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arXiv:1006.5882 · Physical Review Letters

Characterizing Quantum Properties of a Measurement Apparatus: Insights from the Retrodictive Approach

Taoufik Amri, Julien Laurat, Claude Fabre

Published Jan 11, 2011• 34 citations
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Abstract

Using the retrodictive approach of quantum physics, we show that the state retrodicted from the response of a measurement apparatus is a convenient tool to fully characterize its quantum properties. We translate in terms of this state some interesting aspects of the quantum behavior of a detector, such as the nonclassicality or the non-gaussian character of its measurements. We also introduce estimators--the projectivity, the ideality, the fidelity, or the detectivity of measurements performed by the apparatus--which directly follow from the retrodictive approach. Beyond their fundamental significance for describing general quantum measurements, these properties are crucial in several protocols, in particular, in the conditional preparation of nonclassical states of light or in measurement-driven quantum information processing.

DOI
10.1103/physrevlett.106.020502

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Using the retrodictive approach of quantum physics, we show that the state retrodicted from the response of a measurement apparatus is a convenient tool to fully characterize its quantum properties. We translate in terms of this state some interesting aspects of the quantum behavior of a detector, such as the nonclassicality or the non-gaussian character of its measurements. We also introduce estimators--the projectivity, the ideality, the fidelity, or the detectivity of measurements performed by the apparatus--which directly follow from the retrodictive approach. Beyond their fundamental significance for describing general quantum measurements, these properties are crucial in several protocols, in particular, in the conditional preparation of nonclassical states of light or in measurement-driven quantum information processing.
We precise for the first time the quantum behavior of a measurement apparatus in the framework of the usual interpretation of quantum physics. We show how such a behavior can also be studied by the retrodiction of pre-measurement states corresponding to its responses. We translate in terms of these states some interesting properties of the behavior of an apparatus, such as the projectivity, the fidelity, the non-Gaussian character, or the non-classicality of measurements performed by this one. We also propose an experimental procedure allowing the tomography of these pre-measurement states for optical detectors. We illustrate the relevance of these new notions for measurements, by evaluating them for two detectors widely used in quantum optics: the avalanche photodiode and the homodyne detection.
What knowledge can be obtained from the record of a continuous measurement about the quantum state of the measured system at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously addressed in quantum measurement theory through retrodictive positive operator-valued measures (POVMs). This introduction to this general framework presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements and applies it to optomechanical systems. We identify and characterize achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner.
Measurements in the quantum domain can exceed classical notions. This concerns fundamental questions about the nature of the measurement process itself, as well as applications, such as their function as building blocks of quantum information processing protocols. In this paper we explore the notion of entanglement for detection devices in theory and experiment. A method is devised that allows one to determine nonlocal quantum coherence of positive-operator-valued measures via negative contributions in a joint distribution that fully describes the measurement apparatus under study. This approach is then applied to experimental data for detectors that ideally project onto Bell states. In particular, we describe the reconstruction of the aforementioned entanglement quasidistributions from raw data and compare the resulting negativities with those expected from theory. Therefore, our method provides a versatile toolbox for analyzing measurements regarding their quantum-correlation features for quantum science and quantum technology.
When can one interpret the outcomes of a quantum measurement as revealing a pre-existing objective property? Using the recently developed formalism of quantum measurement retrodiction, we provide a quantitative treatment of this question: for any POVM and faithful prior state, we construct a positive semidefinite bilinear form that quantifies the non-objectivity of every real-valued outcome feature through the disagreement between its predictive value and its retrodictive counterpart. We show that this form decomposes exactly into the sum of two positive semidefinite bilinear forms: an unsharpness form and an asymmetry form given by Wigner--Yanase skew information. The total form vanishes precisely on those outcome features that can be interpreted, relative to the prior, as revealing pre-existing properties; in particular, it vanishes identically if and only if the POVM is sharp and commutes with the prior. Finally, under maps that preserve the prior and are covariant under its modular group, asymmetry cannot increase, and any loss of asymmetry is offset by at least as much unsharpness, so that total non-objectivity cannot decrease.
Any nonclassical effect noticed in measured statistics is usually attributed just to the light state being observed. According to Born's rule, quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. We demonstrate that the nonclassicality of the measurement is a necessary condition to obtain nonclassical statistics. To this end, we present a detector characterization for two-observable joint detection processes based on detector tomography. This is particularized to the most common signatures of nonclassical light.