User:Mschirm/Mutually Unbiased Bases

{under construction}

In Quantum Information theory, a pair of orthonormal bases $$ |e_a\rangle $$ and $$ |f_b\rangle $$ in a Hilbert space $$ \mathbb{C}^d $$ are said to be mutually unbiased if the inner product between any basis vector $$ |e_a\rangle $$ with any other basis vector $$ |f_b \rangle $$ is equal to the inverse of the dimension of the Hilbert space, or more notably


 * $$ |\langle e_a | f_b \rangle|^2 = \frac{1}{d},  0\ge a,b \ge d-1 $$

The bases are called mutually unbiased since any measurement made in one basis is completed unrelated to any measurement made in the other basis.

Overview
The notion of mutually unbiased bases were first introduced by Schwinger in 1960, although the first mention of term mutually unbiased bases is unknown. The first person to consider the use of mutually unbiased bases was Ivanovic, in the problem of state determination. Mutually unbiased bases have uses in quantum state tomography and various cryptographic protocols. In general, Mutually Unbiased Bases are useful for finding or hiding information. They permit things that are not normally permitted classically.

Mathematical Formulation
Given two orthonormal bases $$ |e_a\rangle $$ and $$ |f_b\rangle $$ in a vector space $$ \mathbb{C}^d $$, if


 * $$ |\langle e_a | f_b \rangle|^2 = \frac{1}{d},  0\ge a,b \ge d-1 $$

then $$ |e_a\rangle $$ and $$ |f_b\rangle $$ are said to be mutually unbiased. It is important to note that this result is independent of $$ a $$ and $$ b $$: It is true for the inner product between any basis vector $$ |e_a\rangle $$ and any basis vector $$ |f_b\rangle $$. The number of Mutually Unbiased Bases in a vector space $$\mathbb{C}^d$$ is denoted by $$ \mathfrak{M}(d) $$, where the value of $$ \mathfrak{M}(d) $$ depends upon whether or not $$ d $$ is an integer power of a prime number.

Mutually unbiased bases for vector spaces where $$ d $$ is an integer power of a prime number
If the dimension of a Hilbert space $$ d $$ is an integer power of a prime number, then it is possible to find $$ N+1$$ mutually unbiased bases within the Hilbert space.

Mutually unbiased bases for vectors spaces where $$ d $$ is not an integer power of a prime number
When the dimension of the vector space $$ N $$ is not an integer power of a prime number, then in general the following has been established. If


 * $$ d = p_1^{n_1} p_2^{n_2}...p_k^{n_k} $$

is the prime number decomposition of N, where


 * $$ p_1^{n_1} < p_2^{n_2}<...<p_k^{n_k} $$

then the number of mutually unbiased bases constructed satisfies


 * $$p_1^{n_1} \ge \mathfrak{M}(d) \ge d+1 $$

Methods for finding Mutually Unbiased Bases
Different methods for finding Mutually Unbiased Bases exist.

Weyl group method
For two unitary operators $$ X $$ and $$ Z $$ in a Hilbert space $$ \mathbb{C}^d $$ such that for some phase factor $$ q $$, if q is a primitive root of unity, for example then the eigenbases of $$ X $$ and $$ Z $$ are mutually unbiased bases.
 * $$ XZ = qZX $$
 * $$ q \equiv e^{\frac{2 \pi i}{d}} $$

By choosing the eigenbasis of Z to be the standard basis, then a mutually unbiased basis to the standard basis can be generated using the Fourier matrix The construction depends on if $$ d $$ is a power of an even or an odd prime number. The dimensions of the Hilbert space is also important when generating mutually unbiased bases using Weyl groups, as the number of mutually unbiased bases generated by the Weyl group is highly dependent on the dimension of the space. When $$ d $$ is a prime number, then $$ d+1 $$ mutually unbiased bases can be generated using the Weyl group. When $$ d $$ is not a prime number, then sometimes only 3 mutually unbiased bases can be generated in this manner. It is true that for any $$ d $$ the Fourier matrix exists, which implies that there always exists on basis which is mutually unbiased to the standard basis.
 * $$F_{ab} = q^{ab}, 0 \ge a,b \ge N-1 $$

Hadamard matrix method
Given that one basis in a Hilbert space is represented by the unit matrix, then all bases which are mutually unbiased with respect to the standard basis can be represented by a complex Hadamard matrix multiplied by a normalization factor. For $$ d=3 $$ these matrices would have the form 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} $$
 * $$ U = \frac{1}{\sqrt{d}} \begin{bmatrix}

Therefore, the problem is reduced to finding unequivalent Hadamard matrices which are mutually unbiased to each other. It is important to note that two Hadamard matrices are equivalent if, through permutations of rows and columns, and multiplication of rows and columns by arbitrary phase factors, they can be made equal.

An example of a one parameter family of Hadamard matrices in a 4 dimensional Hilbert space is 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} $$
 * $$ H_4(\phi) = \frac{1}{2} \begin{bmatrix}

The problem of finding $$ \mathfrak{M}(d) $$
The first value for $$ d $$ which is not an integer power of a prime number is the value $$ d=6 $$. This is also the smaller dimension for which the number of mutually unbiased bases is not known. The methods used to determine the number of mutually unbiased bases for when $$ d $$ is an integers power of a prime number cannot be used when $$ d $$ is not.