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Physical Review A

Quantum Fisher information maximization in an unbalanced lossy interferometer

Stefan Ataman, Karunesh Kumar Mishra

Published Jun 11, 2024• 9 citations
Publisher

Abstract

In this work we address the problem of quantum Fisher information (QFI) maximization in an unbalanced lossy Mach-Zehnder interferometer. We implement the scheme from Escher et al. [Nat. Phys. 7, 406 (2011)] allowing us to obtain in closed form the lossy QFI for all involved scenarios. Then, mirroring the lossless case Ataman [Phys. Rev. A 105, 012604 (2022)], by optimizing the transmission coefficient of the first beam splitter, we maximize each QFI in question. We consider this problem for both single- and two-parameter QFI, i.e., for the scenarios with or without access to an external phase reference. Contrary to the lossless case, calculations are much more involved, however, we are able to put forward a large number of results in closed form. We also introduce two concepts, the balanced penalty and the QFI loss rate. Finally, we thoroughly discuss our results through a number of examples, including both Gaussian and non-Gaussian input states.

DOI
10.1103/physreva.109.062605

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In this work we address the problem of quantum Fisher information (QFI) maximization in an unbalanced lossy Mach-Zehnder interferometer. We implement the scheme from Escher et al. [Nat. Phys. 7, 406 (2011)] allowing us to obtain in closed form the lossy QFI for all involved scenarios. Then, mirroring the lossless case Ataman [Phys. Rev. A 105, 012604 (2022)], by optimizing the transmission coefficient of the first beam splitter, we maximize each QFI in question. We consider this problem for both single- and two-parameter QFI, i.e., for the scenarios with or without access to an external phase reference. Contrary to the lossless case, calculations are much more involved, however, we are able to put forward a large number of results in closed form. We also introduce two concepts, the balanced penalty and the QFI loss rate. Finally, we thoroughly discuss our results through a number of examples, including both Gaussian and non-Gaussian input states.
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.
In this paper we address the problem of optimizing an unbalanced Mach-Zehnder interferometer, for a given pure input state and considering a specific detection scheme. While the optimum transmission coefficient of the first beam splitter can be uniquely determined via the quantum Fisher information only [Ataman, Phys. Rev. A 105, 012604 (2022)], the second beam-splitter transmission coefficient is detection-scheme dependent, too. We systematically give analytic solutions for the optimum transmission coefficient of the second beam splitter for three types of widely used detection schemes. We provide detailed examples including both Gaussian and non-Gaussian input states, showing when an unbalanced Mach-Zehnder interferometer can outperform its balanced counterpart in terms of phase sensitivity.
Abstract In this work we address the problem of phase sensitivity optimization for an unbalanced Mach-Zehnder interferometer. While the quantum Fisher information can be employed in order to obtain the optimum transmission coefficient for the first beam splitter, this is no longer true for the second one, whose optimization is detection-scheme dependent. We thus consider three commonly used detection schemes and provide the optimal solution for each case. We also provide applications of the optimization method showing that for some input states a non-balanced Mach-Zehnder interferometer can outperform its balanced counterpart in terms of phase sensitivity.
The Cram\'er-Rao bound and the quantum Fisher information have been tools used extensively for interferometric phase sensitivity. Most scenarios considering a Mach-Zehnder interferometer with two input sources focused on the phase-matched case, when the Fisher information is maximal. Under this constraint, the best sensitivity is achieved for a balanced (50/50) input beam splitter. In this paper, we take a different approach: we allow the beam splitter transmission coefficient as well as the input phase mismatch to be variable parameters. We then search for a pair of these parameters that maximizes the Fisher information. We find that for the double coherent input the maximum Fisher information can always be reached in the unbalanced case for a carefully chosen input phase mismatch. For the coherent plus squeezed vacuum case we find that under certain circumstances a threshold phase mismatch exists, beyond which the optimum Fisher information is found for the degenerate case. For the squeezed-coherent plus squeezed vacuum case we find that the optimum actually occurs when the squeezing angles of the two inputs are in antiphase.
Abstract In this work we address the optimization of two important quantum states in terms of interferometric phase sensitivity, given a constraint on the total average number of input photons, N ¯ We use the quantum Fisher information (QFI) as our main tool. We employ both the single- and two-parameter QFI in order to address detection schemes having, and respectively, not having access to an external phase reference. The first quantum state we consider is the coherent plus squeezed vacuum input and find the optimum coherent fraction that maximizes both the single- and two-parameter QFI. We also discuss detection schemes able to perform almost optimally. Finally, we consider the double squeezed vacuum input and determine the squeezed fraction between the inputs that maximizes the QFI. Similar to the previous input state, we also provide detection schemes able to perform optimally.
We address in this work the phase sensitivity of a Mach-Zehnder interferometer with Gaussian input states. A squeezed-coherent plus squeezed vacuum input state allows us to unambiguously determine the optimal phase-matching conditions in order to maximize the quantum Fisher information. Realistic detection schemes are described and their performance compared with respect to the quantum Cram\'er-Rao bound. The core of this paper discusses in detail the most general Gaussian input state, without any a priori parameter restrictions. Prioritizing the maximization of various terms in the quantum Fisher information has the consequence of imposing the input phase-matching conditions. We discuss in detail when each scenario yields an optimal performance. Realistic detection scenarios are also considered and their performance compared to the theoretical optimum. The impact of the beam splitter types employed on the optimum phase-matching conditions is also discussed. We find a number of potentially interesting advantages of these states over the coherent plus squeezed vacuum input case.