Quantum Computing 1.3 -- Polarizers

A two-state quantum system is called a qubit. A qubit can be realized by a photon which has two different polarization states 0\ket{0^{\circ}} and 90\ket{90^{\circ}} which correspond to polarization along the vertical and horizontal axes, respectively. The ability to precisely and reliably manipulate qubits is key to the advent of large-scale quantum computing

A linearly polarized photon aligned with angle θ\theta can be split into its 0\ket{0^{\circ}} and 90\ket{90^{\circ}} components with amplitudes corresponding to the projection of the wave on the measurement axis. These states are orthogonal, so a pure 0\ket{0^{\circ}} photon has no 90\ket{90^{\circ}} component, and vice versa: θ=cos(θ)0+cos(90θ)90,θ[0,360]. \ket{\theta ^ \circ} = \cos(\theta) \ket{0^{\circ}} + \cos(90 ^ \circ -\theta) \ket{90^{\circ}}, \quad \theta \in [0^\circ,360^\circ].

One way to manipulate a photon qubit is with polarization filters. Each filter measures and absorbs all photons in the α+90\ket{\alpha+90 ^ \circ} state, while all others pass through. Consider a series of three linear polarization filters, each rotated by an angle of α=30,60,\alpha = 30 ^ \circ, 60 ^ \circ, and 90,90 ^ \circ, respectively, with respect to the 0\ket{0^{\circ}} polarization direction.

What is the probability PP that a 0\ket{0^{\circ}} photon passes all three polarizing filters and emerges as a 90\ket{90^{\circ}} photon?

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