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Part of a series on Phase retrieval
Mixed state ptychography is a method of accounting for decoherence effects in a ptychography experiment by considering the probe (or object) as a mixture of quantum states by a density matrix. Considering mixed states in the experiment can drastically improve the resolution of the reconstruction (1,2).
Connection to quantum tomography
Supplementary Material from: Thibault, P. & Menzel, A. Reconstructing state mixtures from diffraction measurements. Nature 494, 68–71 (2013).
Let us take a more quantum mechanical-ly motivated view of ptychography, assuming for the moment a fully coherent experiment. Essentially we have
- an incoming probe state \(\ket{P}\)
- scattered by the object operator \(\hat{O}\)
- producing an exit wave \(\ket{\psi} =
\hat{O}\ket{P}\)
- i.e. \(\hat{O}_j \equiv \ket{\psi_j}\bra{P}\) is a scattering matrix relating the probe to the exit wave
- to be detected in the reciprocal \(\mathbf{k}\) basis as \(I = \left| \braket{\mathbf{k}|\psi} \right|^2\).
This formulation is equivalent to the array picture given that, \(P(x) = \braket{\mathbf{x}|P}\) is the \(\mathbf{x}\) basis representation of \(\ket{P}\), and \(O(\mathbf{x}) = \bra{\mathbf{x}} \hat{O} \ket{\mathbf{x}}\) with \(\hat{O}\) diagonal in the \(\mathbf{x}\) basis.
Then from the last line,
\[ \begin{align} I_j(\mathbf{k}) &= \left|\bra{\mathbf{k}}\hat{O}_j\ket{P}\right|^2 \\ \\ &= \bra{P}\hat{O}_j^{\dagger}\ket{\mathbf{k}} \bra{\mathbf{k}}\hat{O}_j\ket{P} \\ \\ &= \sum_{\mathbf{k}^\prime} \braket{P|\mathbf{k}^\prime} \bra{\mathbf{k}^\prime}\hat{O}_j^{\dagger}\ket{\mathbf{k}} \bra{\mathbf{k}}\hat{O}_j\ket{P} \\ \\ &= \sum_{\mathbf{k}^\prime} \bra{\mathbf{k}^\prime} \left[\hat{O}_j^{\dagger}\ket{\mathbf{k}}\bra{\mathbf{k}}\hat{O}_j\ket{P}\bra{P}\right] \ket{\mathbf{k}^\prime}\\ \\ &\equiv \operatorname{Tr}{\left(\mathcal{I_{j\mathbf{k}}\ \rho}\right)} \end{align} \]
Where we have defined the terms in the square brackets as:
- \(\mathcal{I}_{j\mathbf{k}} \equiv \hat{O}_j^{\dagger}\ket{\mathbf{k}}\bra{\mathbf{k}}\hat{O}_j\) is an observable operator corresponding to the measured intensity at \(\mathbf{k}\)
- \(\rho \equiv \ket{P}\bra{P}\) is the density matrix for the pure state \(\ket{P}\)
Expressed in this way, it is clear that the detection \(I = \operatorname{Tr}{(\mathcal{I}\rho)}\) is the ensemble average of \(\mathcal{I}\) over the probe states \(\rho\). Naturally, this formulation is easily generalized to a treatment of an incoherent probe, i.e. a probe with mixed states.
“The goal of quantum state tomography is to solve for \(\rho\) the set of equations given by \(I = \operatorname{Tr}{(\mathcal{I}\rho)}\). Quantum state tomography was initially used on photon states and is now the standard method to measure qubits. It can be seen that ptychography is a special form of quantum state tomography, where not only the photon but also parameters of the measurement operator, i.e. the object, are concurrently reconstructed.”
From Supplementary Material, (3).
Mixed probe states
\[\rho = \sum_{m=1}^M \lambda_m \ket{P_m} \bra{P_m}\]