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arXiv:quant-ph/0402179 · Europhysics Letters (EPL)

Quantum tomography for solid-state qubits

Yu-xi Liu, L. F Wei, Franco Nori

Published Sep 1, 2004• 24 citations
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Abstract

We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g. , with Heisenberg-, XXZ -, or XY -type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data.

DOI
10.1209/epl/i2004-10154-1

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We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g. , with Heisenberg-, XXZ -, or XY -type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally, orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First, we show how nonorthogonal measurements allow the reconstruction of the state of the system provided the measurements span the Hilbert space. We then detail how quantum-state tomography can be performed for multiqudits with a specific example illustrating how to achieve this in one- and two-qutrit systems.
We investigate a range of methods to perform tomography in a solid-state qubit device, for which a priori initialization and measurement of the qubit is restricted to a single basis of the Bloch sphere. We explore and compare several methods to acquire precise descriptions of additional states and measurements, quantifying both stochastic and systematic errors, ultimately leading to a tomographically complete set that can be subsequently used in process tomography. We focus in detail on the example of a spin qubit formed by the singlet-triplet subspace of two electron spins in a GaAs double quantum dot, although our approach is quite general.
We propose an approach to reconstruct any superconducting charge qubit state by using quantum state tomography. This procedure requires a series of measurements on a large enough number of identically prepared copies of the quantum system. The experimental feasibility of this procedure is explained and the time scales for different quantum operations are estimated according to experimentally accessible parameters. Based on the state tomography, we also investigate the possibility of the process tomography.