We consider $n$ identically prepared qubits and study the asymptotic properties of the joint state ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}n}$. We show that for all individual states, $\ensuremath{\rho}$ situated in a local neighborhood of size $1∕\sqrt{n}$ of a fixed state ${\ensuremath{\rho}}^{\mathbf{0}}$, the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exists physical transformations ${T}_{n}$ (trace preserving quantum channels) which map the qubits states asymptotically close to their corresponding oscillator state, uniformly over all states in the local neighborhood. A few consequences of the main result are derived. We show that the optimal joint measurement in the Bayesian setup is also optimal within the point-wise approach. Moreover, this measurement converges to the heterodyne measurement which is the optimal joint measurement of position and momentum for the quantum oscillator. A problem of local state discrimination is solved using local asymptotic normality.We consider $n$ identically prepared qubits and study the asymptotic properties of the joint state ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}n}$. We show that for all individual states, $\ensuremath{\rho}$ situated in a local neighborhood of size $1∕\sqrt{n}$ of a fixed state ${\ensuremath{\rho}}^{\mathbf{0}}$, the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exists physical transformations ${T}_{n}$ (trace preserving quantum channels) which map the qubits states asymptotically close to their corresponding oscillator state, uniformly over all states in the local neighborhood. A few consequences of the main result are derived. We show that the optimal joint measurement in the Bayesian setup is also optimal within the point-wise approach. Moreover, this measurement converges to the heterodyne measurement which is the optimal joint measurement of position and momentum for the quantum oscillator. A problem of local state discrimination is solved using local asymptotic normality.

arXiv:quant-ph/0512075Physical Review AMay 17, 200686 citations

Asymptotically optimal purification and dilution of mixed qubit and Gaussian states

Peter Bowles, Mădălin Guţǎ, Gerardo Adesso

Given an ensemble of mixed qubit states, it is possible to increase the purity of the constituent states using a procedure known as state purification. The reverse operation, which we refer to as dilution, produces a larger ensemble, while reducing the purity level of the systems. In this paper we find asymptotically optimal procedures for purification and dilution of an ensemble of independently and identically distributed mixed qubit states, for some given input and output purities and an asymptotic output rate. Our solution involves using the statistical tool of local asymptotic normality, which recasts the qubit problem in terms of attenuation and amplification of a single-mode displaced Gaussian state. Therefore, to obtain the qubit solutions, we must first solve the analogous problems in the Gaussian setup. We provide full solutions to all of the above for the (global) trace-norm figure of merit.Given an ensemble of mixed qubit states, it is possible to increase the purity of the constituent states using a procedure known as state purification. The reverse operation, which we refer to as dilution, produces a larger ensemble, while reducing the purity level of the systems. In this paper we find asymptotically optimal procedures for purification and dilution of an ensemble of independently and identically distributed mixed qubit states, for some given input and output purities and an asymptotic output rate. Our solution involves using the statistical tool of local asymptotic normality, which recasts the qubit problem in terms of attenuation and amplification of a single-mode displaced Gaussian state. Therefore, to obtain the qubit solutions, we must first solve the analogous problems in the Gaussian setup. We provide full solutions to all of the above for the (global) trace-norm figure of merit.

arXiv:1102.2341Physical Review AAug 16, 20117 citations

Nonlocality of cluster states of qubits (5 pages)

Valerio Scarani

Physical Review AJan 1, 20050 citations

Attaining the Ultimate Precision Limit in Quantum State Estimation

Yuxiang Yang, Giulio Chiribella, Masahito Hayashi

We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator’s deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a useful asymptotic characterization of identically prepared states in terms of Gaussian states. We first prove our results for the mean square error of a special class of models, called D -invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, i.e. parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator’s deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a useful asymptotic characterization of identically prepared states in terms of Gaussian states. We first prove our results for the mean square error of a special class of models, called D -invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, i.e. parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.

arXiv:1802.07587Communications in Mathematical PhysicsApr 26, 2019117 citations

We consider

n

identically prepared qubits and study the asymptotic properties of the joint state ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}n}$. We show that for all individual states, $\ensuremath{\rho}$ situated in a local neighborhood of size

1∕\sqrt{n}

of a fixed state ${\ensuremath{\rho}}^{\mathbf{0}}$, the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exists physical transformations

{T}_{n}

(trace preserving quantum channels) which map the qubits states asymptotically close to their corresponding oscillator state, uniformly over all states in the local neighborhood. A few consequences of the main result are derived. We show that the optimal joint measurement in the Bayesian setup is also optimal within the point-wise approach. Moreover, this measurement converges to the heterodyne measurement which is the optimal joint measurement of position and momentum for the quantum oscillator. A problem of local state discrimination is solved using local asymptotic normality.