Abstract:Funes 3350, 7600 Mar del Plata, Argentina CONICET An iterative algorithm for state determination is presented that uses as physical input the probability distributions for the eigenvalues of two or more observables in an unknown state Φ.Starting form an arbitrary state Ψ0, a succession of states Ψn is obtained that converges to Φ or to a Pauli partner. This algorithm for state reconstruction is efficient and robust as is seen in the numerical tests presented and is a useful tool not only for state determinatio… Show more
“…It was also shown to be equivalent to an alternating-gradient search algorithm. Successful reconstructions were obtained in realistic noisy scenarios, which makes the method amenable to future experiments and a concrete alternative to standard tomographic techniques 12–15 .…”
Section: Resultsmentioning
confidence: 97%
“…By setting the MI to perform , the ptychographic data would be collected simply by shifting the mode filters n times at the input ports and recording the counts at the output ports. For comparison, to reconstruct these states by measuring four or five observables 12–15 , the mode filters would not be necessary, but one would have to reconfigure the whole MI for each measurement basis employed. This shows a nice feature of the ptychographic method: the measurements are effectively performed in a single basis while the probe projectors are “shifted” through the Hilbert space.Figure 5Scheme for ptychographic reconstruction of 8-dimensional quantum states in a multiport interferometer.…”
Section: Discussionmentioning
confidence: 99%
“…However, under prior information the process is simplified. For example, if an unknown state is known to be pure, four 12–14 or five 13,15 measurement bases and simple postprocessing suffice for determining it on any finite dimension. Yet, to implement the measurements in a variety of bases (or, equivalently, to implement various unitary operations) may not be straightforward in all experiments.…”
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.
“…It was also shown to be equivalent to an alternating-gradient search algorithm. Successful reconstructions were obtained in realistic noisy scenarios, which makes the method amenable to future experiments and a concrete alternative to standard tomographic techniques 12–15 .…”
Section: Resultsmentioning
confidence: 97%
“…By setting the MI to perform , the ptychographic data would be collected simply by shifting the mode filters n times at the input ports and recording the counts at the output ports. For comparison, to reconstruct these states by measuring four or five observables 12–15 , the mode filters would not be necessary, but one would have to reconfigure the whole MI for each measurement basis employed. This shows a nice feature of the ptychographic method: the measurements are effectively performed in a single basis while the probe projectors are “shifted” through the Hilbert space.Figure 5Scheme for ptychographic reconstruction of 8-dimensional quantum states in a multiport interferometer.…”
Section: Discussionmentioning
confidence: 99%
“…However, under prior information the process is simplified. For example, if an unknown state is known to be pure, four 12–14 or five 13,15 measurement bases and simple postprocessing suffice for determining it on any finite dimension. Yet, to implement the measurements in a variety of bases (or, equivalently, to implement various unitary operations) may not be straightforward in all experiments.…”
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.
“…This is because maximal sets of mutually unbiased bases are informationally complete. In the case of d = 3 the complete set of Pauli partners have been found [18]. The example shown above can be generalized to any dimension for a general set of observables: Therefore, we realize that bifurcations of the physical imposition operator are the responsible to have multiple solutions to the Pauli problem when an informationally incomplete set of observables is considered.…”
A powerful tool for studying geometrical problems in Hilbert space is
developed. In particular, we study the quantum pure state tomography problem in
finite dimensions from the point of view of dynamical systems and bifurcations
theory. First, we introduce a generalization of the Hellinger metric for
probability distributions which allows us to find a geometrical interpretation
of the quantum state tomography problem. Thereafter, we prove that every
solution to the state tomography problem is an attractive fixed point of the
so-called physical imposition operator. Additionally, we demonstrate that
multiple states corresponding to the same experimental data are associated to
bifurcations of this operator. Such a kind of bifurcations only occurs when
informationally incomplete set of observables are considered. Finally, we prove
that the physical imposition operator has a non-contractive Lipschitz constant
2 for the Bures metric. This value of the Lipschitz constant manifests the
existence of the quantum tomography problem for pure states.Comment: 16 pages, 2 figures. Submitted to Journal of Mathematical Physic
“…However, under prior information the process is simplified. For example, if an unknown state is known to be pure, four [12][13][14] or five [13,15] measurement bases and simple postprocessing suffice for determining it on any dimension. Yet, to implement the measurements in a variety of bases (or, equivalently, to implement various unitary operations) may not be straightforward in all experiments.…”
The quantum analogue of ptychography, a powerful coherent diffractive
imaging technique, is a simple method for reconstructing
d
-dimensional pure states. It relies on
measuring partially overlapping parts of the input state in a single
orthonormal basis and feeding the outcomes to an iterative phase
retrieval algorithm for postprocessing. We provide a proof of concept
demonstration of this method by determining pure states given by
superpositions of
d
transverse spatial modes of an
optical field. A set of
n
rank-
r
projectors, diagonal in the spatial
mode basis, is used to generate
n
partially overlapping parts of the
input, and each part is projectively measured in the Fourier
transformed basis. For
d
up to 32, we successfully
reconstructed hundreds of random states using
n
=
5
and
n
=
d
rank-
⌈
d
/
2
⌉
projectors. The extension of quantum
ptychography for other types of photonic spatial modes is
outlined.
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