Self-Organized Error Correction in Random Unitary Circuits with Measurement

• URL: http://arxiv.org/abs/2002.12385v1
• Date: Thu, 27 Feb 2020 19:00:42 GMT
• Title: Self-Organized Error Correction in Random Unitary Circuits with Measurement
• Authors: Ruihua Fan, Sagar Vijay, Ashvin Vishwanath and Yi-Zhuang You
• Abstract summary: We quantify a universal, subleading logarithmic contribution to the volume law entanglement entropy. We find that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=\kappa L_A+\frac{3}{2}\log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I(\{x\}:\bar{A})\propto x^{-3/2}$ with distance $x$ from the region's boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can 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 $d$: $p_{c}\log[(d^{2}-1)({p_{c}^{-1}-1})]\le \log[(1-p_{c})d]$. The bound is saturated at $p_c(d\rightarrow\infty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) \le 0.1893$.

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