How well can one determine the parameters of a quantum state? A simple example of a quantum state is the two-level system, which describes a number of physical quantum systems including the polarisation of a photon and the spin of an electron. Even though this state is simple, to fully specify it requires the determination of its Bloch vector which has three parameters (σ_x,σ_y and σ_z). This means that the determination of a two level system is already a multi-paramter estimation problem corresponding to estimating three non-commuting observables. In this case, a direct attempt at measuring σ_x will change the values for both σ_y and σ_z. This is an example of the famous Heisenberg uncertainty principle. If we wish to attain the maximum possible information about the state, we will need to use a more sophisticated measurement technique that can extract all three parameters at the same time.
Recently, we have developed a semi-definite program that can find the ultimate precision for multi-parameter estimation. This new theory puts a fundamental limit to how well we can perform state estimation on a variety of quantum systems. When an ensemble of identical quantum states are available, a combined measurement on two independent copies of the quantum state will give a higher precision compared to two separate individual measurements on each copy. This type of measurement is called a quantum collective measurement. Quantum collective measurements are very important for quantum computation and quantum communication applications because they represent a class of optimal measurements on quantum systems. As such, this topic is a active area of research for many quantum information theorists.
This project will begin with by computing the precision limits for estimating the state of simple quantum systems, such as a two-level system. Subsequently, the student will design quantum circuits that can perform collective measurements. These circuits will be examined to see if they can saturate the multi-parameter estimation limit. The developed algorithm, if it reaches the limit, is then guaranteed to be the most precise estimation possible. Finally, students will be asked to run the algorithms on a real quantum computer, such as on an IBM or Google quantum computer.