## Abstract

Construction of Mutually Unbiased Bases (MUBs) is a very challenging combinatorial problem in quantum information theory with several long standing open questions in this domain. With certain relaxations, the object Approximate Mutually Unbiased Bases (AMUBs) is defined in this context. In this paper we provide a method to construct upto \((\sqrt {d} + 1)\) many AMUBs in dimension *d* = *q*^{2}, where *q* is a positive integer. The result is particularly important when *q* ≡ 0 mod 4, as we obtain Approximate Real MUBs (ARMUBs) assuming the cases where a Hadamard matrix of order *q* exists. In this construction, we also characterize the inner product values between the elements of two different bases. In particular, when *d* is of the form (4*x*)^{2} where *x* is a prime, we obtain \((\frac {\sqrt {d}}{4}+1)\) ARMUBs such that for any two vectors *v*_{1},*v*_{2} belonging to different bases, \(|{\left \langle \left . v_{1} \right | v_{2} \right \rangle }| \leq \frac {4}{\sqrt {d}}\).

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## Acknowledgments

We sincerely thank the anonymous reviewer for extremely detailed comments that improved the editorial as well as the technical quality of this paper. The second author acknowledges the project (2016–2021) “Cryptography & Cryptanalysis: How far can we bridge the gap between Classical and Quantum Paradigm”, awarded by the Scientific Research Council of the Department of Atomic Energy (DAE-SRC), the Board of Research in Nuclear Sciences (BRNS).

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Kumar, A., Maitra, S. & Mukherjee, C.S. On approximate real mutually unbiased bases in square dimension.
*Cryptogr. Commun.* (2021). https://doi.org/10.1007/s12095-020-00468-6

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### Keywords

- (Real Approximate) Mutually Unbiased Bases
- Hadamard matrices
- Quantum information theory
- Qudits
- Square dimension

### Mathematics Subject Classification (2010)

- 81P68