Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits

Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements. We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers. We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system. We find that while Rényi entropies grow sub-ballistically as $\sqrt{t}$ in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time $\sim t$. We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes. Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.

Popular Summary

The “scrambling” of quantum information has emerged as a fundamental ingredient in the understanding of quantum thermalization, chaos, and spreading of errors in quantum computations. The robustness of this scrambling to external interference has been studied by subjecting the system to random measurements. These studies have led to new insights into the role of unitary evolution as an efficient quantum encoder that protects information from outside interference. Most studies until now have been restricted to systems having no symmetry. However, physical systems often obey symmetries that give rise to conservation laws, such as conservation of total electrical charge. Here, we provide the first step to bridge this gap by theoretically investigating a generic quantum system with a conserved charge.

Previous studies on systems with no symmetry have found that a phase transition, characterized by the system’s ability (or inability) to scramble extensive amounts of information, occurs as we increase the rate of the measurements. In our work, we consider a system of quantum spins subject to measurements of the charge taken randomly in space and time with some fixed probability. In addition to the transition mentioned above, we find another transition at a lower rate of measurement wherein the measurements can efficiently identify the global charge of the system. We call this a “charge-sharpening” transition.

Though the focus of our work is on systems with charge conservation arising from a particular symmetry, we argue that our results are much more general. A better understanding of the newly found charge-sharpening transition and extensions to different symmetries represent a clear challenge for future work.

Figure 1

Phase diagram and crossover timescales in U(1) symmetric monitored quantum circuits. Our numerical results indicate that there are two distinct phases in the entangling (volume-law) regime $p<p_c$, separated by a charge-sharpening critical point as $p=p_{\#}$. In the charge-fuzzy phase ($p<p_{\#}$), we identify three relevant timescales in the dynamics: For a large enough system, first (1) average Rényi entropies crossover from diffusive $[S_{n>1}]\sim \sqrt{t}$ to ballistic $\sim t$ scaling over a timescale $\sim p^{-3/2}$, then (2) charge sharpens after the crossover timescale $t_{\#}\sim L$, and finally (3) the system purifies over a much longer timescale $t_{\pi }\sim {\mathrm{e}}^L$.

Figure 2

Entanglement transition in qubit chains. (a) Data and collapse of the tripartite mutual information ${\mathcal{I}}_{3,n=1}$ used to determine the critical point of the entanglement transition, $p_c=0.105(3)$, and the correlation length exponent, $\nu =1.32(6)$. The error bars in $p_c$ and $\nu$ are estimated using the region in parameter space near the minimum such that ${\chi }^2<1.3{\chi }_{\min }^2$ [28]; see Fig. 5. (b) Data and collapse of the entanglement transition order parameter $[S_{1,E}]$ used as an alternative method to determine the critical point of the entanglement transition, $p_c=0.110(3)$, and the correlation length exponent, $\nu =1.42(16)$. The critical point and the exponent for $[S_{1,E}]$ are estimated with a finite size scaling analysis as explained in Appendix pp4. (c) At the critical point, the bipartite entanglement entropy shows logarithmic scaling with the system size. The coefficient of the logarithm has strong Rényi index dependence that can be described by a functional form $\alpha (n)=0.65(1)[1+(1/n)]+0.04(1)$. The error bars in $\alpha (n)$ are obtained from the standard deviation of the fit parameters in $\alpha (n)\sim a[1+(1/n)]+b$ while the error in the data points themselves take into account the standard deviation of the fit parameters in $S_n(p_c,L)\sim \alpha (n)\ln L$ and the error in $p_c$. This closely resembles the standard result for the ground state of a CFT, but has an offset slightly larger than zero. Error bars on the data points are obtained from the error on the mean by computing the standard deviation.

Figure 3

Charge-sharpening transition in qubit chains. (a) Data and collapse of $N_0$, the fraction of trajectories with $\delta {\mathcal{Q}}^2<\epsilon$ used to determine the critical point of the charge-sharpening transition, $p_{\#}=0.094(3)$, and correlation length exponent, ${\nu }_{\#}=2.0(3)$. The value of $\epsilon ={10}^{-2}$ is chosen such that it maximizes the quality of collapse at $t/L=4$. The error bars in $p_{\#}$ and ${\nu }_{\#}$ are estimated using the region in parameter space near the minimum such that ${\chi }^2<1.3{\chi }_{\min }^2$ [28]; see Fig. 5. (b) Data and collapse of the charge-sharpening transition order parameter $[S_{1,Q}]$ used as an alternative method to determine the critical point of the sharpening transition, $p_{\#}=0.088(3)$, and the correlation length exponent, ${\nu }_{\#}=2.15(15)$. (c) Data and collapse of the fraction of trajectories where the ancilla qubit is purified $N_{\mathrm{pure}}$ at the charge-sharpening transition. The transition point $p_{\#}=0.087(4)$ and the correlation length exponent ${\nu }_{\#}=2.1(2)$ are consistent with the ancilla probe. The critical point and the exponent obtained from $[S_{1,Q}]$ and $N_{\mathrm{pure}}$ are estimated with a finite size scaling analysis as explained in Appendix pp4. Error bars on the data points are obtained from the error on the mean by computing the standard deviation.

Figure 4

Dynamical exponent. Plot of the rescaled time dependence of the ancilla-circuit entanglement entropy for the entanglement transition at $p=p_c\approx 0.11$ (blue curve) and the charge-sharpening transition at $p=p_{\#}\approx 0.088$ (red curve). The finite size collapse indicates a dynamical exponent $z=1$ for both transitions.

Figure 5

Two transitions. Plot of the 68% confidence interval of the critical point $p$ and the correlation length exponent $\nu$ for both the entanglement and charge transitions. The mutual information $I_{3,1}$ (solid blue circle) and the ancilla probe (dashed blue circle) are for the entanglement transition, while the fraction of trajectories $N_0$ (solid red circle) and the ancilla probe (dashed red circle) are for the charge-sharpening phase transition. The two transitions appear to be different with statistical significance, although we cannot exclude systematic finite size effects that would change this conclusion in the thermodynamic limit.

Figure 6

Statistical mechanics model. (a) The average of $\mathcal{U}=U^{\otimes Q}\otimes U^{*\otimes Q}$ over Haar gates is nonzero if and only if the conjugate (bra) replicas are permutations of the nonconjugate (ket) replicas. Hence we can conveniently write each leg in the circuit as a set of $Q$ copies of nonconjugate states combined with a permutation group element [see Eq. (a1)]. In the large-$d$ limit, the permutation group elements for ingoing and outgoing legs become locked together in a single permutation, and the corresponding permutation group element $\sigma$ can be associated with a vertex (one per gate), while the charge states live on links. The U(1) charges $\alpha$, $\beta$ are constrained by charge conservation. (b) The charge dynamics in each replica are given by an effective 6-vertex model with weights $v$, corresponding to a symmetric exclusion process constrained by the measurements and entanglement cut. (c) Example of charge configuration.

Figure 7

Entanglement transition in the statistical model qubit contribution. (a) Plot of the qubit entanglement entropy $S_1$ versus $p$. The inset shows the finite size collapse using the ansatz discussed in the main text, with $\nu \sim 1.3$ and $p_{\#}\approx 0.29$. (b) Plot of $S_1$ versus $L$ near the critical point. We find that at the critical point $S_1$ grows logarithmically with $L$. (c) Tripartite mutual information ${\mathcal{I}}_{3,1}$ versus $p$, showing a crossing around $p_{\#}\approx 0.29$. The inset shows the finite size collapse using the same correlation length exponent as in (a). Error bars on the data points are obtained from the error on the mean by computing the standard deviation.

Figure 8

Charge-sharpening transition in the statistical mechanics model. (a) Plot of $[\delta {\mathcal{Q}}^2]$ versus $p$ plotted for $t/L=2$. We find that with increasing $L$ the value of $[\delta {\mathcal{Q}}^2]$ approaches zero for $p>p_{\#}\approx 0.3$. This is consistent with the entanglement transition in the qubit sector where we observed area-law scaling for $⟨S_n⟩$ for $p>p_{\#}$. Inset: charge variance in half of the system. (b) Charge variance $[\delta {\mathcal{Q}}^2]/L$ versus time $t$. For $p\lesssim p_{\#}$, $[\delta {\mathcal{Q}}^2]/L$ decays exponentially with a decay rate decreasing with $L$. For $p>p_{\#}$, the decay rate is the same for all $L$ suggesting that $⟨{\sigma }^2⟩/L$ goes to zero faster with increasing system size (faster in units of $t/L$). (c) Histogram of the charge variance in the charge-fuzzy phase. (d) Plot of $N_0$ versus $p$ with finite size collapse in the inset. $N_0$ was calculated at $t/L=2$. We find excellent collapse for $p_{\#}=0.315$ and $\nu =1.3$. (e) Time evolution of $N_0$. We clearly see a reversal in trend with system size $L$ around $p_{\#}\approx 0.31$. At the transition $p=p_{\#}$ we find that $N_0\sim h(t/L)$, with $h(x)$ some scaling function, consistent with a dynamical exponent $z=1$. (f) Histogram of the charge variance in the charge-sharp phase. The peak at 0.25 is due to trajectories with superposition of two charge sectors $\mathcal{Q}$ and $\mathcal{Q}+1$. The peak is stronger, and more stable, in the fuzzy phase than in the sharp phase. Error bars in (a) and (b) are obtained from the error of the mean using standard deviations.

Figure 9

Ancilla probe in the statistical mechanics model. Top: entanglement entropy of the ancilla qubit, which behaves as an order parameter for the charge-sharpening transition. Bottom: finite size scaling of the number of trajectories where the ancilla qubit is purified $N_{\mathrm{pure}}$, probing the charge-sharpening transition.

Figure 10

Entanglement dynamics in the statistical mechanics model. Ratio $R_s=(S_1/S_{\infty })$ versus $\sqrt{t}$ for $L=12$. We find that $R_s$ saturates implying that $S_{\infty }$ and $S_1$ are growing at the same rate $\sim t$. As expected, the saturation time is longer at low $p$. Error bars are from error in mean using standard deviations and standard error propagation.

Figure 11

Replicated statistical mechanics model. (a) The replicated statistical model is defined on a tilted square lattice, with permutation degrees of freedom ${\sigma }_a\in {\mathcal{S}}_Q$ living on vertices, and charge degrees of freedom $\{{\alpha }^i{\}}_{i=1,\dots ,Q}$. The Boltzmann weights have contributions from both vertices $V_a$, see Eq. (a10), and links $W_{ab}$, Eq. (a9). (b) Fixed boundary conditions in $Z_{\varnothing }$ at the top layer (ending the circuit at a given time $t$), with all permutations fixed to $e$. (c) The partition function $Z_A$ differs from $Z_{\varnothing }$ by the boundary condition fixed to ${\sigma }_0$ in the entanglement interval $A$. This creates a domain wall that follows a minimal cut in the limit $d\to \infty$.

Figure 12

Entanglement dynamics in the statistical mechanics model. (a) Plot of $S_1$, $S_2$, $S_{\infty }$, and $-\ln P_0$ versus $\sqrt{t}$ for $p=0$ obtained using the statistical model with a fixed vertical minimal cut. We clearly see different growth of $S_1$ and $S_{\infty }$, $S_2$. The curve of $-\ln P_0$ exactly overlaps the $S_{\infty }$ curve as argued in the main text. (b) Plot showing $[-\ln P_0]$ versus $\sqrt{t}$ for various $p$, and $L=12$. We find that this quantity grows linearly with time for any nonzero $p$. We also plot the average $-\ln [P_0]$ for various $p$ (dashed curves), which grows as $\sqrt{t}$ independently of $p$. Error bars are obtained using standard deviations.

Figure 13

Dynamics of charge sharpening in qubit chains. Fraction of trajectories with $\delta {\mathcal{Q}}^2<\epsilon$ with $\epsilon ={10}^{-3}$ at $p=0.085$, inside the fuzzy phase.

Figure 14

Dynamics of charge sharpening in the statistical model. Fraction of trajectories with $\delta {\mathcal{Q}}^2<\epsilon$ with $\epsilon ={10}^{-2}$ at $p=0.24$, inside the fuzzy phase. Inset: different threshold $\epsilon ={10}^{-10}$, showing a similar scaling of the charge sharpening over a timescale $t\sim L$.

Figure 15

Charge variance distribution in the qubit model. Distribution of the charge variance at $t/L=4$ (a) in the charge-fuzzy phase $p=0.05$, (b) near the critical point $p=0.10$, and (c) in the charge-sharp phase $p=0.14$.

Figure 16

Critical dynamics of charge-sharpening transition in qubit chains. Comparison of the dynamics of the ancilla entanglement entropy $S_{1,Q}$ and the quantity $1-N_0$, where $N_0$ is the fraction of trajectories with $\delta {\mathcal{Q}}^2<0.01$ and the ancilla entanglement entropy $S_{1,Q}$. After rescaling the overall amplitude, both observables collapse to the same exponential function in the longtime limit.

Figure 17

Universal decay rate of charge-sharpening transition in qubit chains. Data and collapse of the decay rate in the vicinity of the charge-sharpening phase transition. The transition point $p=0.088$ and the critical exponent $\nu =2.15$ established in the main text are used to collapse the curves.