We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel--Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all the 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized Pauli operators, where twelve of them can be locally measured, and the other four require nonlocal Bell measurements. Our method corresponds to selectively measuring, one by one, all the real and imaginary elements of an unknown two-qubit density matrix. We describe two experimentally feasible setups of this protocol for the optimal reconstruction of two photons in an unknown polarization state using conventional detectors and linear-optical elements. Moreover, we define the operators for the optimal reconstruction of the states of multiqubit or multilevel (qudit) systems.
We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel--Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all the 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized Pauli operators, where twelve of them can be locally measured, and the other four require nonlocal Bell measurements. Our method corresponds to selectively measuring, one by one, all the real and imaginary elements of an unknown two-qubit density matrix. We describe two experimentally feasible setups of this protocol for the optimal reconstruction of two photons in an unknown polarization state using conventional detectors and linear-optical elements. Moreover, we define the operators for the optimal reconstruction of the states of multiqubit or multilevel (qudit) systems.
In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.
Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is presented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to minimize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems.
In quantum information transformation and quantum computation, the most critical issues are security and accuracy. These features, therefore, stimulate research on quantum state characterization. A characterization tool, quantum state tomography, reconstructs the density matrix of an unknown quantum state. Theoretically, reconstructing an unknown state using this method can be arbitrarily accurate. However, this is not so practical owing to the huge burden of measurements and data processing for large numbers of qubits. Even comprising an efficient estimator and a precise algorithm, an optimal tomographic framework can also be overburdened owing to the exponential growth of the measurements. Moreover, the consequential postprocessing of huge amounts of data challenges the capacity of computers. Thus, it is crucial to build an efficient framework that requires fewer measurements but yields an expected accuracy. To this end, we built a tomography schema by which only a few Pauli measurements enable an accurate tomographic reconstruction. Subsequently, this schema was verified as efficient and accurate through numerical simulations on the tomography of multiqubit quantum states. Furthermore, it was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternative effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.
Scalable characterization of quantum processors is crucial for mitigating noise and imperfections. While randomized measurement protocols enable efficient access to local observables, inferring a globally consistent description of multi-qubit processes remains challenging. Here we introduce a local-to-global reconstruction framework for one-dimensional multi-qubit states and channels. The method is efficient provided that correlations, as quantified by the conditional mutual information, decay exponentially. In particular, we prove that under this assumption, the required number of samples scales polynomially with the system size and the desired global reconstruction error. Our approach is based on combining local shadow tomography with locally optimal recovery maps obtained by convex optimization. We supplement these rigorous guarantees by studying the performance of the protocol numerically for a system evolving under a local Lindbladian and a noisy, shallow circuit. By employing a tensor networ representation, we reconstruct channels acting on up to 50 qubits and accurately recover global diagnostics such as the process fidelity, the Choi state purity, and Pauli-weight-resolved process matrix elements. Our work thus extends the powerful toolbox local shadow tomography to scalable channel characterization with access to global properties.
We study the determination of an unknown mixed state of a d-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d2 for d large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3 projections required by standard tomography.
In quantum information transformation and quantum computation, the most critical issues are security and accuracy. These features, therefore, stimulate research on quantum state characterization. A characterization tool, Quantum state tomography, reconstructs the density matrix of an unknown quantum state. Theoretically, reconstructing an unknown state using this method can be arbitrarily accurate. However, this is less practical owing to the huge burden of measurements and data processing for large numbers of qubits. Even comprising an efficient estimator and a precise algorithm, an optimal tomographic framework can also be overburdened owing to the exponential growth of the measurements. Moreover, the consequential postprocessing of huge amounts of data challenges the capacity of computers. Thus, it is crucial to build an efficient framework that requires fewer measurements but yields an expected accuracy. To this end, we built a tomography schema by which only a few Pauli measurements enable an accurate tomographic reconstruction. Subsequently, this schema was verified as efficient and accurate through numerical simulations on the tomography of multi-qubit quantum states. Furthermore, this schema was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternatively effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.