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arXiv:quant-ph/0208091 · Physics Letters A

Simple optical measurement of the overlap and fidelity of quantum states

Martin Hendrych, Miloslav Dušek, Radim Filip, Jaromı́r Fiurášek

Published Apr 1, 2003• 35 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(03)00262-7

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Two measures of fidelity are proposed for postselecting devices, the retrodictive conditional probability that the state in the measurement arm is the one indicated by the detectors, and the probability that the device produces the state that it would produce if working perfectly. The first is the natural quantity that one wishes to maximize to improve the device operation. The second corresponds more closely with the accurate operation of the device than the more usual overlap-based fidelity. The results are particularly applicable to the types of state preparation and logic gate operation in linear optical quantum computing.
Two measures of fidelity are proposed for postselecting devices, the retrodictive conditional probability that the state in the measurement arm is the one indicated by the detectors, and the probability that the device produces the state that it would produce if working perfectly. The first is the natural quantity that one wishes to maximise to improve the device operation. The second corresponds more closely with the accurate operation of the device than the more usual overlap-based fidelity. The results are particularly applicable to the types of state preparation and logic gate operation in linear optical quantum computing.
A simple and flexible scheme for high-dimensional linear quantum operations is demonstrated on optical discrete spatial modes in the transverse plane. Quantum-state tomography via symmetric informationally complete positive-operator-valued measures and quantum Fourier transformation are implemented with dimensionality of 15. The statistical fidelity of symmetric informationally complete positive-operator-valued measures and the fidelity of quantum-state tomography are approximately 0.97 and up to 0.853, respectively, while the matrix fidelity of quantum Fourier transformation is 0.85. We believe that our approach has the potential for further exploration of high-dimensional spatial entanglement provided by spontaneous parametric down-conversion in nonlinear crystals.