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arXiv:2012.13402 · Acta Physica Polonica A

Quantum Tomography of Pure States with Projective Measurements Distorted by Experimental Noise

Artur Czerwiński

Published Feb 1, 2021• 9 citations
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Abstract

The article undertakes the problem of pure state estimation from projective measurements based on photon counting.Two generic frames for qubit tomography are considered -one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases.Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements.The accuracy of each frame is quantified by the average fidelity and purity.The efficiency of the frames is compared and discussed.The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.

DOI
10.12693/aphyspola.139.164

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The article undertakes the problem of pure state estimation from projective measurements based on photon counting.Two generic frames for qubit tomography are considered -one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases.Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements.The accuracy of each frame is quantified by the average fidelity and purity.The efficiency of the frames is compared and discussed.The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.
The article undertakes the problem of pure state estimation based on projective measurements. Two generic frames for qubit tomography are considered -- one defined by the vectors from the mutually unbiased bases (MUBs) and the other composed of the elements of the SIC-POVM. Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.
In this article, we introduce a framework for quantum state tomography of qutrits by projective measurements.The framework is based on photon-counting with measurement results distorted due to the Poisson noise and dark counts.Two different measurement schemes are investigated numerically and compared in terms of their efficiency for distinct numbers of photons per measurement.The accuracy of state reconstruction is quantified by figures of merit which are presented on graphs versus the amount of noise.
In this paper, we introduce a numerical framework for quantum tomography and entanglement quantification of three-qubit generalized Werner states. The scheme involves the single-qubit symmetric, informationally complete, positive operator-valued measure (SIC-POVM), which is then generalized to perform three-qubit measurements. We introduce random errors into the scheme by imposing the Poisson noise on the measured photon counts. The precision of state estimation is quantified and presented on graphs. As a special case, we compare the efficiency of the framework for the pure three-qubit generalized Werner state and the W state. The reconstructed states are presented graphically and discussed.
We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary} d−d-d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only} d+1d+1d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of d=6−d=6-d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 424242 to only 777.
Finite frames have many applications in quantum informatics, communications, coding theory, and other fields. In this article we investigate their use in quantum tomography. The family of vectors comprising a frame can be used to define a protocol of quantum tomography. To do this one estimates the probabilities of projective measurements on quantum states corresponding to the frame vectors. The state under investigation can then be reconstructed by one of several quantum tomography techniques. Since quantum tomography performance depends greatly on the choice of protocol, the problem of finding an optimal protocol is of considerable interest. WeexamineafamilyofunitnormframesconsistingofnumericalsolutionstoFeketepackingproblemincomplex space. Notable members of this family are Mutually Unbiased Bases (MUBs), which were computed in all investigated prime power dimensions, as well as Symmetrical Informationally Complete Positive Operator Valued Measures(SIC-POVM) in all dimensions. This family contains other uniform tight frames, in which the number of vectors lies between that of full sets of Mutually Unbiased Bases and Symmetrical Informationally Complete POVMs, and which all share the same internal structure. We detail some properties of these frames and their performance as quantum tomography protocols.
We present a quantum state tomography method that enables the reconstruction of arbitrary d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using only d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are spatially multiplexed on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.