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arXiv:quant-ph/0202052 · Journal of Physics A Mathematical and General

Advancement of estimation fidelity in continuous quantum measurement

Lajos Diósi

Published Mar 15, 2002• 5 citations
arXivHTMLPDF

Abstract

We estimate an unknown qubit from the long sequence of n random polarization measurements of precision Δ. Using the standard Ito stochastic equations of the a posteriori state in the continuous measurement limit, we calculate the advancement of fidelity. We show that the standard optimum value 2/3 is achieved asymptotically for n ≫ Δ 2 /96 ≫ 1. We append a brief derivation of novel Ito equations for the estimate state.

DOI
10.1088/0305-4470/35/12/311

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We estimate an unknown qubit from the long sequence of n random polarization measurements of precision Δ. Using the standard Ito stochastic equations of the a posteriori state in the continuous measurement limit, we calculate the advancement of fidelity. We show that the standard optimum value 2/3 is achieved asymptotically for n ≫ Δ 2 /96 ≫ 1. We append a brief derivation of novel Ito equations for the estimate state.
Abstract When estimating an unknown single pure qubit state, the optimum fidelity is 2/3. As it is well known, the value 2/3 can be achieved in one step, by a single ideal measurement of the polarization along a random direction. I analyze the opposite strategy which is the long sequence of unsharp polarization measurements. The evolution of the qubit under the influence of repeated measurements is quite complicated in the general case. Fortunately, in a certain limit of very unsharp measurements the qubit will obey simple stochastic evolution equations known for long under the name of time‐continuous measurement theory. I discuss how the outcomes of the very unsharp measurements will asymptotically contribute to our knowledge of the original qubit. It is reassuring that the fidelity will achieve the optimum 2/3 for long enough sequences of the unsharp measurements.
When estimating an unknown single pure qubit state, the optimum fidelity is 2/3. As it is well known, the value 2/3 can be achieved in one step, by a single ideal measurement of the polarization along a random direction. I analyze the opposite strategy which is the long sequence of unsharp polarization measurements. The evolution of the qubit under the influence of repeated measurements is quite complicated in the general case. Fortunately, in a certain limit of very unsharp measurements the qubit will obey simple stochastic evolution equations known for long under the name of time-continuous measurement theory. I discuss how the outcomes of the very unsharp measurements will asymptotically contribute to our knowledge of the original qubit. It is reassuring that the fidelity will achieve the optimum 2/3 for long enough sequences of the unsharp measurements. 1
We discuss a non-linear stochastic master equation that governs the time-evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. For the first time, we can prove that the calculated estimate almost always converges to the true state, also at low-efficiency measurements. We show that our single-state theory can be adapted to weak continuous ensemble measurements as well.
We discuss a non-linear stochastic master equation that governs the time-evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. For the first time, we can prove that the calculated estimate almost always converges to the true state, also at low-efficiency measurements. We show that our single-state theory can be adapted to weak continuous ensemble measurements as well.
We present filtering equations for single-shot parameter estimation using continuous quantum measurement. By embedding parameter estimation in the standard quantum filtering formalism, we derive the optimal Bayesian filter for cases when the parameter takes on a finite range of values. Leveraging recent convergence results [R. van Handel, e-print arXiv:0709.2216], we give a condition which determines the asymptotic convergence of the estimator. For cases when the parameter is continuous valued, we develop quantum particle filters as a practical computational method for quantum parameter estimation.
This dissertation studies the statistics and modeling of a quantum system probed by a coherent laser field. We focus on an ensemble of qubits dispersively coupled to a traveling wave light field. The first research topic explores the quantum measurement statistics of a quasi-monochromatic laser probe. We identify the shortest timescale that successive measurements approximately commute. Our model predicts that for a probe in the near infrared, noncommuting measurement effects are apparent for subpicosecond times. The second dissertation topic attempts to find an approximation to a conditional master equation, which maps identical product states to identical product states. Through a technique known as projection filtering, we find such a equation for an ensemble of qubits experiencing a diffusive measurement of a collective angular momentum projection, and global rotations. We then test the quality of the approximation through numerical simulations. In the presence of strong randomized rotations, the approximation reproduces the exact expectation values to within 95%. The final topic applies the projection filter to the problem of state reconstruction. We find an initial state estimate based on a single continuous measurement of an identically prepared atomic ensemble. Given the ability to make a continuous collective measurement and simultaneously applying time varying controls, it is possible to find an accurate estimate given based upon a single measurement realization. Here we explore the fundamental limits of this protocol by studying an idealized model for pure qubits, which is limited only by measurement backaction. Using the exact dynamics to produce simulated measurements, we then numerically search for a maximum likelihood estimate based on the approximate expression. Our estimation technique nearly achieves an average fidelity bound set by an optimum POVM.