M.Sc Thesis

M.Sc StudentElron Noam Lev
SubjectExtensions of the Quantum Detection Problem: Optimal
Encoding and Detection with Uncertainty
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROF. Yonina Eldar


Quantum detection, also known as quantum hypothesis testing, is the problem of designing a quantum mechanical measurement to discriminate between a finite set of specified quantum states (code states). A popular performance criterion is the probability of correct detection. For this measure, the problem is essentially solved: necessary and sufficient conditions for optimality are known, polynomial time algorithms for numerical solution exist, and there are many special cases for which closed form solutions have been derived.

In this thesis we consider two extensions of the quantum detection problem. Whereas the original problem is the design of a detector for prespecified code states, we examine situations where one can either choose the code-states, or has incomplete knowledge of the code states.

In the first extension, we incorporate the encoding stage into the design and find the code states which maximize the probability of correct detection for a specified measurement. We then perform joint optimization on both ends of the link - the code states and the measurement - where the constraints are the dimension n of the quantum system, and the dictionary size and prior probabilities of the data. We show that one cannot outperform "pseudo-classical transmission", in which one transmits n symbols with orthogonal code states, and discards the remaining symbols. However, pseudo-classical transmission is not the only optimum. We fully characterize the collection of optimal setups, and briefly discuss the links between our findings and applications such as quantum key distribution and quantum computing. We conclude with a number of results concerning the design under an alternative optimality criterion, the worst-case posterior probability, which serves as a measure of the retrieval reliability.

The second extension is concerned with specified code-states, which are not fully known. We assume that each of the states is a mixture of a known state and an unknown state, and investigate two criteria for optimality. The first is maximization of the worst-case probability of correct detection. For the second we assume a probability distribution on the unknown states, and maximize the expected correct detection probability. We find that under both criteria, the optimal detectors are equivalent to the optimal detectors of an "effective ensemble". In the worst-case, the effective ensemble is comprised of the known states with altered prior probabilities, and in the average case it is made up of altered states with the original prior probabilities.