Theory of the phase transition in random unitary circuits with measurements (1908.04305v3)

Abstract: We present a theory of the entanglement transition tuned by measurement strength in qudit chains evolved by random unitary circuits and subject to either weak or random projective measurements. The transition can be understood as a nonanalytic change in the amount of information extracted by the measurements about the initial state of the system, quantified by the Fisher information. To compute the von~Neumann entanglement entropy $S$ and the Fisher information $\mathcal{F}$, we apply a replica method based on a sequence of quantities $\tilde{S}^{(n)}$ and $\mathcal{F}^{(n)}$ that depend on the $n$-th moments of density matrices and reduce to $S$ and $\mathcal{F}$ in the limit $n\to 1$. These quantities with $n\ge 2$ are mapped to free energies of a classical spin model with $n!$ internal states in two dimensions with specific boundary conditions. In particular, $\tilde{S}^{(n)}$ is the excess free energy of a domain wall terminating on the top boundary, and $\mathcal{F}^{(n)}$ is related to the magnetization on the bottom boundary. Phase transitions occur as the spin models undergo ordering transitions in the bulk. Taking the limit of large local Hilbert space dimension $q$ followed by the replica limit $n\to 1$, we obtain the critical measurement probability $p_c=1/2$ and identify the transition as a bond percolation in the 2D square lattice in this limit. Finally, we show there is no phase transition if the measurements are allowed in an arbitrary nonlocal basis, thereby highlighting the relation between the phase transition and information scrambling. We establish an explicit connection between the entanglement phase transition and the purification dynamics of a mixed state evolution and discuss implications of our results to experimental observations of the transition and simulability of quantum dynamics.

• The paper introduces a mapping of random unitary circuits with measurements onto classical spin models, enabling detailed analysis of the entanglement phase transition.
• It leverages replica methods and Fisher information to accurately compute von Neumann entanglement entropy and quantify initial state information.
• The study reveals a critical measurement probability marking the transition from volume-law to area-law entanglement, akin to bond percolation in two-dimensional lattices.

Overview of the Paper: Theory of the Phase Transition in Random Unitary Circuits with Measurements

This paper provides an in-depth theoretical investigation into the entanglement phase transition in quantum systems, specifically those driven by random unitary circuits (RUCs) subject to varying strengths of measurement. The central focus is on elucidating the nature of the phase transition as a function of measurement strength and understanding its implications on the entanglement properties of the quantum systems involved.

Key Contributions

1. Mapping to Classical Spin Models: The authors introduce an innovative approach by mapping the quantum problem of RUCs with measurements onto classical spin models. This mapping is pivotal as it allows for the analysis of phase transitions within a well-established classical framework.
2. Use of Replicas and Fisher Information: To tackle the averaging issues over random unitary operations and measurement outcomes, the paper employs a replica method. This technique is adeptly used to compute quantitatively the von Neumann entanglement entropy and Fisher information related to the initial state information extracted by measurements.
3. Characterization of the Phase Transition: The transition between volume-law and area-law entanglement is characterized in detail. The transition signifies a change in the scaling of entanglement entropy as a function of subsystem size, moving from a phase where it scales with the volume to one scaling with the area (surface boundary).
4. Numerical and Theoretical Insights: The paper tightly integrates analytical predictions with numerical evidence, providing insights into the critical measurement probability (denoted as $p_c$) where this transition occurs. Notably, in the limit of large local Hilbert space dimension $q$, the transition is identified as a type of bond percolation in a two-dimensional square lattice, leading to $p_c = 1/2$.
5. Implications on Information Scrambling and Quantum Purification: The paper highlights the relation between entanglement phase transitions and information dynamics in quantum circuits. Specifically, it ties the transition to the measurement-induced disruption of information scrambling, suggesting that efficient quantum error correction could render quantum dynamics robust against certain rates of measurement.

Theoretical and Practical Implications

The paper's findings have broad implications both for the theoretical understanding of quantum information dynamics and for practical aspects of quantum computing. On the theoretical side, the mapping to spin models enriches the toolkit available for analyzing entanglement transitions in disordered quantum systems. The use of Fisher information also provides an additional perspective on how information about a quantum system can be probed or protected.

Practically, understanding these transitions is vitally important for optimizing the design and operation of quantum devices, particularly those involving noisy intermediate-scale quantum (NISQ) technologies where measurement and control fidelities are constrained. The insights from this paper could help refine error correction protocols by taking advantage of the natural encoding and protection of quantum information through entanglement.

Future Directions

Looking forward, further studies could explore the effect of different types of measurements, including nonlocal measurements, on the critical dynamics of entanglement. Another avenue is to investigate the potential universality classes of such transitions beyond the bond percolation paradigm, especially for finite Hilbert space dimensions $q$. Additionally, applying these theoretical innovations to newer models of quantum computation or varying Hamiltonian dynamics could elucidate whether the observed transition phenomena hold universally.

In summary, this research carves a significant pathway in understanding entanglement transitions via a novel crossover using classical spin models, enhancing our comprehension of quantum measurements' role in complex unitary dynamics.