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 Incoming links: Involution exponential formula

From

A^{2} = I

, induction gives

A^{2 n} = I

and

A^{2 n + 1} = A

. The matrix exponential converges absolutely in finite dimension, so its even and odd terms may be separated:

e^{- i θ A} = [\sum_{n \geq 0} \frac{( - 1 ) ^{n} θ ^{2 n}}{( 2 n )!}] I - i [\sum_{n \geq 0} \frac{( - 1 ) ^{n} θ ^{2 n + 1}}{( 2 n + 1 )!}] A = cos θ I - i sin θ A .

This involution exponential formula requires no Hermiticity assumption on

A

. Hermiticity is needed if the exponential is additionally asserted to be unitary.

σ_{x}^{2} = σ_{y}^{2} = I

and

σ_{x} σ_{y} + σ_{y} σ_{x} = 0

, so

A = cos ϕ σ_{x} + sin ϕ σ_{y}

has

A^{2} = I

. Its entries are

A_{12} = cos ϕ - i sin ϕ = e^{- i ϕ}

and

A_{21} = e^{i ϕ}

e^{- i θ A} = (cos θ - i e^{i ϕ} sin θ - i e^{- i ϕ} sin θ cos θ) .

The drive phase selects the transverse rotation axis, while the quantum pulse area selects its rotation angle.