Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases

Abstract

Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher statement: © 2017 American Physical Society A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.ukOptimal measurements for simultaneous quantum estimation of multiple phases A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this letter we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the maximal theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.Introduction. Quantum metrology is currently attracting considerable interest in the light of its technological applications. Theoretical developments and experimental investigations have, so far, mostly focussed on the estimation of single phase [1][2][3], for which the ultimate sensitivity bounds and explicit conditions for their saturation are well known [4,5]. These studies have been further extended in order to understand the connection between enhancement in phase estimation and particle entanglement [6][7][8][9], as well as the impact of noise and dissipation on the fundamental bounds [10,11]. Several proof-of-principle experiments have demonstrated phase estimation below the classical (shot-noise) limit [2], including applications in fields as diverse as magnetometry [12], atomic clocks [13] and optical detection of gravitational waves [14].Yet, a significant class of problems can not be efficient...

A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging

“…The theory of multiparameter quantum estimation is an extremely challenging field, as witnessed by the fact that investigations on the topic started in the late 1960s [25,31], but new theoretical results are still being found, e.g. [46,52,66,70]. This is related to the fact that there are still many open questions in the field and here we highlight some of them.…”

“…Finding measurements achieving the QCRB is a nontrivial task: an example of one such measurement was reported in [81], ensuring it could provide separate values for each λ µ . Notably, this requires a projector back on the original state; the most general conditions for the optimal measurement have been obtained and discussed in [70].…”

“…Eventually the state is measured by undergoing the second tritter, performing the inverse unitary transformation U † A , such that U † A U A = I. Coincidence counting is then performed between the outputs of the circuit. In a general interferometer, this may not correspond to the optimal measurement, but it shows practical advantages in terms of ease of fabrication and manipulation [70]. The detailed theoretical analysis of the system capabilities has been extensively carried out in [169,170].…”

Time-local optimal control for parameter estimation in the Gaussian regime

“…Interestingly, the optimal strategy we find is analogous to the one found in [39,40] for noiseless Hamiltonian estimation, i.e., applying the inverse of the parameter encoding unitary. This inverse transformation also appears when studying optimal measurements for noiseless multiparameter estimation [58,59]. Since these optimal operations depend on the unknown true value of the parameter, their effectiveness only make sense in the context of an adaptive estimation scheme [60].…”

Symmetric Logarithmic Derivative of Fermionic Gaussian States

“…Simultaneous quantum estimation of multiple parameters provides better precision over individual estimation strategies with equivalent resources [32,33]. This fact has sparked interest in multi-parameter quantum metrology in a variety of scenarios [32][33][34][35].…”