The Effect of Non-commutative Weak Measurement on the Preparation of Long-range Entanglement

Measurement preparation of long-range entangled states is useful in quantum computation for initiating quantum error correction code states and in many-body physics for investigating topologically ordered states. However, real-world measurements are inherently noisy and often characterized by weak measurements. In this work, we analyze the continuous weak measurement preparation of the GHZ state, with a particular emphasis on the effects of non-commuting measurements and the implications of reading out the measurement outcomes. When all measurement results are collected, the pure final state can evolve into a long-range entangled state, provided that the rate of non-commuting transversal field measurements is significantly lower than that of Ising measurements. In one-dimensional systems, the entanglement generation takes O(log L) time where L is the linear size. In contrast, in higher dimensions the process requires only O(1), after which a critical time is reached, marking the transition of the monitored system into a long-range entangled phase. Suppose the transversal field measurement rate is tuned above a critical value, a quantum phase transition occurs, preventing the system from entering a long-range entangled phase. Conversely, if the outcomes are not read and the system is left in a mixed state, a classical order known as strong-to-weak symmetry breaking occurs after the critical time, which is independent of the transversal field decoherence rate.