Scavenging quantum information: Multiple observations of quantum systems

Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, the state of the system collapses into a postmeasurement state from which the same observer cannot obtain further information about the original state of the system. However, the system still encodes a significant amount of information about the original preparation for a second observer who is unaware of the actions of the first one. We study how a series of independent observers can obtain, or can scavenge, information about the unknown state of a system (quantified by the fidelity) when they sequentially measure it. We give closed-form expressions for the estimation fidelity when one or several qudits are available to carry information about the single-qudit state, and we study the classical limit when an arbitrarily large number of observers can obtain (nearly) complete information on the system. In addition to the case where all observers perform most informative measurements, we study the scenario where a finite number of observers estimates the state with equal fidelity, regardless of their position in the measurement sequence and the scenario where all observers use identical measurement apparatuses (up to a mutually unknown orientation) chosen so that a particular observer’s estimation fidelity is maximized.

Figure 1

The observers’ measurement performance ${\Delta }_{\mathrm{eq}}$ as a function of the number of observers $K$. The solution given by the exact recurrence relations [Eq. (86)] (circles). The approximate solution for $K\gg N$ [Eq. (91)] (dotted) and $N\gg K$ [Eq. (97)] (dashed). Stochastic strategy [Eq. (94)] (solid). The solid line also depicts the performance of each observer measuring only a fraction $\tilde{N}=N/K$ of copies.