Quantum state discrimination

Quantum state discrimination uses a measurement in quantum mechanics to identify which candidate density operator was prepared. Minimum-error and unambiguous discrimination impose different objectives; quantum binary hypothesis testing is the two-state case.

Suppose one equally likely action returns a known normalized pure state $|s⟩$ unchanged, while another returns the maximally mixed state $I/d$. A basic measurement in any orthonormal basis containing $|s⟩$ distinguishes them with success $1-1/(2d)$: guess the pure-state action on its basis label and the reset action on the remaining $d-1$ labels. The first action succeeds with probability one and the second with probability $(d-1)/d$.

The Holevo–Helstrom theorem also proves optimality for these returned states. The Hermitian difference $|s⟩⟨s|-I/d$ has eigenvalues $1-1/d$ and $-1/d$ with multiplicity $d-1$, hence trace norm $2(1-1/d)$. Its optimal equal-prior discrimination success is $1/2+\Vert {\rho }_0-{\rho }_1{\Vert }_1/4$. Random computational-basis permutations fix the uniform superposition, so they supply an example of the invariant action. This compares the returned states in the stated unassisted protocol; it makes no assertion about a more general channel-discrimination problem with an entangled ancillary probe.^[ref]

A task in which a quantum measurement outputs a label of a preparation that can be ruled out. If preparations are ${\rho }_j$ and the outcome labelled $j$ has effect $E_j$, error-free exclusion requires $\mathrm{Tr}(E_j{\rho }_j)=0$ for every $j$, together with $E_j\ge 0$ and ${\sum }_j{E}_j=I$. A family admitting a complete error-free exclusion measurement consists of antidistinguishable quantum states. Exclusion need not identify the actual preparation.

For $d$ labelled pure states in dimension $d$, an orthonormal basis $\{|b_j⟩{\}}_{j=0}^{d-1}$ supplies error-free projective state exclusion when $⟨b_j|{\psi }_j⟩=0$ for every $j$. The Born rule gives zero probability to reporting $j$ on input ${\psi }_j$; orthonormal-basis completeness ensures that an outcome is always obtained. Conversely, zero error for this labelled basic measurement forces precisely those zero overlaps. The matrix of transition probabilities is therefore a matrix of column probability distributions with zero diagonal.

For candidate density matrices ${\rho }_h$ with prior probabilities $p_h$, a projective measurement has outcome likelihood $p(m\mid h)=\mathrm{Tr}(P_m{\rho }_h)$. Bayes' theorem gives the displayed posterior whenever the denominator is positive. An outcome certifies one candidate only when every other candidate has zero likelihood there. If all candidates have the same outcome likelihood, observing it leaves their prior probabilities unchanged.