Sequential Hypothesis Testing of Quantum States

We consider sequential hypothesis testing among multiple samples of one among <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> pure quantum states in an equidistant ensemble, i.e., those with identical pair-wise inner products. Each measurement in the sequence is a binary projective measurement that collapses the sample of the state measured at that instant into a linear span of a collection of states from the ensemble or its orthogonal complement. The algorithm adaptively decides if additional samples are needed or sufficient observation has been gathered. We show that our sequential measurement algorithm outperforms the sequential testing (ST) receiver, whose error-probability exponent is known to achieve the quantum Chernoff bound asymptotically in the limit of large (and fixed) number of samples. Even though our algorithm does not attain the quantum limit of minimum error probability (the Helstrom limit), it paves the way for future research on more advanced sequential quantum hypothesis tests, e.g., those that go beyond binary projective measurements on each sample.