Qumode-Qubit teleportation through a homodyne measurement

Here we go through a quick example to show interesting dynamics that go beyond simple qubits.

A quarter-period Jaynes–Cummings interaction maps $|Z_1F_0\rangle$ to $(|Z_1F_0\rangle-i|Z_2F_1\rangle)/\sqrt{2}$. Measuring the $\hbar=2$ quadrature $\hat{q}=a+a^\dagger$ with result $q$ leaves the qubit proportional to $|Z_1\rangle-iq|Z_2\rangle$. An $X$ rotation removes this known back-action. This short measurement-and-feed-forward primitive is related to transduction and teleportation.

using QuantumSavoryregister = Register([Qubit(), Qumode()])initialize!(register[1:2], Z1 ⊗ F0)interaction = Pm ⊗ Create + Pp ⊗ Destroy# some common operators also have Unicode definitions, for instance:# interaction = σ₋ ⊗ Create + σ₊ ⊗ Destroyquarter_period = exp(-im * (π / 4) * interaction)apply!(register[1:2], quarter_period)q = project_traceout!(register[2], HomodyneMeasurement(0.0))if !isapprox(q, 0; atol = 1e-12)    correction = exp(im * atan(q) * X)    apply!(register[1], correction)endfidelity = observable(register[1], SProjector(Z1))

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