相互に偏りのない基地

3つの基地

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

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

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

C ²における相互に偏りのない基底の最も単純な例を示します。上記の基底は、パウリ スピン行列の固有ベクトルで構成されています。${\displaystyle \sigma _{z},\sigma _{x}}$そして彼らの製品${\displaystyle \sigma _{x}\sigma _{z}}$、 それぞれ。

d  = 4の場合、各基底がM _j、0 ≤ j ≤ 4で表されるd + 1 = 5 個の相互に無偏な基底 の例が次のように与えられます。^{[ 12 ]}    

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

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

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

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

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

させて${\displaystyle {\hat {X}}}$そして${\displaystyle {\hat {Z}}}$ヒルベルト空間C ^dにおける 2 つのユニタリ作用素は、

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

ある位相因子に対して${\displaystyle \omega }$。 もし${\displaystyle \omega }$は、例えば、1の原始根である。${\displaystyle \omega \equiv e^{\frac {2\pi i}{d}}}$すると、​${\displaystyle {\hat {X}}}$そして${\displaystyle {\hat {Z}}}$互いに公平である。

の固有基底を選択することによって${\displaystyle {\hat {Z}}}$標準基底として、フーリエ行列を用いてそれに対して偏りのない別の基底を生成することができる。フーリエ行列の要素は次のように与えられる。

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

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:^[22]

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

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

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

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.^[22] 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.^[11]

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

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