Abstract
We call two non-degenerate observables “mutually unbiased” if every eigenstate of one of them is a state of complete ignorance relative to the other one. For example, for a spin-1/2 particle, the spin components in the x, y, and z directions are all unbiased with respect to each other, since a knowledge of any of these components entails a complete lack of knowledge about the others. It turns out that if, for a given system, there exist a sufficient number of mutually unbiased observables, then measurements of these observables can be used to determine as efficiently as possible the state of an ensemble of such systems. For a system with a finite number N of orthogonal states, the number of mutually unbiased observables one needs is N+1. We have shown that precisely N+1 such observables can indeed be found if N is a power of a prime. We do not know whether this number can be found if N is not a power of a prime.
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References
William K. Wootters and Brian D. Fields, “Optimal State-Determination by Mutually Unbiased Measurements,” submitted to Annals of Physics (1988).
I. D. Ivanovic, J. Phys. A 14 3241 (1981).
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© 1989 Springer Science+Business Media Dordrecht
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Wootters, W.K., Fields, B.D. (1989). Searching for Mutually Unbiased Observables. In: Kafatos, M. (eds) Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Fundamental Theories of Physics, vol 37. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0849-4_9
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DOI: https://doi.org/10.1007/978-94-017-0849-4_9
Publisher Name: Springer, Dordrecht
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