A set of \(n^2\) orthonormal \(n \times n\) matrices with respect to the inner product \(\langle A, B \rangle = \text{Tr}(A^\dagger B)/n\) where \(\text{Tr}(M)\) is the trace of the matrix \(M\) and \(M^\dagger\) is its conjugate transpose.
Example: Pauli Matrices
\(\left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}, \\; \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}, \\ \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}, \\ \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\right\}\)
Conjugate Transpose
\(A^\dagger = A^\text{H} = A^\dagger = \bar{A}^\textsf{T}\)
\(\begin{bmatrix} i & 2+i \\ 1 & 3-2i \\ \end{bmatrix}^\dagger = \begin{bmatrix} -i & 1 \\ 2-i & 3+2i \\ \end{bmatrix}\)
Trace
\(A = [a_{ij}]\)
\(\text{Tr}(A) = \sum_i a_{ii}\)
\(\text{Tr}(A \otimes B) = \text{Tr}(A) \text{Tr}(B)\)
\(\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)\)
\(\text{Tr}(cA) = c\text{Tr}(A)\)
\(\text{Tr}(A) = \sum_i \lambda_i\)
\(\text{Tr}(A) = c\text{Tr}(A)\)
Inner Product
\(A, B\): \(n \times n\) matrices.
\(\langle A, B \rangle = \text{Tr}(A^\dagger B) / n\)
Orthogonality
\(A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\qquad B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}\)
\(A^\dagger = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\)
\(A^\dagger B = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}=\begin{bmatrix} 0 & i \\ i & 0 \\ \end{bmatrix}\)
\(\textsf{tr}(A^\dagger B) = 0 + 0 = 0\)
\(d = 2\)
\(\textsf{tr}(A^\dagger B)/2 = 0/2 = 0\)
Normality
\(A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \qquad A^\dagger = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\)
\(A^\dagger A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}=\begin{bmatrix} -i^2 & 0 \\ 0 & -i^2 \\ \end{bmatrix}\)
\(\textsf{tr}(A^\dagger A) = -i^2 + -i^2 = 2\)
\(d = 2\)
\(\textsf{tr}(A^\dagger B)/2 = 2/2 = 1\)
Equivalence
Unitary matrix: \(U^\dagger U = I\).
\(\mathcal{U}(d)\): group of all \(d \times d\) unitary matrices.
\(\mathcal{E}, \mathcal{E}'\): \(d \times d\) Unitary Error Bases.
\(\mathcal{E} \equiv \mathcal{E}' \Leftrightarrow \exists A, B \in \mathcal{U}(d), f:\mathcal{E} \rightarrow \mathbb{C} : \mathcal{E}' = \{f(E)AEB \mid E \in \mathcal{E}\}\)
Pauli Basis is Unique
Up to equivalence, the Pauli basis is unique in dimension 2.
Let \(\mathcal{E} = \{E_1, E_2, E_3, E_4 \}\) be a \(2 \times 2\) unitary error basis.
Then \(E_1^\dagger \mathcal{E} = \{I, E_1^\dagger E_2, E_1^\dagger E_3, E_1^\dagger E_4 \}\) is also a unitary error basis.
Characterization
- \(U\)
- \(d \times d\) unitary matrix.
- \(E_i(U)\)
- \(\sqrt{d} (U_{i,nd+m})_{n,m=0,\ldots,d-1}\)
\(U = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\)
\E_0(U) = \sqrt{2} \begin{bmatrix} U_{0,0} & U_{0,1} \ U_{0,2} & U_{0,3} \ \end{bmatrix})
TODO: Fix above
Nice Unitary Error Bases
https://people.engr.tamu.edu/andreas-klappenecker/ueb/uebdef.html
\(G\): finite group.
\(|G| = n^2, n \in \mathbb{N}\).
Nice error basis
Image of \(D: G \rightarrow \mathcal{U}(n)\) where:
- \(D(1_G) = I_n\)
- \(\textsf{tr}(D(g)) = 0\) for all \(g \neq 1_G\).
- \(D(g)D(h) = c D(gh)\) where \(c \in \mathbb{C}\) and \(|c| = 1\).
Generators
\(X\): set of generators for \(G\).
\(H\): matrix group generated by \(D(X) = \{ D(x) | x \in X \}\).
\(Z(H) = \{ x | x \in H, \forall h \in H : x h = hx \}\).
Traversal: set of coset representatives of cosets of a subgroup.
Nice error basis is constructed as a traversal of \(Z(H)\) in \(H\).
Example
https://people.engr.tamu.edu/andreas-klappenecker/ueb/uebex.html
Klein Four-Group
Generators
\(x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
Group
\(x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(x^2 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I\)
\(y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(y^2 = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I\)
\(xy = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(xyx = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = y\)
\((xy)^2 = y^2 = I\)
\(xyy = x(y^2) = xI = x\)
\(yx = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = xy\)
\(G = \{ I, x, y, xy \}\)
\(G = \langle x, y \\; | \\; x^2, y^2, (xy)^2 \rangle\)
https://en.wikipedia.org/wiki/Klein_four-group
Centre
\(Z(G) = \{ z \in G \\; | \\; \forall g \in G : gz = zg \}\)
\(xy \neq yx\)
\(\Rightarrow Z(G) = \{ I, x, y, xy \}\)
Group of Order 8
Generators
\(x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
Group
\(x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(x^2 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I\)
\(y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
\(y^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = x^2\)
\(xy = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(xyx = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} = -y\)
\(xyxy = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I\)
\(xyxyx = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = -x\)
\(xyxyxy = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = -xy\)
\(xyxyxyx = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = y\)
\(yx = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = -xy\)
\(yxy = (-xy)y = -x(y^2) = -x\)
\(G = \{ I, x, y, xy, -I, -x, -y, -xy \}\)
\(G = \langle x, y \\; | \\; x^2, y^2, (xy)^4 \rangle\)
Table
\( \begin{array}{c|cccccccc} & I & x & y & xy & -I & -x & -y & -xy \\ \hline I & I & x & y & xy & -I & -x & -y & -xy \\ x & x & I & xy & y & -x & -I & -xy & -y \\ y & y & -xy & I & -x & -y & xy & -I & x \\ xy & xy & -y & x & -I & -xy & y & -x & I \\ -I & -I & -x & -y & -xy & I & x & y & xy \\ -x & -x & -I & -xy & -y & x & I & xy & y \\ -y & -y & xy & -I & x & y & -xy & I & -x \\ -xy & -xy & y & -x & I & xy & -y & x & -I \\ \end{array} \)
Isomorphism
\(I \leftrightarrow e\)
\(xy \leftrightarrow b\)
\(\Rightarrow -I \leftrightarrow b^2\)
\(\Rightarrow -xy \leftrightarrow b^3\)
\(\Rightarrow b^4 \leftrightarrow e\)
\(x \leftrightarrow a\)
\(\Rightarrow a^2 = e\)
\(\Rightarrow ab \leftrightarrow x^2y = y\)
\(\Rightarrow y \leftrightarrow ab\)
\(ba\leftrightarrow xyx = -y\)
\(ab^3 \leftrightarrow x(-xy) = -x^2y = -y \leftrightarrow ba\)
\(\Rightarrow -y \leftrightarrow ab^3 = ba\)
\(\Rightarrow -x \leftrightarrow ab^2\)
\( \begin{array}{c|cccccccc} & e & b & b^2 & b^3 & a & ab & ab^2 & ab^3 \\ \hline e & e & b & b^2 & b^3 & a & ab & ab^2 & ab^3 \\ b & b & e & b^3 & b^2 & ab^3 & a & ab & ab^2 \\ b^2 & b^2 & b^3 & e & b & ab^2 & ab^3 & a & ab \\ b^3 & b^3 & b^2 & b & e & ab & ab^2 & ab^3 & a \\ a & a & ab & ab^2 & ab^3 & e & b & b^2 & b^3 \\ ab & ab & ab^2 & ab^3 & a & b^3 & e & b & b^2 \\ ab^2 & ab^2 & ab^3 & a & ab & b^2 & b^3 & e & b \\ ab^3 & ab^3 & a & ab & ab^2 & b & b^2 & b^3 & e \\ \end{array} \)
https://groupprops.subwiki.org/wiki/Dihedral_group:D8
Centre
\(Z(G) = \{ z \in G \\; | \\; \forall g \in G : gz = zg \}\)
\(ab \neq ba\)
\(\Rightarrow Z(G) = \{ e, b^2 \}\)
Traversal
https://proofwiki.org/wiki/Definition:Transversal_(Group_Theory)
A subset \(S \subseteq G\) where every (left/right) coset contains exactly one element of \(S\).
Nice Error Basis
Traversal of centre of \(H\).
Centre
\(\{ I, -I \}\)
Cosets
\(\{ I, -I \}\) \(\{ x, -x \}\) \(\{ y, -y \}\) \(\{ xy, -xy \}\)
Traversal
\(\{ I, x, y, xy \}\)