Quantum State Discrimination Measurement

Short answer: Quantum state discrimination asks how to tell apart nonorthogonal quantum states, which cannot be done perfectly. One strategy, unambiguous discrimination, never makes an error but sometimes returns an inconclusive result. Research from 2002 to 2005 shows how to design the best such measurement: as a semidefinite programme that can be solved efficiently for sets of linearly independent pure states, and with a derived optimal strategy for the filtering case. These are theoretical results about measurement design.

Why perfect discrimination is impossible

A 2005 Physical Review A paper states that deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. If one does not require success every time and allows inconclusive outcomes, unambiguous discrimination becomes possible with a certain probability of success. A 2002 arXiv paper describes the approach as combining error-free discrimination with a fraction of inconclusive results, pioneered by Ivanovic: design a measurement that sometimes returns an inconclusive result, but whenever it returns an answer the answer is correct.

Optimal measurements as a semidefinite programme

The 2002 paper considers unambiguous discrimination among a set of linearly independent pure states. It shows that the optimal measurement, the one that minimises the probability of an inconclusive result, can be formulated as a semidefinite programme. From that formulation the authors derive necessary and sufficient conditions for an optimal measurement, and show it can be computed in polynomial time using well-known semidefinite programming algorithms, which are known to converge to the global optimum.

The equal-probability measurement

Using the optimality conditions, the same paper derives conditions under which the measurement giving every state the same probability of an inconclusive result, called the equal-probability measurement, is optimal. It shows that for any set of states the prior probabilities can be chosen so that this measurement is optimal. A longer version considers state sets with strong symmetry, geometrically uniform sets defined over a group of unitary matrices and generated by a single vector, and compound sets generated by multiple vectors.

State filtering

The 2005 paper treats a variant called set discrimination, where the states are grouped into sets and the task is to say which set a given pure state belongs to. It studies the simplest case, quantum state filtering, in which N nonorthogonal states are split into two sets, one containing a single state and the other the remaining states. The authors derive the optimal measurement strategy, expressed as a generalised measurement known as a positive-operator-valued measure, with the matching optimal success and failure probabilities. They say an earlier Physical Review Letters paper presented the results, with emphasis on applications to probabilistic quantum algorithms.

What this does not tell you

The semidefinite programme result is for linearly independent pure states, and the filtering result for a specific split of the state set. These are mathematical optimal-measurement results and do not describe an experiment.

FAQ

Why can't nonorthogonal states be told apart perfectly?

Quantum measurement theory forbids deterministic discrimination of nonorthogonal states.

What is unambiguous discrimination?

A strategy that never errs but may return an inconclusive result.

How is the best measurement found?

For linearly independent pure states, by solving a semidefinite programme in polynomial time.

What is state filtering?

Deciding whether a state is one particular state or one of the others.

Does this involve hardware?

No. These papers are theoretical.