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Quantum tomography of pure states with projective measurements generated by MUBs and SIC-POVM

Artur Czerwiński

Published Dec 24, 2020• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
We present a feasible proposal for quantum tomography of qubit and qutrit states via mutually unbiased measurements in dispersively coupled driven cavity QED systems. We first show that measurements in the mutually unbiased bases (MUBs) are practically implemented by projecting the detected states onto the computational basis after performing appropriate unitary transformations. The measurement outcomes can then be determined by detecting the steady-state transmission spectra (SSTS) of the driven cavity. It is found that all the measurement outcomes for each MUB (i.e., all the diagonal elements of the density matrix of each detected state) can be read out directly from only one kind of SSTS. In this way, we numerically demonstrate that the exemplified qubit and qutrit states can be reconstructed with the fidelities 0.952 and 0.961, respectively. Our proposal could be straightforwardly extended to other high-dimensional quantum systems provided that their MUBs exist.
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 the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.