Abstract
The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N→∞, which is called the asymptotic QFI. Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations. It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with N. Here we show that a simple criterion can determine whether the scaling is linear or quadratic. In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program. When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered. When the scaling is linear, we show that the asymptotic QFI is still in general larger than N times the single-channel QFI and, furthermore, that sequential estimation strategies provide no advantage over parallel ones.Received 23 March 2020Revised 20 December 2020Accepted 26 February 2021DOI:https://doi.org/10.1103/PRXQuantum.2.010343Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum error correctionQuantum metrologyQuantum sensingQuantum Information
Highlights
Quantum metrology studies parameter estimation in a quantum system [1,2,3,4,5]
The quantum Fisher information (QFI), which is inversely proportional to the minimum estimation variance, characterizes the amount of information a quantum state carries about an unknown parameter [19,20,21,22]
We focus on computing the asymptotic QFI for single-qubit depolarizing channels, but we do not provide explicit quantum error correction (QEC) protocols achieving the QFI
Summary
Quantum metrology studies parameter estimation in a quantum system [1,2,3,4,5]. Usually, a quantum probe interacts with a physical system and the experimentalist performs measurements on the final probe state and infers the value of the unknown parameter(s) in the system from the measurement outcomes. We answer these two open problems in the setting of entanglement-assisted channel estimation by providing an optimal quantum error correction (QEC) metrological protocol that entangles both the probe and a clean ancillary system. The result could be generalized to any system with a signal Hamiltonian and Markovian noise [33,34] These QEC protocols, can only estimate Hamiltonian parameters and all rely on fast and frequent quantum operations that have limited practical applications. The QFI is a good measure of the amount of information a quantum state ρω carries about an unknown parameter It is defined by F(ρω) = Tr(L2ρω), where L is a Hermitian operator called the symmetric logarithmic derivative (SLD) satisfying ρω. Using the purification-based definition of the QFI [Eq (3)], we have [24,28,29]
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