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.
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 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.
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 N 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.
Optimal probe states and measurements are always required to achieve the best estimation precision in quantum metrology. In the actual physical environment, ubiquitous noise hinders estimation precision and amplifies the challenges of optimizing the probe states and measurements. In this study we extend a theoretical proposal, presented in Phys. Rev. A 103, 032615 (2021), to encompass noisy general Pauli channels for SU(2) phase estimation. By utilizing a superposition of different causal orders of two channels, we establish theoretically a probe-state-independent criterion for the estimation precision. We show that the probe-state-independent property results from the noncommutativity of the Kraus operators of the parameter-encoding channels. Based on this criterion, one can find some parameter-encoding channels without the requirement of precisely preparing the probe state. Moreover, our scheme requires only deterministic projection measurement on the control qubit. In this way, one can simultaneously avoid optimizing both the probe states and measurements. In addition, we also show that the estimation precision and probe-state-independent property of the indefinite causal order scheme are independent of representations of the Kraus operators. This alleviates the experimental challenges in realizing quantum channels. We demonstrate experimentally the advantages of the indefinite causal order for phase estimation of an SU(2) unitary transformation in the three kinds of quantum noise channels. Our result shows that the dynamic evolution of indefinite causal order can outperform the conventional cascaded estimation scenario at high noise levels, exhibiting high robustness and feasibility in practical estimation tasks.
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 ordering (ICO) refers to quantum channels arranged so that the path through them is a super- position of different possible paths. The ICO is called pure when the path superposition is a pure quantum state. Channel probing, in which probes in prepared quantum states are passed through copies of the channel to estimate one or more channel parameters, is aided by ICO. Specifically, pure ICO is known to increase the quantum Fisher information (QFI) in the processed probe state about the unknown parameter(s). We consider the case complementary to pure ICO in which the path state is maximally mixed for a given indefiniteness. Deriving the QFI under this condition by a new approach, we find that mixed ICO-assisted probing, while not as beneficial as pure ICO probing, still yields greater QFI than does the comparable scheme with definite causal order. The d-dimensional quantum depolarizing channel is used as the channel model for this study. While mixed ICO-assisted probing, like pure ICO-assisted probing, is generally advantageous for probing the depolarizing channel, mixed ICO's relative effectiveness decreases with probe dimension, just as for pure ICO-assisted probing, each being most effective for qubit probes.
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.