The optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/N, where N is the number of used measurements.The optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/N, where N is the number of used measurements.

Jan 1, 20128 citations

Experiment design and parameter estimation of Pauli channels using convex optimization

Gábor Balló, Katalin Mária Hangos

A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.

arXiv:1107.0890Jul 5, 20113 citations

Convex Optimization-Based Parameter Estimation and Experiment Design for Pauli Channels

Gábor Balló, Katalin Mária Hangos, D�nes Petz

In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.

IEEE Transactions on Automatic ControlJul 19, 201210 citations

Pauli channel tomography with unknown channel directions

Dániel Virosztek, László Ruppert, Katalin Mária Hangos

In this paper we estimate the parameters of the qubit Pauli channel using the channel matrix formalism. The main novelty of this work is that we do not assume the directions of the Pauli channel to be known, but they are determined through the tomography process, too. The results show that for optimally estimating the contraction parameters and the channel matrix we should have input qubits and measurements in the channel directions. However, for optimally estimating the channel directions, we should use different tomography conditions.In this paper we estimate the parameters of the qubit Pauli channel using the channel matrix formalism. The main novelty of this work is that we do not assume the directions of the Pauli channel to be known, but they are determined through the tomography process, too. The results show that for optimally estimating the contraction parameters and the channel matrix we should have input qubits and measurements in the channel directions. However, for optimally estimating the channel directions, we should use different tomography conditions.

arXiv:1304.4492Apr 16, 20130 citations

Entanglement depth and ancilla efficiency in quantum channel estimation

Javid Naikoo

We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the $k$ -ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most $k$ , and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension $k^*$ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the $k$-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most $k$, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension $k^*$ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.

quant-pharXiv:2608.11042Aug 12, 20260 citations

Quantum parameter estimation of a generalized Pauli channel

Akio Fujiwara, Hiroshi Imai

We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : ( d ) → ( d ), where the parameter θ is regarded as a coordinate system of the probability simplex d 2 −1 . We show that for each degree n of extension (id ⊗ Γ θ ) ⊗ n : (( d ⊗ d ) ⊗ n ) → (( d ⊗ d ) ⊗ n ), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n -tensor product of a maximally entangled state τ ME ∊ ( d ⊗ d ). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇ m -affine in d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ ME )} θ of output states can be identified with d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : ( d ) → ( d ), where the parameter θ is regarded as a coordinate system of the probability simplex d 2 −1 . We show that for each degree n of extension (id ⊗ Γ θ ) ⊗ n : (( d ⊗ d ) ⊗ n ) → (( d ⊗ d ) ⊗ n ), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n -tensor product of a maximally entangled state τ ME ∊ ( d ⊗ d ). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇ m -affine in d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ ME )} θ of output states can be identified with d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.

Journal of Physics A Mathematical and GeneralJul 9, 2003109 citations

We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the

k

-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most

k

, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension

k^*

required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.