Quantum conditional mutual information as a probe of measurement-induced entanglement phase transitions

TL;DR

The paper addresses measurement-induced entanglement phase transitions (MIPTs) in monitored quantum circuits and the difficulty of using entanglement entropy near criticality due to logarithmic scaling. It introduces the quantum conditional mutual information $I(A:C|B)$, evaluated with carefully chosen partitions, as a robust probe that identifies the phase boundary and yields the universal logarithmic coefficient $ ilde{c}$ of the entanglement entropy. Through simulations of one-dimensional stabilizer circuits with variable-range Clifford gates, it presents phase diagrams, data-collapse analyses to extract the critical point $p_c$ and the correlation-length exponent $ u$, and a crossing-point analysis that yields $ ilde{c}=1.519(3)$ for the shortest-range case. The results show transitions for $2\lesssimoldsymbol{ extalpha}\lesssim 3$ with nonconformal criticality at smaller $oldsymbol{ extalpha}$, while $ ilde{c}$ remains robust across partitions for $oldsymbol{ extalpha}\gtrsim 3$, underscoring the efficacy of $I(A:C|B)$ as a diagnostic tool for MIPTs and emergent conformal criticality.

Abstract

We propose that the quantum conditional mutual information (QCMI), computed with a suitably chosen partition of the system, serves as a powerful probe for detecting measurement-induced entanglement phase transitions in monitored quantum circuits. To demonstrate this, we investigate monitored variable-range Clifford circuits and identify the phase boundary between volume-law and area-law entanglement phases by performing finite-size scaling analyses of the QCMI. Assuming that the entanglement entropy exhibits a logarithmic dependence on system size at criticality in short-range interacting cases, we further show that the QCMI allows for the simultaneous determination of both the critical point and the universal coefficient of the logarithmic term in the entanglement entropy via a crossing-point analysis. For the shortest-range interacting case studied, we obtain the thermodynamic-limit value of the coefficient as $\tilde{c}=1.519(3)$, which is significantly smaller than values reported in previous studies.

Figures (12)

• Figure 1: Schematic figure of a monitored variable-range random Clifford circuit. Time flows from bottom to top. Vertical lines represent qubits, dumbbell-shaped symbols indicate two-qubit random Clifford gates, and circles denote projective measurements that occur probabilistically.
• Figure 2: Two distinct partitions of the $L$-qubit system into subsystems $A$ (green), $B$ (white), $C$ (red), and $\overline{A\cup B\cup C}$ (gray). Each circle represents a qubit, and the number of qubits in each segment is indicated. Partition (a) is used for the QCMI in Eq. (\ref{['eq:IACBa']}), while partition (b) is employed for the QCMI in Eq. (\ref{['eq:IACBb']}).
• Figure 3: Phase diagram in the thermodynamic limit. The phase boundary is estimated from a data-collapse analysis of the QCMI $I_a(p, L)$, evaluated using the partitioning scheme shown in Fig. \ref{['fig:ABC']}(a). An equivalent phase diagram is obtained using the QCMI $I_b(p, L)$ based on the alternative partitioning shown in Fig. \ref{['fig:ABC']}(b). The dashed vertical line at $\alpha=2$ is a guide to the eye.
• Figure 4: QCMI $I_\gamma(p, L)$ for different partitions and interaction ranges. Panels (a) and (b) show results for $\gamma=a$, while (c) and (d) correspond to $\gamma=b$. The interaction-range parameter is $\alpha=3.5$ for (a) and (c), and $\alpha=2.5$ for (b) and (d). Vertical dashed lines indicate the critical points estimated from the data-collapse analyses in Figs \ref{['fig:vsLmin-alpha3500']} and \ref{['fig:vsLmin-alpha2500']}.
• Figure 5: Data-collapse fits of the QCMI $I_\gamma(p, L)$. Panels (a) and (b) show results for $\gamma=a$, while (c) and (d) correspond to $\gamma=b$. The interaction-range parameter is $\alpha=3.5$ for (a) and (c), and $\alpha=2.5$ for (b) and (d). The extrapolated values of $p_{\mathrm{c}}$ and $\nu$ in the thermodynamic limit, as obtained in Figs \ref{['fig:vsLmin-alpha3500']} and \ref{['fig:vsLmin-alpha2500']}, are used here.