Pauli matrices

In and, the Pauli matrices are a set of three $2 × 2$   which are  and. Usually indicated by the letter  ($σ$), they are occasionally denoted by  ($τ$) when used in connection with  symmetries. They are
 * $$\begin{align}

\sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\     1&0    \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\     0&-1    \end{pmatrix} \,. \end{align}$$

These matrices are named after the physicist. In, they occur in the which takes into account the interaction of the  of a particle with an external.

Each Pauli matrix is, and together with the identity matrix $I$ (sometimes considered as the zeroth Pauli matrix $σ_{0}$), the Pauli matrices form a for the real  of $2 × 2$ Hermitian matrices. This means that any $2 × 2$ can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

Hermitian operators represent s in quantum mechanics, so the Pauli matrices span the space of observables of the $2$-dimensional complex. In the context of Pauli's work, $σ_{k}$ represents the observable corresponding to spin along the $k$th coordinate axis in three-dimensional  $ℝ^{3}$.

The Pauli matrices (after multiplication by $i$ to make them ) also generate transformations in the sense of s: the matrices $iσ_{1}, iσ_{2}, iσ_{3}$ form a basis for the real Lie algebra $$\mathfrak{su}(2)$$, which  to the special unitary group. The generated by the three matrices $σ_{1}, σ_{2}, σ_{3}$ is  to the  of $ℝ^{3}$, and the algebra generated by $iσ_{1}, iσ_{2}, iσ_{3}$ is isomorphic to the.

Algebraic properties
All three of the Pauli matrices can be compacted into a single expression:



\sigma_a = \begin{pmatrix} \delta_{a3}               &  \delta_{a1} - i\delta_{a2}\\ \delta_{a1} + i\delta_{a2} & -\delta_{a3} \end{pmatrix} $$

where $i = √−1$ is the, and $δ_{ab}$ is the , which equals +1 if $a = b$ and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of $a = 1, 2, 3$, in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are :
 * $$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I$$

where $I$ is the.

The s and s of the Pauli matrices are:
 * $$\begin{align}

\det \sigma_i &= -1, \\ \operatorname{tr} \sigma_i &= 0. \end{align}$$

From which, we can deduce that the of each $σ_{i}$ are $±1$.

With the inclusion of the identity matrix, $I$ (sometimes denoted $σ_{0}$), the Pauli matrices form an orthogonal basis (in the sense of ) of the real of $2 × 2$ complex Hermitian matrices, $$\mathcal{H}_2(\mathbb{C})$$, and the complex Hilbert space of all $2 × 2$ matrices, $$\mathcal{M}_{2,2}(\mathbb{C})$$.

Eigenvectors and eigenvalues
Each of the Pauli matrices has two, $+1$ and $−1$. Using a convention in which prior to normalization, the 1 is placed into the top and bottom positions of the + and – wavefunctions respectively, the corresponding  are:
 * $$\begin{align}

\psi_{x+} &= \frac{1}\sqrt{2} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, & \psi_{x-} &= \frac{1}\sqrt{2} \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \\ \psi_{y+} &= \frac{1}\sqrt{2} \begin{pmatrix} 1 \\ i \end{pmatrix}, & \psi_{y-} &= \frac{1}\sqrt{2} \begin{pmatrix} i \\ 1 \end{pmatrix}, \\ \psi_{z+} &=                 \begin{pmatrix} 1 \\ 0 \end{pmatrix}, & \psi_{z-} &=                 \begin{pmatrix} 0 \\  1 \end{pmatrix}. \end{align}$$

An advantage of using this convention is that the + and – wavefunctions may be related to one another, using the Pauli matrices themselves, by $${{\psi }_{x+}}=i{{\sigma }_{y}}{{\psi }_{x-}}$$, $${{\psi }_{y+}}={{\sigma }_{x}}{{\psi }_{y-}}$$ and $${{\psi }_{z+}}={{\sigma }_{x}}{{\psi }_{z-}}$$.

Pauli vector
The Pauli vector is defined by
 * $$\vec{\sigma} = \sigma_1 \hat{x} + \sigma_2 \hat{y} + \sigma_3 \hat{z} $$

and provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows,
 * $$\begin{align}

\vec{a} \cdot \vec{\sigma} &= \left(a_i \hat{x}_i\right) \cdot \left(\sigma_j \hat{x}_j\right) = a_i \sigma_j \hat{x}_i \cdot \hat{x}_j \\ &= a_i \sigma_j \delta_{ij} = a_i \sigma_i = \begin{pmatrix} a_3 & a_1 - ia_2 \\ a_1 + ia_2 & -a_3 \end{pmatrix} \end{align}$$

using the. Further,
 * $$\det \vec{a} \cdot \vec{\sigma} = -\vec{a} \cdot \vec{a} = -\left|\vec{a}\right|^2,$$

its eigenvalues being $$\pm |\vec{a}| $$, and moreover (see completeness, below)
 * $$\frac{1}{2} \operatorname{tr} \left(\left(\vec{a} \cdot \vec{\sigma}\right) \vec{\sigma}\right) = \vec{a} ~.$$

Its (unnormalized) eigenvectors are

\psi_+ = \begin{pmatrix} a_3 + |\vec{a}| \\ a_1 + ia_2 \end{pmatrix}; \qquad \psi_- = \begin{pmatrix} ia_2 - a_1 \\ a_3 + |\vec{a}| \end{pmatrix}. $$

Commutation relations
The Pauli matrices obey the following relations:
 * $$[\sigma_a, \sigma_b] = 2 i \varepsilon_{a b c}\,\sigma_c \,, $$

and relations:
 * $$\{\sigma_a, \sigma_b\} = 2 \delta_{a b}\,I.$$

where the $ε_{abc}$ is the, Einstein summation notation is used, $δ_{ab}$ is the , and $I$ is the $2 × 2$ identity matrix.

For example,
 * $$\begin{align}

\text{commutators} & & \text{anticommutators} & \, \\ \left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \, & \left\{\sigma_1, \sigma_1\right\} &= 2I\, \\ \left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \, & \left\{\sigma_1, \sigma_2\right\} &= 0\, \\ \left[\sigma_3, \sigma_1\right] &= 2i\sigma_2 \, & \left\{\sigma_1, \sigma_3\right\} &= 0\, \\ \left[\sigma_1, \sigma_1\right] &= 0 \, & \left\{\sigma_2, \sigma_2\right\} &= 2I\,. \\ \end{align}$$

Relation to dot and cross product
Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives
 * $$ \begin{align}

\left[\sigma_a, \sigma_b\right] + \{\sigma_a, \sigma_b\} &= (\sigma_a \sigma_b - \sigma_b \sigma_a ) + (\sigma_a \sigma_b + \sigma_b \sigma_a) \\ 2i\varepsilon_{a b c}\,\sigma_c + 2 \delta_{a b}I &= 2\sigma_a \sigma_b \end{align}$$ so that,

each side of the equation with components of two $3$-vectors $a_{p}$ and $b_{q}$ (which commute with the Pauli matrices, i.e., $a_{p}σ_{q} = σ_{q}a_{p})$ for each matrix $σ_{q}$ and vector component $a_{p}$ (and likewise with $b_{q}$), and relabeling indices $a, b, c → p, q, r$, to prevent notational conflicts, yields
 * $$ \begin{align}

a_p b_q \sigma_p \sigma_q & = a_p b_q \left(i\varepsilon_{pqr}\,\sigma_r + \delta_{pq}I\right) \\ a_p \sigma_p b_q \sigma_q & = i\varepsilon_{pqr}\,a_p b_q \sigma_r + a_p b_q \delta_{pq}I ~. \end{align}$$

Finally, translating the index notation for the and  results in

If $$ i $$ is identified with the pseudoscalar $$ \sigma_x \sigma_y \sigma_z $$ then the right hand side becomes $$ a \cdot b + a \wedge b $$ which is also the definition for the product of two vectors in geometric algebra.

Some trace relations
Following traces can be derived using the commutation and anticommutation relations.


 * $$\begin{align}

\operatorname{tr}\left(\sigma_a\right) &= 0 \\ \operatorname{tr}\left(\sigma_a \sigma_b\right) &= 2\delta_{ab} \\ \operatorname{tr}\left(\sigma_a \sigma_b \sigma_c\right) &= 2i\varepsilon_{abc} \\ \operatorname{tr}\left(\sigma_a \sigma_b \sigma_c \sigma_d\right) &= 2\left(\delta_{ab}\delta_{cd} - \delta_{ac}\delta_{bd} + \delta_{ad}\delta_{bc}\right) \end{align}$$

Exponential of a Pauli vector
For
 * $$\vec{a} = a\hat{n}, \quad |\hat{n}| = 1,$$

one has, for even powers, $$2p, \ \ p = 0, 1, 2, 3, \ldots$$
 * $$(\hat{n} \cdot \vec{\sigma})^{2p} = I $$

which can be shown first for the $$p = 1$$ case using the anticommutation relations. For convenience, the case $$p = 0$$ is taken to be $$I$$ by convention.

For odd powers, $$2q + 1,\ \ q = 0, 1, 2, 3, \ldots$$
 * $$\left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} \, .$$

, and using the ,
 * $$\begin{align}

e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} &= \sum_{k=0}^\infty{\frac{i^k \left[a \left(\hat{n} \cdot \vec{\sigma}\right)\right]^k}{k!}} \\ &= \sum_{p=0}^\infty{\frac{(-1)^p (a\hat{n}\cdot \vec{\sigma})^{2p}}{(2p)!}} + i\sum_{q=0}^\infty{\frac{(-1)^q (a\hat{n}\cdot \vec{\sigma})^{2q + 1}}{(2q + 1)!}} \\ &= I\sum_{p=0}^\infty{\frac{(-1)^p a^{2p}}{(2p)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{q=0}^\infty{\frac{(-1)^q a^{2q+1}}{(2q + 1)!}}\\ \end{align}$$.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

which is to, extended to.

Note that
 * $$\det[i a(\hat{n} \cdot \vec{\sigma})] = a^2$$,

while the determinant of the exponential itself is just $1$, which makes it the generic group element of .

A more abstract version of formula $$ for a general $2 × 2$ matrix can be found in the article on. A general version of $$ for an analytic (at a and &minus;a) function is provided by application of  ,
 * $$f(a(\hat{n} \cdot \vec{\sigma})) = I\frac{f(a) + f(-a)}{2} + \hat{n} \cdot \vec{\sigma} \frac{f(a) - f(-a)}{2} ~.$$

The group composition law of $SU(2)$
A straightforward application of formula $$  provides a parameterization of the composition law of  the group $SU(2)$. One may directly solve for $$ in
 * $$\begin{align}

e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} e^{i b\left(\hat{m} \cdot \vec{\sigma}\right)} &= I(\cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b) + i(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b  )\cdot  \vec{\sigma } \\ &= I\cos{c} + i (\hat{k} \cdot \vec{\sigma}) \sin{c} \\ &= e^{i c \left(\hat{k} \cdot \vec{\sigma}\right)}, \end{align}$$

which specifies the generic group multiplication, where, manifestly,
 * $$\cos c = \cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b~,$$

the. Given $$, then,
 * $$\hat{k} = \frac{1}{\sin c}\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} \sin a \sin b\right) ~.$$

Consequently, the composite rotation parameters in this group element (a closed form of the respective in this case) simply amount to
 * $$e^{ic \hat{k}\cdot \vec{\sigma}}= \exp \left( i\frac{c}{\sin c}  (\hat{n} \sin a \cos b + \hat{m}  \sin b \cos a - \hat{n}\times\hat{m}  ~ \sin a \sin b  )\cdot \vec{\sigma}\right ) ~.$$

(Of course, when $c$ is parallel to $c$, so is $n&#770;$, and $c = a + b$.)

Adjoint action
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle $m&#770;$,

e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma}~ e^{-i a\left(\hat{n} \cdot \vec{\sigma}\right)} = \vec{\sigma}  \cos (2a) + \hat{n} \times \vec{\sigma} ~\sin (2a)+ \hat{n} ~ \hat{n} \cdot \vec{\sigma} ~ (1 - \cos (2a))~ .$$

Completeness relation
An alternative notation that is commonly used for the Pauli matrices is to write the vector index $k&#770;$ in the superscript, and the matrix indices as subscripts, so that the element in row $a$ and column $i$ of the $α$-th Pauli matrix is $σ ^{i}_{αβ}$.

In this notation, the completeness relation for the Pauli matrices can be written
 * $$\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{i=1}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}.$$


 * Proof: The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the complex Hilbert space of all 2 &times; 2 matrices means that we can express any matrix M as
 * $$M = c I + \sum_i a_i \sigma^i$$
 * where c is a complex number, and a is a 3-component complex vector. It is straightforward to show, using the properties listed above, that
 * $$\operatorname{tr}\, \sigma^i\sigma^j = 2\delta_{ij}$$
 * where "tr" denotes the, and hence that
 * $$\begin{align}

c &= \frac{1}{2}\operatorname{tr}\,M,\ \ a_i = \frac{1}{2}\operatorname{tr}\,\sigma^i M ~. \\[3pt] \therefore 2M &= I \operatorname{tr}\, M + \sum_i \sigma^i \operatorname{tr}\, \sigma^i M ~, \end{align}$$
 * which can be rewritten in terms of matrix indices as
 * $$2M_{\alpha\beta} = \delta_{\alpha\beta} M_{\gamma\gamma} + \sum_i \sigma^i_{\alpha\beta} \sigma^i_{\gamma\delta} M_{\delta\gamma}~,$$
 * where over the repeated indices γ and δ.  Since this is true for any choice of the matrix M, the completeness relation follows as stated above.

As noted above, it is common to denote the 2 &times; 2 unit matrix by σ0, so σ0αβ = δαβ. The completeness relation can alternatively be expressed as
 * $$\sum_{i=0}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma}~.$$

The fact that any 2 &times; 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the representation of 2 &times; 2 s' density matrix, (2 &times; 2 positive semidefinite matrices with unit trace.  This can be seen by  first  expressing an arbitrary  Hermitian matrix as a real linear combination of ${ σ_{0}, σ_{1}, σ_{2}, σ_{3}} |undefined$ as above, and then imposing the positive-semidefinite and  $1$ conditions.

For a pure state,  in polar coordinates,  $$\vec a = (\sin \theta \cos \phi, \sin \theta \sin \phi,  \cos \theta )$$,   the idempotent density matrix
 * $$\frac{1}{2}(1\!\! 1 + \vec a \cdot \vec \sigma) =

\begin{pmatrix} \cos^2 \theta/2            &  \sin \theta/2 ~ \cos\theta/2 ~e^{-i\phi} \\ \sin \theta/2 ~ \cos\theta/2 ~e^{i\phi} & \sin^2 \theta/2 \end{pmatrix} $$ acts on the state eigenvector $$ (\cos \theta/2, e^{i\phi} \sin \theta/2) $$ with eigenvalue 1, so like a for it.

Relation with the permutation operator
Let $P_{ij}$ be the (also known as a permutation) between two spins $σ_{i}$ and $σ_{j}$ living in the  space $ℂ^{2} &otimes; ℂ^{2}$,
 * $$P_{ij}|\sigma_i \sigma_j\rangle = |\sigma_j \sigma_i\rangle \,.$$

This operator can also be written more explicitly as ,
 * $$P_{ij} = \frac{1}{2}(\vec{\sigma}_i\cdot\vec{\sigma}_j + 1)\,.$$

Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

SU(2)
The group is the  of  $2 × 2$ matrices with unit determinant; its  is the set of all $2 × 2$ anti-Hermitian matrices  with trace 0. Direct calculation, as above, shows that the  $$\mathfrak{su}_2$$ is the 3-dimensional real algebra  by the set ${iσ_{j}}$. In compact notation,
 * $$ \mathfrak{su}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2, i \sigma_3 \}.$$

As a result, each $iσ_{j}$ can be seen as an of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper, as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is $su(2)$ $β$, so that
 * $$ \mathfrak{su}(2) = \operatorname{span} \left\{\frac{i \sigma_1}{2}, \frac{i \sigma_2}{2}, \frac{i \sigma_3}{2} \right\}.$$

As SU(2) is a compact group, its is trivial.

SO(3)
The Lie algebra $λ =$ is to the Lie algebra $su(2)$, which corresponds to the Lie group, the  of s in three-dimensional space. In other words, one can say that the $so(3)$ are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though $iσ_{j}$ and $su(2)$ are isomorphic as Lie algebras, $so(3)$ and $SU(2)$ are not isomorphic as Lie groups. $SO(3)$ is actually a of $SU(2)$, meaning that there is a two-to-one group homomorphism from $SO(3)$ to $SU(2)$, see.

Quaternions
The real linear span of $SO(3)$ is isomorphic to the real algebra of ${ I, iσ_{1}, iσ_{2}, iσ_{3}} |undefined$. The isomorphism from $ℍ$ to this set is given by the following map (notice the reversed signs for the Pauli matrices):

1 \mapsto I, \quad \mathbf{i} \mapsto - \sigma_2\sigma_3 = - i \sigma_1, \quad \mathbf{j} \mapsto - \sigma_3\sigma_1 = - i \sigma_2, \quad \mathbf{k} \mapsto - \sigma_1\sigma_2 = - i \sigma_3. $$

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,

1 \mapsto I, \quad \mathbf{i} \mapsto i \sigma_3, \quad \mathbf{j} \mapsto i \sigma_2, \quad \mathbf{k} \mapsto i \sigma_1. $$

As the set of s U ⊂ ℍ forms a group isomorphic to $ℍ$, U gives yet another way of describing $SU(2)$. The two-to-one homomorphism from $SU(2)$ to $SU(2)$ may be given in terms of the Pauli matrices in this formulation.

Quaternions form a —every non-zero element has an inverse—whereas Pauli matrices do not.

Classical mechanics
In, Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix P corresponding to the position $$\vec{x}$$ of a point in space is defined in terms of the above Pauli vector matrix,
 * $$P = \vec{x} \cdot \vec{\sigma} = x\sigma_x + y\sigma_y + z\sigma_z ~.$$

Consequently, the transformation matrix $$Q_\theta$$ for rotations about the x-axis through an angle θ  may be written in terms of Pauli matrices and the unit matrix as
 * $$Q_\theta = 1\!\!1 \cos\frac{\theta}{2} + i\sigma_x \sin\frac{\theta}{2} ~.$$

Similar expressions follow for general Pauli vector rotations as detailed above.

Quantum mechanics
In, each Pauli matrix is related to an that corresponds to an  describing the  of a  particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, $SO(3)$ are the generators of a (spin representation) of the  acting on  particles with spin ½. The of the particles are represented as two-component. In the same way, the Pauli matrices are related to the.

An interesting property of spin ½ particles is that they must be rotated by an angle of 4$i$ in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the $1⁄2$ 2, they are actually represented by  vectors in the two dimensional complex.

For a spin ½ particle, the spin operator is given by $iσ_{j}$, the of. By taking s of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting s for higher spin systems in three spatial dimensions,  for arbitrarily large j,  can be calculated using this  and. They can be found in. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.

Also useful in the of multiparticle systems, the general  $J = ħ⁄2&sigma;$ is defined to consist of all $π$-fold  products of Pauli matrices.

Relativistic quantum mechanics
In, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as
 * $$\mathsf{\Sigma}_i = \begin{pmatrix} \mathsf{\sigma}_i & 0 \\ 0 & \mathsf{\sigma}_i \end{pmatrix}.$$

It follows from this definition that $$\mathsf{\Sigma}_i$$ matrices have the same algebraic properties as $$\mathsf{\sigma}_i$$ matrices.

However, is not a three-vector, but a second order. Hence $$\mathsf{\Sigma}_i$$ needs to be replaced by $$\Sigma_{\mu\nu}$$, the generator of. By the antisymmetry of angular momentum, the $$\Sigma_{\mu\nu}$$ are also antisymmetric. Hence there are only six independent matrices.

The first three are the $$ \Sigma_{jk}\equiv \epsilon_{ijk}\mathsf{\Sigma}_i .$$ The remaining three, $$-i\Sigma_{0i}\equiv\mathsf{\alpha}_i$$, where the are defined as


 * $$\mathsf{\alpha}_i = \begin{pmatrix} 0 & \mathsf{\sigma}_i\\ \mathsf{\sigma}_i & 0\end{pmatrix}.$$

The relativistic spin matrices $$\Sigma_{\mu\nu}$$ are written in compact form in terms of commutator of  as
 * $$\Sigma_{\mu\nu} = \frac{i}{2}\left[\gamma_\mu, \gamma_\nu\right]$$.

Quantum information
In, single- s are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the Z–Y decomposition of a single-qubit gate. Choosing a different Cartan pair gives a similar X–Y decomposition of a single-qubit gate.