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Schematic diagram of the energy levels of a spin I = 3 / 2 nucleus in an external magnetic field B 0 due to Zeeman interactions with additional shifts originating from the first-
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We discuss methods of quantum state tomography for solid-state systems with a
large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a
quantum well. Due to quadrupolar interactions, the Zeeman levels of these
nuclear-spin devices become nonequidistant, forming a controllable four-level
quantum system (known as quartit or ququ...
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... interactions (shown in Fig. 1), this system is described in an external magnetic field by a non-equidistant four- level energy spectrum. Thus, this spin-3/2 system can be referred to as a quartit (also called ququart or four-level qudit). The basic set of eigenfunctions of the system can be described with the states | mn ≡ | m A | n B of two logical (or virtual) qubits A and B corresponding to an ensemble of identical spin-1/2 ...
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... implies κ 2 ( C ) = 29 . 87 and κ F ( C ) = 74 . 38. Thus, one can again conclude that the error robustness of the method, as quantified by the condition numbers κ ( C ) and κ ( C ), is slightly worse than that for the original method, given by Eq. (43). The error-robustness analysis is based on the proper- ties of the coefficient matrices C (or A ) and, thus, enables to find experimental setups for the reliable QST even without specific experimental data. The optimization in our approach resides in (i) replacing degenerate multiphoton (multi-quantum) rotations by single-photon ones; and (ii) finding sets of rotations with the minimal number of readouts, which still enable reconstruction of reliable density matrices. Thus, in the following, with the help of the condition numbers σ min ( C ), κ 2 ( C ), and κ F ( C ), we compare the robustness against errors in measurements of various tomographic methods in the quest to find the optimal sets of tomographic rotations. The discussed sets of rotations for QST include single-photon X -rotations ( X 01 , X 12 , X 23 ), two-photon X -rotations ( X 02 , X 13 ), and a three-photon X -rotation ( X 03 ) together with analogous Y -rotations. The methods look very intuitive but they are not the simplest to be realized experimentally due to the degeneracy between ω 03 / 3 and ω 12 if the 2nd order quadrupolar shifts are ne- glected (see Fig. 1). Namely, we want to perform the three-photon rotations Y 03 ( π/ 2) and X 03 ( π/ 2) between levels | 0 and | 3 (for brevity, we say the 0-3 rotation) solely without changing populations between levels | 1 and | 2 . We can effectively rotate 1-2 without rotating 0-3, but we are not able to rotate 0-3 without rotating 1- 2. So a feasible tomographic method should be described without direct rotations 0-3. Under this requirement, it is easy to show analytically that one needs combinations of at least two rotations for some of the operations ...
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... Z = hω 0 I z and Q describe, respectively, the Zeeman and quadrupole splittings (see Fig. 1 in the special case for spin I = 3 / 2). The operator I α (for α = x, y, z ) is the α -component of the spin angular momentum operator, and I ± = I x ± iI y . Moreover, ω 0 = − γB 0 is the nuclear Larmor frequency, and ω k = − γB k is the amplitude (strength) of the k th pulse, where γ is the gyromagnetic ratio. For the example of the nuclei 69 Ga and 71 As of spin I = 3 / 2 in semiconductor GaAs, we can choose the gyromagnetic ratios to be γ ( 69 Ga) = 1 . 17 × 10 7 s − 1 T − 1 and γ ( 71 As) = 7 . 32 × 10 6 s − 1 T − 1 , which are estimated from the spectra measured in Ref. [42]. The Hamiltonian (1) (2) H Q = H Q + H Q + ... describes the quadrupolar interaction as a sum of the first- and second-order quadrupolar terms (as shown in Fig. 1 for spin I = 3 / 2), but also higher-order terms. The first-order quadrupolar splitting parameter (quadrupolar frequency) 2 ω Q is given for solids by ...
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... Z = hω 0 I z and Q describe, respectively, the Zeeman and quadrupole splittings (see Fig. 1 in the special case for spin I = 3 / 2). The operator I α (for α = x, y, z ) is the α -component of the spin angular momentum operator, and I ± = I x ± iI y . Moreover, ω 0 = − γB 0 is the nuclear Larmor frequency, and ω k = − γB k is the amplitude (strength) of the k th pulse, where γ is the gyromagnetic ratio. For the example of the nuclei 69 Ga and 71 As of spin I = 3 / 2 in semiconductor GaAs, we can choose the gyromagnetic ratios to be γ ( 69 Ga) = 1 . 17 × 10 7 s − 1 T − 1 and γ ( 71 As) = 7 . 32 × 10 6 s − 1 T − 1 , which are estimated from the spectra measured in Ref. [42]. The Hamiltonian (1) (2) H Q = H Q + H Q + ... describes the quadrupolar interaction as a sum of the first- and second-order quadrupolar terms (as shown in Fig. 1 for spin I = 3 / 2), but also higher-order terms. The first-order quadrupolar splitting parameter (quadrupolar frequency) 2 ω Q is given for solids by ...
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Citations
... Index of bases improved representational power (71), at the cost of growing training complexity, to further develop the NDO method to learn the open quantum systems. Adding an extra noise layer (40) or using global measurements (72,73) could improve the robustness of state reconstructions against experimental errors. Overall, our approach provides a promising avenue for addressing the challenges associated with verifying open QW. ...
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... To evaluate the efficiency of the tomographic technique, we used the so-called conditional number [16][17][18]. Previously, the parameter was used for a comprehensive comparison of the optical tomographic methods of two polarization qubits [17] or the nuclear magnetic resonance (NMR) tomography of two 1 H spins 1/2 (two qubits) [19] and a single nuclear spin 3/2 (a quartit) in a semiconductor quantum well [20]. We demonstrate that, by an appropriate tuning of a probing light, the conditional number can be minimized (corresponding to an optimized reconstruction), reaching as small value as 2.25. ...
... The fitting values are then used to determine the observables and reconstruct the qutrit density-matrix elements using the linear inversion method given in Eq. (3). However, as this method does not guarantee the reconstructed matrices to be positive semidefinite, we utilize a maximum likelihood method with the Euclidean norm [20,33] to find the closest physical realization of the reconstructed matrix. ...
... The simplest approach is to set negative eigenvalues to zero and renormalize the density matrix. More elaborate techniques find the closest positive semidefinite matrix using different norms using, for example, a maximum likelihood method [20,33]. In the presented work, we use such a method, assuming that a distance between matrices is based on the Euclidean norm, ...
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... Therefore, as a figure of merit for the robustness of a faithful retrieval of x against errors in the measurement outcomes y, as well as possible perturbations in the entries of A, we use the 2-norm condition number κ [15][16][17][18], which is defined as ...
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... The reconstruction equation (44) is similar to the corresponding one for the qubit MUB tomography (15), except that the coefficients are linear combinations of the measured probabilities. The probabilities p λ κ andp μ κ are not independent and apart from the normalization conditions similar to (16), they satisfy the following relations directly followed from (38) and (39), γ∈ (2) pλ +δ κ+γ = γ∈ (2) pλ κ+γ , ...
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