An introduction to quantum state estimation

Quantum state estimation is a key statistical tool for quantum engineering applications. In quantum computation for instance, one applies a certain sequence of transformations (quantum algorithm) to a register of two dimensional quantum systems (qubits), which often produces joint states with special correlation properties (entanglement). In order to verify that the preparation procedure has been successful, one needs to measure the final state, to obtain a random outcome whose distribution depends on the quantum state. By repeating the procedure over many independent preparations, one collects statistical data that can be used to statistically reconstruct the state.

In this talk I will introduce the basic concepts required to formulation the state estimation problem (also known as quantum tomography), and present a recently proposed method for doing this. The Projected Least Squares (PLS) estimator consists of first computing a traditional least square estimator, which is subsequently projected onto the space of quantum states (positive trace-one matrices). This method is faster that traditional maximum likelihood estimation, and has good statistical behaviour. Indeed, using matrix concentration inequalities we show that PLS attains fundamental lower bounds for the estimation of low rank states with separate measurements.

Time permitting, I will discuss a more fundamental question concerning the highest estimation precision allowed by quantum mechanics. Here the concept of quantum local asymptotic normality (QLAN) gives a precise answer to this question and indicates what measurements are required to achieve this precision.