The concept of mutually unbiased bases is studied for $N$ pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For $N=2$ , the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.The concept of mutually unbiased bases is studied for $N$ pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For $N=2$, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.

arXiv:0802.0394Physical Review AAug 29, 200840 citations

Mutually unbiased bases, Heisenberg-Weyl orbits and the distance between them

Kate Blanchfield

AIP conference proceedingsJan 1, 20122 citations

Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

Rod Gow

We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$ , say, of multiplicative order $q^{2r-1}+1$ , whose $q^{2r-1}+1$ powers $D$ , \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$ . Thus the scaled matrices $q^rD$ , \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$ , we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$ , where $k\geq 3$ . Then $G$ has a real representation $R$ , say, of degree $2^{2^{k-1}}$ such that the elements $R(σ)$ , $σ\in G$ , define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$ , where $d= 2^{2^{k-1}}$ . In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$ . Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$, say, of multiplicative order $q^{2r-1}+1$, whose $q^{2r-1}+1$ powers $D$, \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$. Thus the scaled matrices $q^rD$, \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$, we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$, where $k\geq 3$. Then $G$ has a real representation $R$, say, of degree $2^{2^{k-1}}$ such that the elements $R(σ)$, $σ\in G$, define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$, where $d= 2^{2^{k-1}}$. In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.

arXiv:1410.4059Oct 15, 20141 citation

Mutually unbiased bases, Latin squares and orthogonal pairs(Group Theory and Related Topics)

Yasuo Watatani

Jul 1, 20070 citations

The concept of mutually unbiased bases is studied for

N

pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For

N=2

, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.

We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let

q

be a power of 2 and

r

a positive integer. Then we can find a

q^{2r}\times q^{2r}

real orthogonal matrix

D

, say, of multiplicative order

q^{2r-1}+1

, whose

q^{2r-1}+1

powers

D

, \dots,

D^{q^{2r-1}+1}=I

define

q^{2r-1}+1

mutually unbiased bases in

\mathbb{R}^{q^{2r}}

. Thus the scaled matrices

q^rD

, \dots,

q^rD^{q^{2r-1}}

are

q^{2r-1}

different Hadamard matrices. When we take

q=2

, we achieve the maximum number of real mutually unbiased bases in dimension

2^{2r}

using the elements of a cyclic group. We also prove the following. Let

G

be an arbitrary finite group of odd order

2k+1

, where

k\geq 3

. Then

G

has a real representation

R

, say, of degree

2^{2^{k-1}}

such that the elements

R(σ)

σ\in G

, define

|G|

mutually unbiased bases in

\mathbb{R}^{d}

, where

d= 2^{2^{k-1}}

. In addition, a group of order 5 defines five real mutually unbiased bases in

\mathbb{R}^{16}

and a group of order 3 defines three real mutually unbiased bases in

\mathbb{R}^{4}

. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.