A transform of complementary aspects with applications to entropic uncertainty relations
Abstract
Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols, they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d = 2^n, which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set of MUBs and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension d = 4, which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.
Additional Information
© 2010 American Institute of Physics. Received 8 May 2010; accepted 15 July 2010; published online 31 August 2010. We are grateful to David Gross for pointing us to the relevant literature for the discrete phase space construction. We also thank Lukasz Fidkowski and John Preskill for interesting discussions. P.M. and S.W. are supported by the NSF under Grant No. PHY-0803371.Attached Files
Published - Mandayam2010p11474J_Math_Phys.pdf
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Additional details
- Eprint ID
- 20265
- Resolver ID
- CaltechAUTHORS:20101004-082513387
- PHY-0803371
- NSF
- Created
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2010-11-16Created from EPrint's datestamp field
- Updated
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2021-11-08Created from EPrint's last_modified field