Abstract
After a brief review of the notion of a full set of mutually unbiased bases in an N-dimensional Hilbert space, we summarize the work of Wootters and Fields (W K Wootters and B C Fields, Ann. Phys. 191, 363 (1989)) which gives an explicit construction for such bases for the case N=p r, where p is a prime. Further, we show how, by exploiting certain freedom in the Wootters-Fields construction, the task of explicitly writing down such bases can be simplified for the case when p is an odd prime. In particular, we express the results entirely in terms of the character vectors of the cyclic group G of order p. We also analyse the connection between mutually unbiased bases and the representations of G.
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References
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Lists of irreducible polynomials for low values of p and r together with the order of their roots may be found in [6]. For given p and r, the number of such polynomials is equal to φ(p4−1)/r, where φ is the Euler phi-function
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Chaturvedi, S. Mutually unbiased bases. Pramana - J Phys 59, 345–350 (2002). https://doi.org/10.1007/s12043-002-0126-0
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DOI: https://doi.org/10.1007/s12043-002-0126-0
Keywords
- Mutually unbiased bases
- maximally noncommuting observables
- optimal quantum state determination
- Galois fields determination