Achieving Heisenberg scaling with maximally entangled states: an analytic upper bound for the attainable root mean square error

TL;DR

This paper derives an analytical upper bound on the mean squared error for Heisenberg-limited quantum phase estimation using maximally entangled states, demonstrating the potential for optimal precision without prior knowledge.

Contribution

It provides the first analytical upper bound on the mean squared error for maximally entangled states in quantum metrology, extending understanding of Heisenberg scaling limits.

Findings

Upper bound decreases monotonically with the square of the number of probes

Protocol is non-adaptive and requires only separable measurements for distinguishable probes

Results include analysis under entanglement size limitations and loss conditions

Abstract

In this paper we explore the possibility of performing Heisenberg limited quantum metrology of a phase, without any prior, by employing only maximally entangled states. Starting from the estimator introduced by Higgins et al. in New J. Phys. 11, 073023 (2009), the main result of this paper is to produce an analytical upper bound on the associated Mean Squared Error which is monotonically decreasing as a function of the square of the number of quantum probes used in the process. The analysed protocol is non-adaptive and requires in principle (for distinguishable probes) only separable measurements. We explore also metrology in presence of a limitation on the entanglement size and in presence of loss.