Quantum state discrimination identifies which state was supplied from a known list by choosing measurements suited to that list. For mutually orthogonal states, one experiment can distinguish the possibilities perfectly; nonorthogonal states require different strategies. section 4.2
For orthogonal-state discrimination:
- Represent the possible states as a promised list of orthogonal quantum-register states. section 4.2
- When each state uses distinct computational-basis states, measure in a basis that reveals which listed state was supplied. section 4.2
- For a general set, construct a unitary transformation that maps the states to computational-basis states, then measure. section 4.8
- Complete the transformation matrix with additional columns when fewer than 2ⁿ states are specified, preserving unitarity. section 4.8
Four three-qubit states
The states (|000⟩ + |111⟩)/√2, (|100⟩ + |011⟩)/√2, (|010⟩ + |101⟩)/√2, and (|001⟩ + |110⟩)/√2 form an orthogonal list because each computational-basis state appears in only one member. section 4.2
Nonorthogonal states cannot be distinguished perfectly in one experiment. Minimum-error discrimination incorporates unequal input probabilities, while unambiguous discrimination permits “I don’t know” outcomes to avoid incorrect identifications. section 4.2 section 4.8
