A monitored many-body body can be broadly classified into two dynamic phases, ‘entangled’ or ‘unentangled’, separated by transitions as a function of the rate at which measurements are made in the system. Developing an analytical theory of the transitions induced by this measurement is an open challenge. Recent work has made progress in the context of tree tensor networks. This can be related to all-to-all quantum circuit dynamics with forced (post-selection) measurements. So far, however, there is no exact solution for spin-1/2-degree-of-freedom (qubit) dynamics by “real” measurements where the resulting probabilities are sampled according to the Born rule. Here we use real-world measurements to define a qubit dynamic process with a tree-like spatio-temporal interaction graph that contracts or expands the system as a function of time. In the former case, we get a measurement transition that can be solved exactly. We exploit the recursive structure of the tree to explore these processes analytically and numerically. Compare the “actual” and “forced” measurement cases. Both cases show a transition at a significant value of measured intensity and, in the case of real measurements, a smaller entangled phase. Both exhibit exponential scaling of entanglement near the transition, but differ in the value of the critical exponent. An interesting difference between the two cases is that the actual measurement case lies on the boundary between two different types of critical scaling. Based on our results, we propose a protocol for realizing experimentally measured phase transitions via an extension process.

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