Mutually unbiased bases

The three bases

${\displaystyle M_{0}=\left\{|0\rangle ,|1\rangle \right\}}$

${\displaystyle M_{1}=\left\{{\frac {|0\rangle +|1\rangle }{\sqrt {2}}},{\frac {|0\rangle -|1\rangle }{\sqrt {2}}}\right\}}$

${\displaystyle M_{2}=\left\{{\frac {|0\rangle +i|1\rangle }{\sqrt {2}}},{\frac {|0\rangle -i|1\rangle }{\sqrt {2}}}\right\}}$

provide the simplest example of mutually unbiased bases in C². The above bases are composed of the eigenvectors of the Pauli spin matrices ${\displaystyle \sigma _{z},\sigma _{x}}$ and their product ${\displaystyle \sigma _{x}\sigma _{z}}$, respectively.

For d = 4, an example of d + 1 = 5 mutually unbiased bases where each basis is denoted by M_j, 0 ≤ j ≤ 4, is given as follows:[12]

${\displaystyle M_{0}=\left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}}$

${\displaystyle M_{1}=\left\{{\frac {1}{2}}(1,1,1,1),{\frac {1}{2}}(1,1,-1,-1),{\frac {1}{2}}(1,-1,-1,1),{\frac {1}{2}}(1,-1,1,-1)\right\}}$

${\displaystyle M_{2}=\left\{{\frac {1}{2}}(1,-1,-i,-i),{\frac {1}{2}}(1,-1,i,i),{\frac {1}{2}}(1,1,i,-i),{\frac {1}{2}}(1,1,-i,i)\right\}}$

${\displaystyle M_{3}=\left\{{\frac {1}{2}}(1,-i,-i,-1),{\frac {1}{2}}(1,-i,i,1),{\frac {1}{2}}(1,i,i,-1),{\frac {1}{2}}(1,i,-i,1)\right\}}$

${\displaystyle M_{4}=\left\{{\frac {1}{2}}(1,-i,-1,-i),{\frac {1}{2}}(1,-i,1,i),{\frac {1}{2}}(1,i,-1,i),{\frac {1}{2}}(1,i,1,-i)\right\}}$

Let ${\displaystyle {\hat {X}}}$ and ${\displaystyle {\hat {Z}}}$ be two unitary operators in the Hilbert space C^d such that

${\displaystyle {\hat {Z}}{\hat {X}}=\omega {\hat {X}}{\hat {Z}}}$

for some phase factor ${\displaystyle \omega }$. If ${\displaystyle \omega }$ is a primitive root of unity, for example ${\displaystyle \omega \equiv e^{\frac {2\pi i}{d}}}$ then the eigenbases of ${\displaystyle {\hat {X}}}$ and ${\displaystyle {\hat {Z}}}$ are mutually unbiased.

By choosing the eigenbasis of ${\displaystyle {\hat {Z}}}$ to be the standard basis, we can generate another basis unbiased to it using a Fourier matrix. The elements of the Fourier matrix are given by

${\displaystyle F_{ab}=\omega ^{ab},0\leq a,b\leq N-1}$

Other bases which are unbiased to both the standard basis and the basis generated by the Fourier matrix can be generated using Weyl groups.[11] The dimension of the Hilbert space is important when generating sets of mutually unbiased bases using Weyl groups. When d is a prime number, then the usual d + 1 mutually unbiased bases can be generated using Weyl groups. When d is not a prime number, then it is possible that the maximal number of mutually unbiased bases which can be generated using this method is 3.

When d = p is prime, we define the unitary operators ${\displaystyle {\hat {X}}}$ and ${\displaystyle {\hat {Z}}}$ by

${\displaystyle {\hat {X}}=\sum _{k=0}^{d-1}|k+1\rangle \langle k|}$

${\displaystyle {\hat {Z}}=\sum _{k=0}^{d-1}\omega ^{k}|k\rangle \langle k|}$

where ${\displaystyle \{|k\rangle |0\leq k\leq d-1\}}$ is the standard basis and ${\displaystyle \omega =e^{\frac {2\pi i}{d}}}$ is a root of unity.

Then the eigenbases of the following d + 1 operators are mutually unbiased:[19]

${\displaystyle {\hat {X}},{\hat {Z}},{\hat {X}}{\hat {Z}},{\hat {X}}{\hat {Z}}^{2}...{\hat {X}}{\hat {Z}}^{d-1}.}$

For odd d, the t-th eigenvector of the operator ${\displaystyle {\hat {X}}{\hat {Z}}^{k}}$ is given explicitly by[13]

${\displaystyle |\psi _{t}^{k}\rangle ={\frac {1}{\sqrt {d}}}\sum _{j=0}^{d-1}\omega ^{-jt}\omega ^{kj(j-1)/2}|j\rangle .}$

When ${\displaystyle d=p^{r}}$ is a power of a prime, we make use of the finite field ${\displaystyle \mathbb {F} _{d}}$ to construct a maximal set of d + 1 mutually unbiased bases. We label the elements of the computational basis of C^d using the finite field: ${\displaystyle \{|a\rangle |a\in \mathbb {F} _{d}\}}$.

We define the operators ${\displaystyle {\hat {X_{a}}}}$ and ${\displaystyle {\hat {Z_{b}}}}$ in the following way

${\displaystyle {\hat {X_{a}}}=\sum _{c\in \mathbb {F} _{d}}|c+a\rangle \langle c|}$

${\displaystyle {\hat {Z_{b}}}=\sum _{c\in \mathbb {F} _{d}}\chi (bc)|c\rangle \langle c|}$

where

${\displaystyle \chi (\theta )=\exp \left[{\frac {2\pi i}{p}}\left(\theta +\theta ^{p}+\theta ^{p^{2}}+\cdots +\theta ^{p^{r-1}}\right)\right],}$

is an additive character over the field and the addition and multiplication in the kets and ${\displaystyle \chi (\cdot )}$ is that of ${\displaystyle \mathbb {F} _{d}}$.

Then we form d + 1 sets of commuting unitary operators:

${\displaystyle \{{\hat {Z_{s}}}|s\in \mathbb {F} _{d}\}}$ and ${\displaystyle \{{\hat {X_{s}}}{\hat {Z_{sr}}}|s\in \mathbb {F} _{d}\}}$ for each ${\displaystyle r\in \mathbb {F} _{d}}$

The joint eigenbases of the operators in one set are mutually unbiased to that of any other set.[19] We thus have d + 1 mutually unbiased bases.

Given that one basis in a Hilbert space is the standard basis, then all bases which are unbiased with respect to this basis can be represented by the columns of a complex Hadamard matrix multiplied by a normalization factor. For d = 3 these matrices would have the form

${\displaystyle U={\frac {1}{\sqrt {d}}}{\begin{bmatrix}1&1&1\\e^{i\phi _{10}}&e^{i\phi _{11}}&e^{i\phi _{12}}\\e^{i\phi _{20}}&e^{i\phi _{21}}&e^{i\phi _{22}}\end{bmatrix}}}$

The problem of finding a set of k+1 mutually unbiased bases therefore corresponds to finding k mutually unbiased complex Hadamard matrices.

An example of a one parameter family of Hadamard matrices in a 4-dimensional Hilbert space is

${\displaystyle H_{4}(\phi )={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&e^{i\phi }&-1&-e^{i\phi }\\1&-1&1&-1\\1&-e^{i\phi }&-1&e^{i\phi }\end{bmatrix}}}$