Local asymptotic normality for qubit states

Abstract

We consider n identically prepared qubits and study the asymptotic properties of the joint state \rho^{\otimes n}. We show that for all individual states \rho situated in a local neighborhood of size 1/\sqrt{n} of a fixed state \rho^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 exist 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 set-up is also optimal within the pointwise 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.Comment: 16 pages, 3 figures, published versio

Local Asymptotic Normality in Quantum Statistics

“…experiment ϕ n θ0+u/ √ n to a quantum Gaussian shift experiment φ u , which is the main result of the paper. This theorem holds for smooth families of states on matrix algebras of arbitrary finite dimension, and it is complementary to the result of [14] concerning strong convergence for qubit states. For pedagogical reasons we first prove the result for a unitary family of states in Section 5.1, which could be seen as a purely quantum experiment, after which we allow the change in eigenvalues leading to the presence of a classical Gaussian component in the limit experiment.…”

“…Among the many applications in mathematical statistics, local asymptotic normality is essential in asymptotic optimality theory and explains the asymptotic normality of certain estimators such as the maximum likelihood estimator. Based on the same principle, the paper [14] shows that a similar phenomenon occurs in quantum statistics: the family of joint states of n identically prepared qubits converges to a family of Gaussian states of a quantum oscillator with unknown displacement. More precisely, there exists a physical transformation (quantum channel) which maps the joint state of the spins into the oscillator state, such that local rotations around a fixed spin direction correspond to displacements of a thermal equilibrium state.…”

“…However, in a quantum theory of experiments the Le Cam distance should play a central role and some encouraging results in this direction exist already. In [14,13] it is shown that the quantum version of the local asymptotic normality with the Le Cam type convergence holds for identically prepared qubits with the limit experiment being a family of displaced thermal equilibrium states. In [15], the problem of optimal cloning of mixed quantum Gaussian states is solved along lines similar to the solution of the classical problem of finding the deficiency between two Gaussian shift experiments.…”

“…In particular this means that there exists a classical statistical decision problem for which the minimax risk of the experiment E is strictly smaller than the minimax risk of the experiment F . An example of such decision problem [14], is that of distinguishing between two states φ u and φ −u with u = 0 for which the optimal measurement is different from the measurement of L.…”

“…This paper together with the closely related works [14,13] extend the results of Hayashi and Matsumoto, and aim at developing quantum statistical analogues of fundamental concepts and tools in asymptotic statistics, such as convergence of statistical experiments and local asymptotic normality. The idea of approximating a sequence statistical models by a family of Gaussian distributions appeared in [54], and was fully developed by Le Cam [28] who introduced the term "local asymptotic normality".…”