Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0609218 · Journal of Physics A Mathematical and General

N-body-extended channel estimation for low-noise parameters

M Hotta, T Karasawa, M Ozawa

Published Nov 1, 2006• 15 citations
arXivHTMLPDF

Abstract

The notion of low-noise channels was recently proposed and analyzed in detail in order to describe noise-processes driven by environment [M. Hotta, T. Karasawa and M. Ozawa, Phys. Rev. A72, 052334 (2005)]. An estimation theory of low-noise parameters of channels has also been developed. In this report, we address the low-noise parameter estimation problem for the $N$-body extension of low-noise channels. We perturbatively calculate the Fisher information of the output states in order to evaluate the lower-bound of the mean-square error of the parameter estimation. We show that the maximum of the Fisher information over all input states can be attained by a factorized input state in the leading order of the low-noise parameter. Thus, to achieve optimal estimation, it is not necessary for there to be entanglement of the $N$ subsystems, as long as the true low-noise parameter is sufficiently small.

DOI
10.1088/0305-4470/39/46/015

Related papers

Ancilla-assisted enhancement of channel estimation for low-noise parameters

Masahiro Hotta, Tokishiro Karasawa, Masanao Ozawa

In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this ``ancilla-assisted enhancement'' for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by $3∕2$. Some conditions for the attainability are also given with illustrative examples.

arXiv:quant-ph/0507055•Physical Review A•Nov 29, 2005•33 citations

Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information Matrix

Hyukgun Kwon, Seok Hyung Lie, Liang Jiang

We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.

quant-pharXiv:2602.21510•PRX Quantum•Feb 26, 2026•0 citations

Channel estimation theory of low-noise multiple parameters: Attainability problem of the Cramér-Rao bounds

Masahiro Hotta, Tokishiro Karasawa

We consider the achievability problem of the Cram\'er-Rao bound for multiparameter estimations. In general, it is not achievable due to the noncommutativity of optimal measurements for the corresponding parameters. However, we show that, under certain conditions, it can always be attained up to the leading order in the parameters as long as $D\ensuremath{\leqslant}N\ensuremath{-}1$, where $D$ and $N$ denote the number of parameters and the dimension of the system, respectively. After proving that, we discuss the achievability in the context of channel estimation for a general channel called a low-noise channel, which is very useful for investigating parameter estimation in the leading order. This allows us to find an ancilla-assisted enhancement effect: if entangled input states with an ancilla system are utilized for the channel estimation together with collective measurements on those output states, the bound becomes achievable for $D\ensuremath{\leqslant}{N}^{2}\ensuremath{-}1$.

arXiv:0803.0613•Physical Review A•Jul 18, 2008•3 citations

Estimation of a generalized amplitude-damping channel

Akio Fujiwara

The problem of finding the optimal strategy for estimating a generalized amplitude damping channel ${\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ by means of the extension $\mathrm{id}\ensuremath{\bigotimes}{\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ is addressed. We first evaluate the quantum Fisher information of output states based on the symmetric logarithmic derivative and specify all pure-state inputs that maximize the quantum Fisher information. We next investigate the ${\ensuremath{\nabla}}^{e}$ autoparallelity of output state manifolds and characterize the condition for the existence of an efficient estimator. A comparison of these results concludes that, while there is no uniformly optimal input for all $p$ and $\ensuremath{\eta}$, a maximally entangled input is an admissible one under a nonasymptotic setting.

Physical Review A•Jul 29, 2004•66 citations

Geometric Rényi divergence and RLD Fisher information under quantum processes without definite causal order

Zixin Huang, Mark M. Wilde

Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in $(0,1)\cup(1,2]$ and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.

quant-pharXiv:2610.39217•Oct 30, 2026•0 citations

Entanglement-assisted quantum parameter estimation from a noisy qubit pair: A Fisher information analysis

François Chapeau‐Blondeau

Physics Letters A•Feb 28, 2017•31 citations
The notion of low-noise channels was recently proposed and analyzed in detail in order to describe noise-processes driven by environment [M. Hotta, T. Karasawa and M. Ozawa, Phys. Rev. A72, 052334 (2005)]. An estimation theory of low-noise parameters of channels has also been developed. In this report, we address the low-noise parameter estimation problem for the NNN-body extension of low-noise channels. We perturbatively calculate the Fisher information of the output states in order to evaluate the lower-bound of the mean-square error of the parameter estimation. We show that the maximum of the Fisher information over all input states can be attained by a factorized input state in the leading order of the low-noise parameter. Thus, to achieve optimal estimation, it is not necessary for there to be entanglement of the NNN subsystems, as long as the true low-noise parameter is sufficiently small.
In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this ``ancilla-assisted enhancement'' for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by 3∕23∕23∕2. Some conditions for the attainability are also given with illustrative examples.
We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.
We consider the achievability problem of the Cram\'er-Rao bound for multiparameter estimations. In general, it is not achievable due to the noncommutativity of optimal measurements for the corresponding parameters. However, we show that, under certain conditions, it can always be attained up to the leading order in the parameters as long as $D\ensuremath{\leqslant}N\ensuremath{-}1$, where DDD and NNN denote the number of parameters and the dimension of the system, respectively. After proving that, we discuss the achievability in the context of channel estimation for a general channel called a low-noise channel, which is very useful for investigating parameter estimation in the leading order. This allows us to find an ancilla-assisted enhancement effect: if entangled input states with an ancilla system are utilized for the channel estimation together with collective measurements on those output states, the bound becomes achievable for $D\ensuremath{\leqslant}{N}^{2}\ensuremath{-}1$.
The problem of finding the optimal strategy for estimating a generalized amplitude damping channel ${\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ by means of the extension $\mathrm{id}\ensuremath{\bigotimes}{\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ is addressed. We first evaluate the quantum Fisher information of output states based on the symmetric logarithmic derivative and specify all pure-state inputs that maximize the quantum Fisher information. We next investigate the ${\ensuremath{\nabla}}^{e}$ autoparallelity of output state manifolds and characterize the condition for the existence of an efficient estimator. A comparison of these results concludes that, while there is no uniformly optimal input for all ppp and $\ensuremath{\eta}$, a maximally entangled input is an admissible one under a nonasymptotic setting.
Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in (0,1)∪(1,2](0,1)\cup(1,2](0,1)∪(1,2] and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.