Ptychography of pure quantum states

Abstract

Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors “scans” overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d / 2, we performed numerical simulations for single systems in an economic (n = 4) and a costly (n = d) scenario, as well as for multiqubit systems (n = 6logd). This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10−2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data—key for its success—are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.

Ptychographic reconstruction of pure quantum states

“…Recently, two of us have proposed a quantum analogue of ptychography as a simple and robust method to reconstruct pure states [11]. Ptychography [12] is a form of coherent diffractive imaging where a localized illumination is shifted through partially overlapping parts of an object and generates multiple diffraction patterns; these patterns are processed by an iterative phase retrieval algorithm (e.g., the ptychograhic iterative engine (PIE) [12,13]), which recover the complex-valued object transmission function.…”

“…In our proposal [11], a d-dimensional (d > 2) pure quantum state is the object of interest and the role of the shifting illumination is played by a set of n rank-r projectors. These projectors, one at a time, "slice" the state into n partially overlapping parts and each part is projectively measured in the same orthonormal basis.…”

“…Comparing with other tomographic approaches for pure states, which require projective measurements in a fixed number of bases [6][7][8][9][10], the ptychographic technique [11] resorts to a flexible number (n) of projective measurements in a single basis. It also uses simpler settings in the experimental setup, since the rank-r projectors are usually easy to implement.…”

“…An arbitrary pure quantum state in a d-dimensional Hilbert space H d can be written as |ψ = ∑ d−1 k=0 c k |k , where {|k } d−1 k=0 is an orthonormal basis for H d (here called computational basis) and {c k } d−1 k=0 (with ∑ k |c k | 2 = 1) the set of complex coefficients which completely specify |ψ . The ptychographic method to determine these coefficients requires a set of n projectors { P } n−1 =0 of rank r (> 1), where each P has a partial overlap with at least one other partner and all levels of H d are addressed at least once [11]. After specifying them, the protocol goes as follows: first, one applies the -th projector on the input ensemble of quantum systems described by an unknown |ψ .…”

“…As shown in [11], a suitable choice for { P } n−1 =0 comprises projectors diagonal in the computational basis given by…”