Critical behaviors of magic and participation entropy at measurement induced phase transitions

Abstract

We study the participation and stabilizer entropy of non-unitary quantum circuit dynamics, focusing on the critical line that separates the low-entanglement spin-glass phase and the paramagnetic phase. Along this critical line, the entanglement has a logarithmic scaling, which enables us to access the critical regime using large-scale matrix product state simulations with modest bond dimension. We find that both the participation entropy and stabilizer entropy exhibit critical slowing down: their saturation time scales linearly with the system size, in stark contrast to purely unitary dynamics, where saturation occurs on logarithmic time scales. In addition, we study bipartite participation and stabilizer mutual information, and find that it shows similar scaling behavior to the entanglement entropy. Finally, by analyzing the participation entropy of several paradigmatic Clifford circuits, we identify similar slow dynamical behavior near their respective critical points.

Figures (10)

• Figure 1: An example of the hybrid circuit dynamics described in the text. Time progresses from left to right. The blue rectangles correspond to Clifford unitaries selected randomly from the self-dual ensemble described in the text. The red rhombuses correspond to weak measurements with strength $\beta$ in the basis of either $XX$ or $ZI$, with probability $p$ and $1 - p$, respectively.
• Figure 2: (a) A schematic picture of the phase diagram. The model with $Z_2$ symmetry exhibits three phases: a paradigmatic volume-law phase (in red), a paramagnetic area-law phase (in blue), and a spin-glass area-law phase (in green). The transition between the paramagnetic and spin-glass phase occurs at $p = 0.5$. The gray region separating the volume-law from the area-law phases indicates uncertainty in $\beta_c$. (b) We show the entanglement as a function of $\log x$ for a small system of $L=24$ qubits and $p = 0.5$. The results are computed by equilibrating the system for $t_{eq} = 100$ steps, and then sampling the entanglement every $t_m = 5$ steps for an additional $t_s = 100$ steps. (c) We show the SRE density at $\beta = 0.8$ for various system sizes and as $p$ is varied. We see that the SRE is extensive for all $p$. We set $t_{eq} = 500$, $t_m = 25$, $t_s = 500$. (d) We show the converged entanglement entropy as a function of the max bond dimension imposed on the system for $p = 0.5$ and $L = 128$. We set $t_{eq} = 500$, $t_m = 25$, $t_s = 500$.
• Figure 3: Growth of various entropies and bipartite mutual information as functions of time and subsystem size for a system of size $L = 128$ at $p = 0.5$ and $\beta = 0.8$. (Left) Steady-state scaling of the von Neumann EE, BSMI, and BPMI as the bipartition size $\ell$ is varied. (Right) The same quantities exhibit logarithmic growth in time. The scaling coefficients of the EE are $\alpha_t^{\mathrm{EE}} = 0.41$ and $\alpha_s^{\mathrm{EE}}=0.42$, yielding $z^{\mathrm{EE}} = 1.02$. The scaling coefficients of the SMI are $\alpha_t^{\mathrm{SRE}} = 0.22$ and $\alpha_s^{\mathrm{SRE}} = 0.18$, yielding $z^{\mathrm{SRE}} = 0.81$. The deviation from logarithmic behavior observed in the PE is discussed in the main text.
• Figure 4: We show the approach to equilibrium by the SRE and PE. The measurement strength parameter $\beta = 0.8$. For each reported value, the entropies are calculated according to the methods described in Apps. \ref{['app:stabilizer_entropy']}-\ref{['app:participation_entropy']}. The left column shows the data on a log-log plot, emphasizing that at early times, the deviation from the steady state scales according to a power-law, while the right column is on a semi-log plot, emphasizing the crossover to an exponential approach to equilibrium at later times. In both columns, we have added dotted lines to serve as a guide to linear behavior indicating power-law (exponential) behavior on the left (right) column. On the left, the slope is fixed at 1 to emphasize the conformal symmetry.
• Figure 5: The growth of the entanglement and BPMI for the Clifford dynamics described in Sec. \ref{['sec:clifford_circuit']} at $\gamma = 1$ and $L=256$, both displaying logarithmic growth in time and space. The scaling coefficients for entanglement are $\alpha_t^{\mathrm{EE}} = 0.31$, $\alpha_s^{\mathrm{EE}}=0.33$, yielding $z^\mathrm{EE}=1.06$. The scaling coefficients for the BPMI are $\alpha_t^\mathrm{PE} = 0.168$ and $\alpha_s^{\mathrm{PE}} = 0.167$, yielding $z^\mathrm{PE}=0.99$.