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Journal of Modern Optics

Robustness of homodyne tomography to phase-insensitive noise

Giacomo Mauro D’Ariano, Nicoletta Sterpi

Published Nov 1, 1997• 7 citations
Publisher

Abstract

We study the effects of phase-insensitive noise on homodyne measurements of a radiation density matrix. We prove that this noise has an effect equivalent to a non-unit quantum efficiency at detectors. The overall effective quantum efficiency η∗ of the measurement is evaluated in terms of the quantum efficiency at detectors and of the average number of noise photons added to the radiation field. For pure Gaussian-displacement noise, we show that half a photon of noise is enough to prevent the homodyne measurement of the density matrix.

DOI
10.1080/09500349708231880

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We study the effects of phase-insensitive noise on homodyne measurements of a radiation density matrix. We prove that this noise has an effect equivalent to a non-unit quantum efficiency at detectors. The overall effective quantum efficiency η∗ of the measurement is evaluated in terms of the quantum efficiency at detectors and of the average number of noise photons added to the radiation field. For pure Gaussian-displacement noise, we show that half a photon of noise is enough to prevent the homodyne measurement of the density matrix.
We study both systematic and statistical errors in radiation density matrix measurements. First we estimate the minimum number of scanning phases needed to reduce systematic errors to below a fixed threshold. Then, we calculate the statistical errors, intrinsic in the procedure that gives the density matrix. We present a detailed study of such errors versus the detectors' quantum efficiency and the matrix indices in the number representation, for different radiation states. For unit quantum efficiency, and for both coherent and squeezed states, the statistical errors of the diagonal matrix elements saturate for large n. In contrast, off-diagonal errors increase with the distance from the diagonal. For non-unit quantum efficiency the statistical errors along the diagonal do not saturate and increase dramatically versus both and the matrix indices.
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We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.