In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.

Feb 3, 20051 citation

On the mathematical foundations of mutually unbiased bases

Koen Thas

Quantum Information ProcessingDec 21, 20173 citations

Constructions of Approximately Mutually Unbiased Bases

Igor E. Shparlinski, Arne Winterhof

Lecture notes in computer scienceJan 1, 200625 citations

Constructions on Real Approximate Mutually Unbiased Bases

Minghui Yang, Aixian Zhang, Jiejing Wen +1 more

Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in $\mathbb{C}^d$ for some dimension $d$ and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in $\mathbb{R}^{2d}$ from any complex AMUB in $\mathbb{C}^d$ . From this method we present many new series of real AMUB's with parameters better than previous results.Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in $\mathbb{C}^d$ for some dimension $d$ and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in $\mathbb{R}^{2d}$ from any complex AMUB in $\mathbb{C}^d$. From this method we present many new series of real AMUB's with parameters better than previous results.

arXiv:2110.06665Oct 13, 20211 citation

On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions $d \not\equiv 0 \bmod 4$ On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions $d \not\equiv 0 \bmod 4$

Ajeet Kumar, Rakesh Kumar, Subhamoy Maitra +1 more

It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension $d > 2$ which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct $> \lceil \sqrt{d} \rceil$ many ARMUBs for certain odd dimensions $d$ of the form $d = (4n-t)s$ , $t = 1, 2, 3$ , where $n$ is a natural number and $s$ is an odd prime power. Our method exploits any available $4n \times 4n$ real Hadamard matrix $H_{4n}$ (conjectured to be true) and uses this to construct an orthogonal matrix ${Y}_{4n-t}$ of size $(4n - t) \times (4n - t)$ , such that the absolute value of each entry varies a little from $\frac{1}{\sqrt{4n-t}}$ . In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be $\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2$ , for proper choices of parameters, the class of dimensions $d$ being infinitely large.It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension $d > 2$ which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct $> \lceil \sqrt{d} \rceil$ many ARMUBs for certain odd dimensions $d$ of the form $d = (4n-t)s$, $t = 1, 2, 3$, where $n$ is a natural number and $s$ is an odd prime power. Our method exploits any available $4n \times 4n$ real Hadamard matrix $H_{4n}$ (conjectured to be true) and uses this to construct an orthogonal matrix ${Y}_{4n-t}$ of size $(4n - t) \times (4n - t)$, such that the absolute value of each entry varies a little from $\frac{1}{\sqrt{4n-t}}$. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be $\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2$, for proper choices of parameters, the class of dimensions $d$ being infinitely large.

arXiv:2507.07028Jul 9, 20250 citations

Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in

\mathbb{C}^d

for some dimension

d

and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in

\mathbb{R}^{2d}

from any complex AMUB in

\mathbb{C}^d

. From this method we present many new series of real AMUB's with parameters better than previous results.

On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions

d \not\equiv 0 \bmod 4

It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension

d > 2

which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct

> \lceil \sqrt{d} \rceil

many ARMUBs for certain odd dimensions

d

of the form

d = (4n-t)s

t = 1, 2, 3

, where

n

is a natural number and

s

is an odd prime power. Our method exploits any available

4n \times 4n

real Hadamard matrix

H_{4n}

(conjectured to be true) and uses this to construct an orthogonal matrix

{Y}_{4n-t}

of size

(4n - t) \times (4n - t)

, such that the absolute value of each entry varies a little from

\frac{1}{\sqrt{4n-t}}

. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be

\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2

, for proper choices of parameters, the class of dimensions

d

being infinitely large.