^{1,a)}, Stephanie Wehner

^{1,b)}and Niranjan Balachandran

^{2,c)}

### 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 mutually unbiased bases (MUBs) in dimension , 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 , 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.

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.

I. INTRODUCTION

A. Entropic uncertainty relations

B. Mutually unbiased bases

C. Min-entropic uncertainty relations

II. SYMMETRIC MUBS

A. Clifford algebra

B. Construction

C. Examples

III. UNCERTAINTY RELATIONS

A. Entropic quantities

B. Min-entropy and symmetry

1. Symmetries

2. Discrete Wigner function

C. A simple bound

IV. CONCLUSIONS AND OPEN QUESTIONS

### Key Topics

- Fourier transforms
- 14.0
- Uncertainty principle
- 14.0
- Entropy
- 12.0
- Eigenvalues
- 11.0
- Shannon entropy
- 11.0

## Figures

Average min-entropy for different sets of MUBs in dimension . The crosses denote numerically computed minima of the average min-entropy for MUBs obtained using our construction. The bound in (37) is clearly tight for both and MUBs. The second analytical bound in (41) is stronger than (37) for bases. The circle denotes the average min-entropy for the invariant states given in (43). For four MUBs in , the minimum of the average min-entropy is indeed attained by states invariant under .

Average min-entropy for different sets of MUBs in dimension . The crosses denote numerically computed minima of the average min-entropy for MUBs obtained using our construction. The bound in (37) is clearly tight for both and MUBs. The second analytical bound in (41) is stronger than (37) for bases. The circle denotes the average min-entropy for the invariant states given in (43). For four MUBs in , the minimum of the average min-entropy is indeed attained by states invariant under .

Average min-entropy for different sets of MUBs in dimension . The bound in (37) is close to tight for MUBs in dimension . The second analytical bound in (41) is stronger than (37) for bases. The circle denotes the average min-entropy for invariant states constructed in dimension , similar to the states described in (43). For six MUBS in , the minimum of the average min-entropy is nearly attained by states invariant under .

Average min-entropy for different sets of MUBs in dimension . The bound in (37) is close to tight for MUBs in dimension . The second analytical bound in (41) is stronger than (37) for bases. The circle denotes the average min-entropy for invariant states constructed in dimension , similar to the states described in (43). For six MUBS in , the minimum of the average min-entropy is nearly attained by states invariant under .

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