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arXiv:quant-ph/0012071 · Physical Review Letters

Quantum Tomography for Measuring Experimentally the Matrix Elements of an Arbitrary Quantum Operation

Giacomo Mauro D’Ariano, Paoloplacido Lo Presti

Published May 7, 2001• 305 citations
arXivHTMLPDF

Abstract

Quantum operations describe any state change allowed in quantum mechanics, including the evolution of an open system or the state change due to a measurement. We present a general method based on quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. As input the method needs only a single entangled state. The feasibility of the technique for the electromagnetic field is shown, and the experimental setup is illustrated based on homodyne tomography of a twin beam.

DOI
10.1103/physrevlett.86.4195

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The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to revealing quantum properties relies on the reconstruction of entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of the matrix components of the measurement operators, which makes it possible to simplify the characterization process. We propose a general framework to directly obtain individual matrix elements of the measurement operators by sequentially measuring two noncompatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the required information. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix elements. The investigation of the measurement precision indicates the good feasibility of our protocol for arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix components of the measurement operators.
We suggest and demonstrate a tomographic method to characterise homodyne detectors at the quantum level.The positive operator measure associated with the detector is expanded in a quadrature basis and probed with a set of coherent states.The coefficients of the expansion are then retrieved using a least squares algorithm.Our model is general enough to describe different implementations of the homodyne setup, and it has proven capable of effectively describing the detector response to different tomographic sets.We validate the reconstructed operator measure on nonclassical states and exploit results to estimate the overall quantum efficiency of the detector.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
We show that the information collected in the course of a generic quantum tomography experiment can be used for verifying experimenters' assumptions about the state preparation and measurement. In particular, systematic errors, such as drifts and instabilities inherent in the tomography setup, can be identified without the need for any specific measurements designed to detect such problems. This is done by statistical analysis of available tomography data. As an example, we discuss the detection and characterization of unexpected changes in the signal state preparation during the tomography measurement and argue that minimal tomography schemes, such as symmetric informationally complete measurements, are more difficult to validate than overcomplete schemes. The method is illustrated with the analysis of experimental quantum homodyne tomography.