Post-selected Criticality in Measurement-induced Phase Transitions

Abstract

Information-theoretic phase transitions, such as the measurement-induced phase transition (MIPT), characterize the robustness of quantum dynamics to local monitoring and are naturally formulated in terms of trajectories conditioned on typical measurement outcomes, which are naively accessible only through post-selection. Here we implement forced measurements to investigate how explicit post-selection alters the nature of the transition. We find that post-selection fundamentally alters the universality class by reweighting trajectories that are otherwise rare. In particular, we obtain a correlation-length exponent $ν\approx 2.1$ larger than that of the standard MIPT and a negative effective central charge $c_\mathrm{eff}\approx -0.4$. We also compare the post-selected MIPT to the entanglement transition of Random Tensor Networks (RTN), and demonstrate that their universality class is the same. This setup further allows time-periodic, translationally-invariant circuits with post-selected weak measurements. In both models, we find that an onsite dimension of at least 3 (qutrits but not qubits) is necessary to induce a transition.

Figures (8)

• Figure 1: (a-b): Schematic diagrams of both the models (a) Circuit diagram of the Random Quantum Circuit (RQC) (b) Circuit diagram of the Random Tensor Network (RTN) (c-f): Scaling collapse of various entanglement measures. Different system sizes are shown in shades of blue (Random Tensor Network) and red (Random Quantum Circuit). (c) Tripartite mutual information, $I_3$ near the critical point $p_c$. (d) Time evolution of the von Neumann entropy of an ancilla maximally entangled with $L$ qubits. (e) Von Neumann entropy of an ancilla entangled with a single qubit at $t > 4L$ (f) Mutual information between ancillas entangled at sites $r$ and $r'$ such that $|r - r'| = L/2$.
• Figure 2: (a) Schematic of the central charge $c(k)$ of the replica partition function $Z_k$. $k=0,1$ correspond to the post-selected and Born rule MIPTs respectively, and $k=2$ is the Ising model which has $c=1/2$.(b) Free-energy density $f(L)$ (see Eq. \ref{['eq:casimir']}) vs $1/L^2$.
• Figure 3: Tripartite mutual information $I_3$ for translationally-invariant post-selected weak-measurement variants.
• Figure 4: Coefficients $\alpha(n)$ of the Renyi entropies (blue triangles: RTN; red circles: RQC). Blue dotted and red dashed lines show the $n$ dependence, and the horizontal lines denote $\alpha(\infty)$.
• Figure 5: Scaling collapse of the ancilla entropy $S_a$ near the critical point ($p_c$ for RQC, $\chi_c$ for RTN; red circles: RQC, blue triangles: RTN).