Abstract¶ Details, properties and applications of the discrete Weyl operators
Keywords: Weyl Operators Complete Basis. ¶ The discrete Weyl operators are a set of unitary operators that form a complete basis for the space of complex matrices. Within quantum information, they are used to define discrete Wigner functions Wootters (1987) Veitch et al. (2014)
Definition ¶ The discrete Weyl operators are defined as
W a , b = X a Z b , a , b ∈ { 0 , 1 , . . . , d − 1 } , W_{a,b} = X^a Z^b, \hspace{2mm} a,b \in \{0, 1, ~... ~,d-1\}, W a , b = X a Z b , a , b ∈ { 0 , 1 , ... , d − 1 } , where
X = ∑ j = 0 d − 1 ∣ j + 1 mod d ⟩ ⟨ j ∣ , Z = ∑ j = 0 d − 1 Ω j ∣ j ⟩ ⟨ j ∣ , Ω = e 2 π i d , X = \sum_{j=0}^{d-1} \vert j+1 ~ \textrm{mod} ~d \rangle \langle j \vert, ~ ~ Z= \sum_{j=0}^{d-1} \Omega^j \vert j \rangle \langle j \vert, ~ ~ \Omega = e^{\frac{2 \pi i}{d}}, X = j = 0 ∑ d − 1 ∣ j + 1 mod d ⟩ ⟨ j ∣ , Z = j = 0 ∑ d − 1 Ω j ∣ j ⟩ ⟨ j ∣ , Ω = e d 2 πi , such that the ( a , b ) (a,b) ( a , b )
W a , b = ∑ c Ω b c ∣ c + a mod d ⟩ ⟨ c ∣ . W_{a,b} = \sum_{c} \Omega^{bc} \ket{c + a ~ \textrm{mod} ~d}\bra{c}. W a , b = c ∑ Ω b c ∣ c + a mod d ⟩ ⟨ c ∣ . The X X X Z Z Z
W 0 , 0 = I , W 0 , 1 = σ z , W 1 , 0 = σ x , W 1 , 1 = − i σ y , W_{0,0} = \mathbb{I}, ~ ~ W_{0,1} = \sigma_z, ~ ~ W_{1,0} = \sigma_x, ~ ~ W_{1,1} = -i \sigma_y, W 0 , 0 = I , W 0 , 1 = σ z , W 1 , 0 = σ x , W 1 , 1 = − i σ y , where σ x , σ y , σ z \sigma_x, \sigma_y, \sigma_z σ x , σ y , σ z X , Y X, Y X , Y Z Z Z
Identities ¶ Here we list some identities for the Weyl operators.
The following identities are for how the discrete Weyl operators are transformed via transposition, adjoint, and multiplication:
( W a , b ) t = Ω − a b W − a , b , ( W a , b ) † = Ω a b W − a , − b , W a , b W c , d = Ω b c W a + c , b + d = Ω b c − a d W c , d W a , b . \begin{split}
(W_{a, b}) ^ {t} &= \Omega^{-ab}W_{-a, b}, \\
(W_{a, b}) ^ {\dagger} &= \Omega^{ab}W_{-a, -b}, \\
W_{a, b}W_{c, d} &= \Omega^{bc}W_{a + c, b + d} = \Omega^{bc - ad}W_{c, d}W_{a, b}. \\
\end{split} ( W a , b ) t ( W a , b ) † W a , b W c , d = Ω − ab W − a , b , = Ω ab W − a , − b , = Ω b c W a + c , b + d = Ω b c − a d W c , d W a , b . The following identities are for the trace and (Hilbert-Schmidt) inner product of Weyl operators:
tr [ W a , b ] = d δ a , 0 δ b , 0 , tr [ W a , b † W c , d ] = d δ a , c δ b , d , \begin{split}
\textrm{tr}\big[ W_{a,b} \big] &= d ~ \delta_{a,0} \delta_{b,0}, \\
\textrm{tr}\big[ W_{a,b}^\dagger W_{c,d} \big] &= d ~ \delta_{a,c} \delta_{b,d},
\end{split} tr [ W a , b ] tr [ W a , b † W c , d ] = d δ a , 0 δ b , 0 , = d δ a , c δ b , d , where δ a , b \delta_{a,b} δ a , b
A Complete Operator Basis ¶ The set
{ 1 d W a , b : a , b ∈ { 0 , 1 , . . . , d − 1 } } , \bigg\{ \frac{1}{\sqrt{d}} W_{a,b} : a,b \in \{0, 1, ~... ~,d-1\} \bigg\}, { d 1 W a , b : a , b ∈ { 0 , 1 , ... , d − 1 } } , form a complete orthonormal basis for the space of d × d d \times d d × d
Hence, if A A A d × d d \times d d × d p a , b ∈ C p_{a,b} \in \mathbb{C} p a , b ∈ C
A = 1 d ∑ a , b = 0 d − 1 p a , b W a , b . A = \frac{1}{\sqrt{d}}\sum_{a,b=0}^{d-1} p_{a,b} W_{a,b}. A = d 1 a , b = 0 ∑ d − 1 p a , b W a , b . Generating A Maximally Entangled Basis ¶ Let ∣ Φ 00 + ⟩ = d − 1 / 2 ∑ i = 0 d − 1 ∣ i i ⟩ ∈ H d \ket{\Phi^+_{00}} = d^{-1/2} \sum_{i=0}^{d-1} \ket{ii} \in \mathcal{H}^d ∣ Φ 00 + ⟩ = d − 1/2 ∑ i = 0 d − 1 ∣ ii ⟩ ∈ H d ∣ Φ 00 + ⟩ \ket{\Phi^+_{00}} ∣ Φ 00 + ⟩
The following set is an orthonormal basis for the space H d ⊗ H d \mathcal{H}^d \otimes \mathcal{H}^d H d ⊗ H d
{ ∣ Φ a , b + ⟩ = ( I ⊗ W a , b ) ∣ Φ 00 + ⟩ : a , b ∈ { 0 , 1 , . . . , d − 1 } } \big\{ \ket{\Phi_{a,b}^+} = (\mathbb{I} \otimes W_{a,b}) \ket{\Phi_{00}^+} ~ : ~ a,b \in \{0, 1, ~... ~,d-1\} \big\} { ∣ Φ a , b + ⟩ = ( I ⊗ W a , b ) ∣ Φ 00 + ⟩ : a , b ∈ { 0 , 1 , ... , d − 1 } } We note that this holds for any basis of unitaries that are orthogonal under the Hilbert Schmit inner product.
Wootters, W. K. (1987). A Wigner-function formulation of finite-state quantum mechanics. Annals of Physics , 176 (1), 1–21. 10.1016/0003-4916(87)90176-x Veitch, V., Hamed Mousavian, S. A., Gottesman, D., & Emerson, J. (2014). The resource theory of stabilizer quantum computation. New Journal of Physics , 16 (1), 013009. 10.1088/1367-2630/16/1/013009