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On the Distinguishability of Random Quantum States

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Abstract

We develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states: a bound based on the pairwise inner products of the states, and a bound based on the eigenvalues of their Gram matrix. We use the latter, and results from random matrix theory, to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions, where n/d approaches a constant. In particular, for almost all ensembles of n states in n dimensions, p > 0.72. An application to distinguishing Boolean functions (the “oracle identification problem”) in quantum computation is given.

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Authors and Affiliations

  1. Department of Computer Science, University of Bristol, Woodland Road, Bristol, BS8 1UB, UK

    Ashley Montanaro

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  1. Ashley Montanaro
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Correspondence to Ashley Montanaro.

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Communicated by M.B. Ruskai

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Montanaro, A. On the Distinguishability of Random Quantum States. Commun. Math. Phys. 273, 619–636 (2007). https://doi.org/10.1007/s00220-007-0221-7

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  • DOI: https://doi.org/10.1007/s00220-007-0221-7

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