Abstract
Precise device characterisation is a fundamental requirement for a large range of applications using photonic hardware, and constitutes a multi-parameter estimation problem. Estimates based on measurements using single photons or classical light have precision which is limited by shot noise, while quantum resources can be used to achieve sub-shot-noise precision. However, there are many open questions with regard to the best quantum protocols for multi-parameter estimation, including the ultimate limits to achievable precision, as well as optimal choices for probe states and measurements. In this paper, we develop a formalism based on Fisher information to tackle these questions for setups based on linear-optical components and photon-counting measurements. A key ingredient of our analysis is a mapping for equivalent protocols defined for photonic and spin systems, which allows us to draw upon results in the literature for general finite-dimensional systems. Motivated by the protocol in Zhou, et al Optica 2, 510 (2015), we present new results for quantum-enhanced tomography of unitary processes, including a comparison of Holland-Burnett and NOON probe states.
Highlights
Advances in precision measurement are playing an ever more important role in technological development
This is true in conventional multi-parameter estimation using single-photon probes [8]. There are situations, such as when probing delicate samples, where the use of high N is undesirable [1]. It is well known for the idealised case, when the effects of particle losses and decoherence can be ignored, that quantum resources enable up to quadratic improvement in precision for both single-parameter estimation and multi-parameter estimation
On the other hand, when the restriction is made that probe states can only be prepared with correlations on up to N spin particles, the optimal precision using our protocol versus repeated use of a single type of probe state is the same, with tr(Ia-b1)∣min = 3 [2N (N + 2)] when 3N spin particles are used
Summary
Advances in precision measurement are playing an ever more important role in technological development. There are situations, such as when probing delicate samples, where the use of high N is undesirable [1] It is well known for the idealised case, when the effects of particle losses and decoherence can be ignored, that quantum resources enable up to quadratic improvement in precision for both single-parameter estimation and multi-parameter estimation. One of the first experimental demonstrations of Heisenberg scaling for a general linear-optical process was recently performed [11] (for N = 4 ) It was based on a new protocol for characterising an unknown two-mode linear-optical process, using Holland-Burnett states [12] and photon-number-counting measurements. This is equivalent to estimating the independent parameters of an unknown SU(2) matrix.
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