holographic framework to study dynamical phases in random quantum circuits with a global symmetry G. Viewing the circuit as a tensor network, we decompose it into two parts: a symmetric layer, which defines an emergent gauge wavefunction in one higher dimension, and a non-symmetric layer, composed of random multiplicity tensors. For G=ZN symmetric circuits consisting of local unitary gates interspersed with local symmetric noise channels, averaging over the non-symmetric layer yields a dynamically generated noisy ZN surface code. This allows us to interpret ZN symmetric circuits in the volume-law phase as quantum error-correcting codes with a distinguished set of logical spin states that inherit the topological protection of the bulk code. By establishing equality of bulk and boundary coherent information, we show that quantum information encoded in these logical states is maximally protected against symmetric noise up to a finite threshold. We further study weakly monitored ZN symmetric circuits which exhibit a charge-sharpening transition. We show that the point at which the observer gains classical information about the global charge coincides with the point at which measurements destroy the underlying quantum information encoded in the bulk surface code. This also allows for a natural interpretation of the sharpening transition as a confinement transition in the gauge theory. For N≤4, weak measurements drive a single transition from a charge-fuzzy phase with exponential sharpening time t#∼eL to a charge-sharp phase with t#∼O(1). On the other hand, for N>4, the circuit can enter an intermediate phase with a linear sharpening time t#∼O(L). In this regime, the bulk gauge theory realizes a Coulomb phase with emergent gapless photons.