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arXiv:quant-ph/0201074 · Physical Review A

Minimum-error discrimination between three mirror-symmetric states

Erika Andersson, Stephen Mark Barnett, Claire R Gilson, Kieran Hunter

Published Apr 19, 2002• 107 citations
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Abstract

We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation ${|+\stackrel{\ensuremath{\rightarrow}}{〉}|+〉,|\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{〉}\ensuremath{-}|\ensuremath{-}〉}$ leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.

DOI
10.1103/physreva.65.052308

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We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation ∣+〉\ensuremath→∣+〉,∣\ensuremath−〉\ensuremath→\ensuremath−∣\ensuremath−〉{|+\stackrel{\ensuremath{\rightarrow}}{〉}|+〉,|\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{〉}\ensuremath{-}|\ensuremath{-}〉}∣+〉\ensuremath→​∣+〉,∣\ensuremath−〉\ensuremath→​\ensuremath−∣\ensuremath−〉 leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.
We provide an exact solution of finding the optimum measurement strategy for distinguishing three mixed quantum states with mirror symmetry. The signal states reside in two-dimensional Hilbert space. By mirror symmetry, two of the three states have the same prior probabilities and transform into each other under the mirror symmetry. We find that the optimum measurement strategy may consist of three, two, or a single nonzero probability operator measure elements depending upon the prior probabilities of the quantum signals.
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We present the first experimental demonstration of the maximum confidence measurement strategy for quantum state discrimination. Applying this strategy to an arbitrary set of states assigns to each input state a measurement outcome which, when realized, gives the highest possible confidence that the state was indeed present. The theoretically optimal measurement for discriminating between three equiprobable symmetric qubit states is implemented in a polarization-based free-space interferometer. The maximum confidence in the measurement result is 2/3. This is the first explicit demonstration that an improvement in the confidence over the optimal minimum error measurement is possible for linearly dependent states.
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We formulate an expression for the accessible information in mirror-symmetric ensembles of real qubit states. This expression is used to make a detailed study of optimum quantum measurements for extracting the accessible information in three-state, mirror-symmetric ensembles. Distinct measurement regimes are identified for these ensembles with optimal measurement strategies involving different numbers of measurement operators, similar to results found for minimum error discrimination. Our results generalize known results for the accessible information in two pure states and in the trine ensemble.