Quantum signal processing for quantum phase estimation: Fourier transform versus maximum likelihood approaches
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Quantum signal processing for quantum phase estimation: Fourier transform versus maximum likelihood approaches Franc¸ois Chapeau-Blondeau1
· Etienne Belin1
Received: 18 March 2020 / Accepted: 20 August 2020 © Institut Mines-T´el´ecom and Springer Nature Switzerland AG 2020
Abstract The phase in quantum states is an essential information carrier for quantum telecommunications, signal processing, and computation. Quantum phase estimation is therefore a fundamental operation to extract and control useful information at the quantum level. Here, we analyze various approaches to quantum phase estimation, when a phase parameter characterizing a quantum process gets imprinted in a relative phase attached to a quantum state serving as a probe signal. The estimation approaches are based on standard concepts of signal processing (Fourier transform, maximum likelihood), yet operated in the quantum realm. We also exploit the Fisher information, both in its classical and its quantum forms, in order to assess the performance of each approach to quantum phase estimation. We demonstrate a possibility of enhanced estimation performance, inaccessible classically, which is obtained via optimized quantum entanglement. Beyond their significance to quantum phase estimation, the results illustrate how standard concepts of signal processing can contribute to the ongoing developments in quantum information and quantum technologies. Keywords Quantum signal · Quantum phase · Quantum estimation · Quantum Fourier transform · Maximum likelihood · Fisher information
1 Introduction Quantum methodologies and quantum technologies hold large potentialities for information processing, telecommunications, and computation [1]. Provably secure telecommunications, high-precision metrology, high-sensitivity sensing, quantum computers, and quantum networking represent diverse application areas, which are diversely advanced and currently under intense development, and where quantum approaches can offer decisive contributions for enhanced performance [2–8]. In this context, quantum states constitute information-carrying signals for quantum telecommunications, information processing, and computation. Quantum states are normalized state vectors defined on a complex
Franc¸ois Chapeau-Blondeau
[email protected] Etienne Belin [email protected] 1
Laboratoire Angevin de Recherche en Ing´enierie des Syst`emes (LARIS), Universit´e d’Angers, 62 avenue Notre Dame du Lac, 49000, Angers, France
Hilbert space, and they remain with unit norm throughout any valid quantum processing. Beyond its unit norm, the phase associated with a quantum state stands as an essential carrier for the information contained in a quantum state. The phase or relative phases in quantum states condition their properties of coherence, interference, and interaction, and determine their ability for information processing and computation [9, 10]. Quantum states when interacting with quantum processes or devices often experience an alteration in their phase; and such phase m
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