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 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 specific mutually unbiased bases is logd∕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.
It is known that there exists 2N+ 1 mutually unbiased bases for N qubits system. Between the different MUB construction algorithms of the three-qubit case, we focus on Wootters method with discrete phase space that leads naturally to a complete set of 2N+ 1 mutually unbiased bases for the state space. We construct discrete Wigner function using mutually unbiased bases from the discrete phase space for three-qubit system by explicitly calculating the Wigner functions for exemplary three-qubit pure states such as the GHZ state, the W state and the embedded Bell state. We also highlight some quasi-probability characteristics of these entangled states.