M.Sc Thesis | |

M.Sc Student | Elron Noam Lev |
---|---|

Subject | Extensions of the Quantum Detection Problem: Optimal Encoding and Detection with Uncertainty |

Department | Department 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.