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Locking with three mutually unbiased bases

Manuel A. Ballester, Stephanie Wehner

Published Jun 29, 2006• 0 citations
OpenAlex

Abstract

We show that using more mutually unbiased bases does not always lead to a better locking effect. In particular, we prove that the optimal bound for the accessible information using three specific mutually unbiased bases is n/2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect, and we need to look for additional properties. We also prove an entropic uncertainty relation that may be of independent interest.

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We show that using more mutually unbiased bases does not always lead to a better locking effect. In particular, we prove that the optimal bound for the accessible information using three specific mutually unbiased bases is n/2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect, and we need to look for additional properties. We also prove an entropic uncertainty relation that may be of independent interest.
We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] for two such measurements can in fact be tight for up to d\sqrt{d}d​ measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to d\sqrt{d}d​ specific mutually unbiased bases is logd∕2\mathrm{log}\phantom{\rule{0.2em}{0ex}}d∕2logd∕2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional properties.
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions (d=p2d=p^{2}d=p2) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of p+1p+1p+1 MUBs that saturate both a Shannon (H1H_{1}H1​) and a collision (H2H_{2}H2​) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.