Quantum binary hypothesis testing

Quantum binary hypothesis testing chooses a two-outcome measurement to distinguish two candidate density operators, trading the probabilities of the two kinds of error.

For $k$ equiprobable quantum code states decoded by a POVM $\{E_m\}$, testing whether the classical message label agrees with the decoded output gives a joint binary POVM with $F_0={\sum }_m|m⟩⟨m|\otimes E_m$. Its acceptance probability is the decoding success $1-ϵ$ on the joint classical-quantum state and $1/k$ on the product of its marginals. Applying quantum relative entropy monotonicity to this test gives the one-shot classical-quantum coding converse.

Quantum channel discrimination feeds a common input into one of several candidate quantum channels and measures the output. An entangled quantum ancilla permits inputs ${\rho }_{HR}$ and compares $(T_j\otimes {\mathrm{id}}_R)({\rho }_{HR})$; optimal binary discrimination is governed by the diamond norm of the weighted channel difference.

For the transposition map $\Theta :M_d\to M_d$, the induced trace norm is $\Vert \Theta {\Vert }_1=1$, while stabilization givesThus an entangled quantum ancilla can amplify the distinguishability witnessed by transposition by a factor of $d$.

For hypotheses ${\rho }_0,{\rho }_1$ with priors $p,1-p$, the optimal success probability isAn optimal measurement projects onto the positive and negative spectral subspaces of the weighted difference.

This is the minimum-error optimum in binary quantum state discrimination.

For equiprobable qubit states with Bloch vectors $r,s$, their density-matrix difference is $(r-s)\cdot \sigma /2$, with eigenvalues $\pm |r-s|/2$. An effect $0\le E\le I$ declaring the first state yields success $1/2+\mathrm{Tr}[E({\rho }_r-{\rho }_s)]/2$. Selecting its positive-eigenvalue eigenspace maximizes this trace and proves the displayed result. Measure along $r-s$ and attach the corresponding state labels; if the vectors coincide, no measurement improves on guessing.