Measurement-induced phase transition with a time-varying number of qubits

The measurement-induced phase transition (MIPT) is typically described in two different ways: (i) as a transition in the dynamical behavior of the entanglement of a pure state, or (ii) as a transition in the purification rate of an initially mixed state. These two definitions of the MIPT coincide for systems with a fixed number of qubits. Here we consider a system in which the total number of qubits is not fixed in time. For example, one can add a finite probability that a qubit is discarded after being measured. We show that in this case the definitions (i) and (ii) represent two distinct transitions that do not, in general, coincide. We examine the space of measurement probability and discard probability for a 1+1D system and find the existence of a new "disentangling and non-purifying" phase. The phase space admits two distinct transitions, an entanglement transition and a purification transition, that belong to distinct universality classes.