Memory in a sequence of weak and short duration measurements of non-commuting observables

Sequential projective measurements of non-commuting observables destroy information about previous measurement outcomes, but weak measurements conducted over short durations do not cause as much disturbance to the system, and as a result, memory of previous outcomes may not be entirely eliminated. We use Quantum State Diffusion to unravel a Lindblad equation, generating numerical, stochastic trajectories for a spin 1/2 system and a system of two entangled spin 1/2 subsystems, each undergoing a sequence of measurements of Sz and Sₓ spin observables. The autocorrelation function of completed Sz measurement outcomes for a range of lag times is then found. In a second set of trajectories, the simulation is terminated if the system reaches the vicinity of a chosen set of eigenstates of Sz and Sₓ: the `terminating eigenstates', during the respective measurement intervals of the two observables. From the generated trajectories, the `probability of termination' that the system will arrive at one of the terminating eigenstates after a certain number of measurements have been performed, can be calculated, as well as the mean first passage, which is the number of measurements performed, on average, before the simulation ends. Results can then be compared with projective measurement Born rule statistics. We demonstrate that with short and weak measurements the autocorrelation function of Sz can be non-zero for a range of lag times, while it vanishes for strong, longer duration measurements in agreement with the Born rule. The probability of termination and the mean first passage increase under weaker and shorter measurements, revealing that the system is able to avoid particular eigenstates of Sz and Sₓ for extended periods of time. A bias can develop in favour of a return to the same eigenstate visited in the previous measurement(s) of a given observable: a memory effect.