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Symmetry

Symmetries of Quantum Fisher Information as Parameter Estimator for Pauli Channels under Indefinite Causal Order

Francisco Delgado

Published Sep 1, 2022• 5 citations
Publisher

Abstract

Quantum Fisher Information is considered in Quantum Information literature as the main resource to determine a bound in the parametric characterization problem of a quantum channel by means of probe states. The parameters characterizing a quantum channel can be estimated until a limited precision settled by the Cramér–Rao bound established in estimation theory and statistics. The involved Quantum Fisher Information of the emerging quantum state provides such a bound. Quantum states with dimension d=2, the qubits, still comprise the main resources considered in Quantum Information and Quantum Processing theories. For them, Pauli channels are an important family of parametric quantum channels providing the most faithful deformation effects of imperfect quantum communication channels. Recently, Pauli channels have been characterized when they are arranged in an Indefinite Causal Order. Thus, their fidelity has been compared with single or sequential arrangements of identical channels to analyse their induced transparency under a joint behaviour. The most recent characterization has exhibited important features for quantum communication related with their parametric nature. In this work, a parallel analysis has been conducted to extended such a characterization, this time in terms of their emerging Quantum Fisher Information to pursue the advantages of each kind of arrangement for the parameter estimation problem. The objective is to reach the arrangement stating the best estimation bound for each type of Pauli channel. A complete map for such an effectivity is provided for each Pauli channel under the most affordable setups considering sequential and Indefinite Causal Order arrangements, as well as discussing their advantages and disadvantages.

DOI
10.3390/sym14091813

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Quantum Fisher Information is considered in Quantum Information literature as the main resource to determine a bound in the parametric characterization problem of a quantum channel by means of probe states. The parameters characterizing a quantum channel can be estimated until a limited precision settled by the Cramér–Rao bound established in estimation theory and statistics. The involved Quantum Fisher Information of the emerging quantum state provides such a bound. Quantum states with dimension d=2, the qubits, still comprise the main resources considered in Quantum Information and Quantum Processing theories. For them, Pauli channels are an important family of parametric quantum channels providing the most faithful deformation effects of imperfect quantum communication channels. Recently, Pauli channels have been characterized when they are arranged in an Indefinite Causal Order. Thus, their fidelity has been compared with single or sequential arrangements of identical channels to analyse their induced transparency under a joint behaviour. The most recent characterization has exhibited important features for quantum communication related with their parametric nature. In this work, a parallel analysis has been conducted to extended such a characterization, this time in terms of their emerging Quantum Fisher Information to pursue the advantages of each kind of arrangement for the parameter estimation problem. The objective is to reach the arrangement stating the best estimation bound for each type of Pauli channel. A complete map for such an effectivity is provided for each Pauli channel under the most affordable setups considering sequential and Indefinite Causal Order arrangements, as well as discussing their advantages and disadvantages.
Parameter estimation for devices containing or supporting quantum systems is a field of quantum metrology using quantum probe states to reach their characterization. Pauli channels are ideal structures where qubits are transmitted or contained, commonly altering them with specific fingerprints. The ultimate limit imposed on such estimation is addressed using the quantum Fisher information, stating a lower bound for it. Although the most simple scheme suggests performing such an estimation directly using the individual channel, other approaches have shown improved outcomes by repeating identical copies of the channel for the characterization, or otherwise those connected inside of specific circuit arrangements. These connections commonly include path superposition or causal indefinite architectures. In addition, other improvements have been observed in concrete channels when complementary unitary controls are included. The current research analyses the complete set of Pauli channels under some of those architectures in a comparative approach to reach a better estimation, thus stating hierarchies. It is observed that the use of those unitary controls notably improves previous outcomes by several orders of magnitude.
Quantum Parameter Estimation (QPE) is commonly led using quantum probe states for the characterization of quantum systems. For these purposes, Quantum Fisher Information (QFI) plays a crucial role by imposing a lower bound for the parametric estimation of quantum channels. Several schemes for obtaining QFI lower bounds have been proposed, particularly for Pauli channels regarding qubits. Those schemes commonly employ either the individual channel, multiple copies of it, or arrangements including communication architectures. The present work aims to propose an architecture involving path superposition and causal indefinite order in superposition. Thus, by controlling the symmetry balance of this superposition, it reaches notable improvements in quantum parameter estimation. The proposed architecture has been tested to find the best possible QPE bounds for a representative and emblematic set of Pauli channels. Further, for the most reluctant channels, it was revisited testing the architecture again under a primary path superposition (using double teleportation) and also using entangled probe states to recombine their outputs with the original undisturbed state. Notable outcomes practically near zero were found for the QPE bounds, stating a hierarchy between the approaches, but anyway reaching a perfect theoretical QPE, particularly for the last path superposition including the proposed architecture.
Indefinite causal order has introduced disruptive procedures to improve the fidelity of quantum communication by introducing the superposition of { orders} on a set of quantum channels. It has been applied to several well characterized quantum channels as depolarizing, dephasing and teleportation. This work analyses the behavior of a parametric quantum channel for single qubits expressed in the form of Pauli channels. Combinatorics lets to obtain affordable formulas for the analysis of the output state of the channel when it goes through a certain imperfect quantum communication channel when it is deployed as a redundant application of it under indefinite causal order. In addition, the process exploits post-measurement on the associated control to select certain components of transmission. Then, the fidelity of such outputs is analysed to characterize the generic channel in terms of its parameters. As a result, we get notable enhancement in the transmission of information for well characterized channels due to the combined process: indefinite causal order plus post-measurement.
Abstract Quantum Fisher information is the principal tool used to give the ultimate precision bound on the estimation of parameters for quantum channels. In this work, we present analytical expressions for the quantum Fisher information with three noisy channels for the case where the channels are in superposition of causal orders. We found that the quantum Fisher information increases as the number of causal orders increases for certain combinations. We also show that certain combinations of causal orders attain higher precision on bounds than others for the same number of causal orders. Based on our results, we chose the best combinations of causal orders with three channels for probing schemes using indefinite causal structures.
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in NNN identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when