Self-organized error correction in random unitary circuits with measurement

Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a subthermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. We quantify these notions by identifying a power-law decay of the mutual information I ({x } :A ¯) ∝x^{−3 /2} in the volume-law-entangled phase, between a qudit located a distance x from the boundary of a region A , and the complement A ¯, which implies that a measurement of this qudit will retrieve very little information about A ¯. We also find a universal logarithmic contribution to the volume law entanglement entropy S^{(2 )}(A ) =κ L_A+3/2 logL_A which is intimately related to the first observation. We obtain these results by relating the entanglement dynamics to the imaginary time evolution of an Ising model, to which we apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength p_c as a function of the qudit dimension d : p_clog[(d²−1 ) (p_c⁻¹−1 ) ] ≤log[(1 −p_c) d ] . The bound is saturated at p_c(d →∞ ) =1 /2 and provides a reasonable estimate for the qubit transition: p_c(d =2 ) ≤0.1893 .