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Physical Review A

Loss-error compensation in quantum-state measurements and the solution of the time-reversed damping equation

Ulrike Herzog

Published Mar 1, 1996• 30 citations
Publisher

Abstract

Making use of a generating-function approach, an effective mathematical procedure is derived for the compensation of detection-loss induced errors in quantum-state measurements at any value of the overall detection efficiency \ensuremath{\eta}. The connection with the solution of the time-reversed damping equation is discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.1245

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Reply to “Loss-error compensation in quantum-state measurements”

T. Kiss, Ulrike Herzog, Ulf Leonhardt

The authors of the preceding Comment [G. M. D'Ariano and C. Macchiavello, preceding paper, Phys. Rev. A 57, 3131 (1998)] tried to reestablish a $0.5$ efficiency bound for loss compensation in optical homodyne tomography. In our reply we demonstrate that neither does such a rigorous bound exist nor is the bound required for ruling out the state reconstruction of an individual system [G. M. D'Ariano and H. P. Yuen, Phys. Rev. Lett. 76, 2832 (1996)].

arXiv:quant-ph/9801028•Physical Review A•Apr 1, 1998•18 citations
Making use of a generating-function approach, an effective mathematical procedure is derived for the compensation of detection-loss induced errors in quantum-state measurements at any value of the overall detection efficiency \ensuremath{\eta}. The connection with the solution of the time-reversed damping equation is discussed. \textcopyright{} 1996 The American Physical Society.
Quantum state engineering plays a vital role in various applications in the field of quantum information. Different strategies, including drive-and-dissipation, adiabatic cooling, and measurement-based steering, have been proposed for state generation and manipulation, each with its upsides and downsides. Here, we address a class of measurement-based state engineering protocols where a sequence of generalized measurements is employed to steer a quantum system toward a desired (pure or mixed) target state. Previously studied measurement-based protocols relied on idealized procedures and avoided exploration of the effects of various errors stemming from imperfections of experimental realizations and external noise. We employ the quantum trajectory formalism to provide a detailed analysis of the robustness of these steering protocols against multiple classes of errors. We study a set of realistic errors that can be classified as dynamic or static, depending on whether they remain unchanged while running the protocol. More specifically, we investigate the impact of the erroneous choice of detector-system coupling, erroneous re-initialization of the detector state following a measurement step, fluctuating steering directions, and environmentally induced errors in the detector-system interaction. We show that the protocol remains fully robust against the erroneous choice of detector-system coupling parameters and presents reasonable robustness against other types of errors. Our analysis employs various quantifiers such as fidelity, trace distance, and linear entropy to characterize the protocol's robustness and provide analytical results for these quantifiers against various errors. We introduce averaging hierarchies of stochastic equations describing individual quantum trajectories associated with detector readouts. Subsequently, we demonstrate the commutation between the classical expectation value and the time-ordering operator of the exponential of a Hamiltonian with multiplicative white noise, as well as the commutation of the expectation value and the partial trace with respect to detector outcomes. Our ideas are implemented and demonstrated for a specific class of steering platforms, addressing a single qubit. Published by the American Physical Society 2024
In the two papers [T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995); U. Herzog, Phys. Rev. A 53, 1245 (1996)] with titles similar to the one given above, the authors assert that in some cases it is possible to compensate a quantum efficiency $\ensuremath{\eta}<~1/2$ in quantum-state measurements, violating the lower bound 1/21/21/2 proved in a preceding paper [G. M. D'Ariano, U. Leonhardt, and H. Paul, Phys. Rev. A 52, R1801 (1995)]. Here we reestablish the bound for homodyning any quantum state, and show how the proposed loss-compensation method would fail in a real measurement outside the $\ensuremath{\eta}>1/2$ regime.
Quantum estimation of parameters defining open-system dynamics may be enhanced by using ancillas that are entangled with the probe but are not submitted to the dynamics. Here we consider the important problem of estimation of transmission of light by a sample, with losses due to absorption and scattering. We show, through the determination of the quantum Fisher information, that the ancilla strategy leads to the best possible precision in single-mode estimation—the one obtained for a Fock-state input—through joint photon counting of probe and ancilla, which are modes of a bimodal squeezed state produced by an optical parametric amplifier. This proposal overcomes the challenge of producing and detecting high-photon-number Fock states, and it is quite robust in the presence of additional noise: We show that it is immune to phase noise and the precision does not change if the incoming state gets disentangled. Furthermore, the quantum gain is still present under moderate photon losses of the input beams. We also discuss an alternative to joint photon counting, which is readily implementable with present technology and approaches the quantum Fisher information result for weak absorption, even with moderate photon losses of the input beams before the sample is probed: a time-reversal procedure, placing the sample between two optical parametric amplifiers, with the second undoing the squeezing produced by the first one. The precision of estimation of the loss parameter is obtained from the average outgoing total photon number and its variance. In both procedures, the state of the probe and the detection procedure are independent of the value of the parameter. Published by the American Physical Society 2024
The authors of the preceding Comment [G. M. D'Ariano and C. Macchiavello, preceding paper, Phys. Rev. A 57, 3131 (1998)] tried to reestablish a 0.50.50.5 efficiency bound for loss compensation in optical homodyne tomography. In our reply we demonstrate that neither does such a rigorous bound exist nor is the bound required for ruling out the state reconstruction of an individual system [G. M. D'Ariano and H. P. Yuen, Phys. Rev. Lett. 76, 2832 (1996)].