Projected least squares quantum tomography by Jonas Kahn
Quantum statistics require a mathematical framework different from that
of classical statistics: if we can measure A or B, we cannot in general,
even theoretically, measure A and B. We'll start the talk by introducing
the quantum statistics framework.
The key object is the state, encoded by a semi-definite positive matrix
with trace one. In practice it will often be low-rank. In particular,
the states usually considered in physics classes, namely the pure
states, are the rank-one states.
We will show that a very simple and computationally efficient procedure
allows rate-optimal estimation of a state or of a channel (quantum
transformation), while being adaptive to the rank, and allowing the
experimentalist to know in real-time if the rank is low enough to stop
acquiring more data.