Quantum Fisher information maximization in an unbalanced interferometer

TL;DR

This paper analytically investigates how to optimize the transmission coefficient of an unbalanced Mach-Zehnder interferometer to maximize quantum Fisher information for various input states, revealing scenarios where unbalance offers significant advantages.

Contribution

It provides analytical solutions for maximizing quantum Fisher information in unbalanced interferometers for both single- and two-parameter cases, highlighting when unbalance improves metrological precision.

Findings

Balanced interferometers often optimize two-parameter QFI for many states.

Unbalanced interferometers can significantly enhance single-parameter QFI.

Conditions are identified under which external phase references do not provide additional advantage.

Abstract

In this paper we provide the answer to the following question: given an arbitrary pure input state and a general, unbalanced, Mach-Zehnder interferometer, what transmission coefficient of the first beam splitter maximizes the quantum Fisher information (QFI)? We consider this question for both single- and two-parameter QFI, or, in other words, with or without having access to an external phase reference. We give analytical results for all involved scenarios. It turns out that, for a large class of input states, the balanced (50/50) scenario yields the optimal two-parameter QFI, however this is far from being a universal truth. When it comes to the single-parameter QFI, the balanced scenario is rarely the optimal one and an unbalanced interferometer can bring a significant advantage over the balanced case. We also state the condition imposed upon the input state so that no metrological advantage can be exploited via an external phase reference. Finally, we illustrate and discuss our assertions through a number of examples, including both Gaussian and non-Gaussian input states.