Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough

TL;DR

This paper investigates how to maximize information gain in weak or continuous measurements of qudits, showing that optimal control protocols can significantly improve purification rates, with implications for quantum measurement strategies.

Contribution

The study demonstrates that complementarity alone offers limited improvement, but optimal control protocols can achieve near-deterministic information acquisition with quadratic scaling in qudits.

Findings

Complementarity yields only a 2-fold improvement in purification rate.

Optimal control protocols can reach an $O(D^{2})$ scaling in information gain.

Results contrast with protocols based solely on complementarity in qubit registers.

Abstract

To maximize average information gain for a classical measurement, all outcomes of an observation must be equally likely. The condition of equally likely outcomes may be enforced in quantum theory by ensuring that one's state $\rho$ is maximally different, or complementary, to the measured observable. This requires the ability to perform unitary operations on the state, conditioned on the results of prior measurements. We consider the case of measurement of a component of angular momentum for a qudit (a $D$-dimensional system, with $D=2J+1$). For weak or continuous-in-time (i.e. repeated weak) measurements, we show that the complementarity condition ensures an average improvement, in the rate of purification, of only 2. However, we show that by choosing the optimal control protocol of this type, one can attain the best possible scaling, $O(D^{2})$, for the average improvement. For this protocol the acquisition of information is nearly deterministic. Finally we contrast these results with those for complementarity-based protocols in a register of qbits.