Optimal reconstruction of the states in qutrit systems

Yan, Fei; Yang, Ming; Cao, Zhuo‐Liang

doi:10.1103/physreva.82.044102

Cited by 13 publications

(13 citation statements)

References 38 publications

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“…In this sense, whenever they exist, mutually unbiased bases (MUBs) are known to be optimal [1,2]. Research since has focused mainly on proving or disproving existence of, and implement MUBs in various dimensions [2][3][4]. Other work, [5,6] considered OED based on the Cramér-Rao bound.…”

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confidence: 99%

Adaptive Bayesian quantum tomography

2012

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In this letter we revisit the problem of optimal design of quantum tomographic experiments. In contrast to previous approaches where an optimal set of measurements is decided in advance of the experiment, we allow for measurements to be adaptively and efficiently re-optimised depending on data collected so far. We develop an adaptive statistical framework based on Bayesian inference and Shannon's information, and demonstrate a ten-fold reduction in the total number of measurements required as compared to non-adaptive methods, including mutually unbiased bases.

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confidence: 99%

Adaptive Bayesian quantum tomography

2012

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“…This geometrical representation is useful in providing a visualization of quantum states and their transformations, particularly in the case of NMR-based quantum computation, where the spin-1 2 magnetization and its transformation through NMR rf pulses is visualized on the Bloch sphere [3]. There have been several proposals for the geometrical representation for higher-level quantum systems [4][5][6][7][8][9][10][11] however, extensions of a Bloch sphere-like picture to higher spins is not straightforward. A geometrical representation was proposed by Majorana in which, a pure state of a spin 's' is represented by '2s' points on the surface of a unit sphere, called the Majorana sphere [12].…”

Section: Introductionmentioning

confidence: 99%

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

Dogra¹,

Dorai²,

Arvind³

2018

J. Phys. B: At. Mol. Opt. Phys.

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We report a study of the Majorana geometrical representation of a qutrit, where a pair of points on a unit sphere represents its quantum states. A canonical form for qutrit states is presented, where every state can be obtained from a one-parameter family of states via SO(3) action. The notion of spin-1 magnetization which is invariant under SO(3) is geometrically interpreted on the Majorana sphere. Furthermore, we describe the action of several quantum gates in the Majorana picture and experimentally implement these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline environment. We study the dynamics of the pair of points representing a qutrit state under various useful quantum operations and connect them to different NMR operations. Finally, using the Gell Mann matrix picture we experimentally implement a scheme for complete qutrit state tomography.

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“…This is also a kind of nondestructive measurement due to the fact that the interaction Hamiltonian between the qutrit and the cavity (i.e., Stark shift term in equation (19)) †  = P S a a j jj int commutes with the qutrit operator P jj , that is, [ ]  P = , 0 jj int . Theoretical analysis and numerical experiments also indicate that multiple peaks emerge in the SSTS (22) with the same feature as the qubit case. This feature is also verified in section 4.3.…”

Section: Ssts Of a Driven Cavity With A Qutritmentioning

confidence: 65%

“…Likewise, we can directly read out the projective measurement outcomes ( ) from figure 3(b), (c), and (d), respectively. Finally, substituting these projective measurement outcomes and the MUBs for d=3 [22] into equation (2), the normalized reconstructed state is obtained as Note that the reconstructed density matrix (25) is also unphysical since it contains a negative eigenvalue.…”

Section: Numerical Demonstration Of Mubs-qst Of Qutrit Statesmentioning

confidence: 99%

“…Due to this distinct property, MUBs have extensive applications in the field of quantum information. Therein, a typical application is QST [18][19][20][21][22][23]. In the context of QST, Wootters and Fields [18] proved that the measurements in the MUBs provide a minimal and optimal way to realize QST (called MUBs-QST hereafter) in the sense of maximizing information extraction from each measurement and minimizing the effects of statistical errors in the measurements.…”

Section: Introductionmentioning

confidence: 99%

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Quantum state tomography via mutually unbiased measurements in driven cavity QED systems

2016

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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.implement MUBs-QST only in optical systems [24][25][26][27]. Additionally, a theoretical scheme has been presented to realize MUBs-QST of two spin qubits in a double quantum dot [28].As a possible physical implementation, in this paper we propose a feasible proposal for MUBs-QST in dispersively coupled driven cavity QED systems. We demonstrate our idea for the cases of qubit and qutrit states. Certainly, our idea is also suitable for other high-dimensional quantum systems if their MUBs exist. The main idea of our proposal is summarized as follows. First, measurements in the MUBs are implemented by projecting the detected states onto the computational basis after performing proper unitary transformations, which can be readily realized by adjusting the classical driving field applied on the qubit or qutrit. Secondly, the projective measurement outcomes can be determined by detecting the steady-state transmission spectra (SSTS) of the driven cavity. Through theoretical analysis and numerical experiments, it is found that multiple peaks appear in the SSTS: each of the detected peaks marks one of the computational basis states and its relative height equals the corresponding superposed probability in the detected state. This manifest advantage allows us to directly read out all the measurement outcomes for each MUB (i.e., all the diagonal elements of the density matrix of each detected state) by only one kind of SSTS. In this manner, MUBs-QST can be realized. Finally, our proposed readout method is of the nondestructive property [29]. This means that what we directly detect is the transmitted photons through the driven cavity, rather than intracavity atom itself. The measurement-induced noises on the atom can be efficiently suppressed. Thus we numerically demonstrate that the fidelities of the exemplified qubit and qutrit states can attain 0.952 and 0.961, respectively.The rest of this paper is organized as follows. In section 2, we briefly introduce the MUBs and the MUBs-QST for d-dimensional quantum system, especially for two minimal prime numbers d=2 a...

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Optimal reconstruction of the states in qutrit systems

Cited by 13 publications

References 38 publications

Adaptive Bayesian quantum tomography

Adaptive Bayesian quantum tomography

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

Quantum state tomography via mutually unbiased measurements in driven cavity QED systems

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