30 research outputs found
Optimal estimation of pure states with displaced-null measurements
We revisit the problem of estimating an unknown parameter of a pure quantum
state, and investigate `null-measurement' strategies in which the experimenter
aims to measure in a basis that contains a vector close to the true system
state. Such strategies are known to approach the quantum Fisher information for
models where the quantum Cram\'{e}r-Rao bound is achievable but a detailed
adaptive strategy for achieving the bound in the multi-copy setting has been
lacking. We first show that the following naive null-measurement implementation
fails to attain even the standard estimation scaling: estimate the parameter on
a small sub-sample, and apply the null-measurement corresponding to the
estimated value on the rest of the systems. This is due to non-identifiability
issues specific to null-measurements, which arise when the true and reference
parameters are close to each other. To avoid this, we propose the alternative
displaced-null measurement strategy in which the reference parameter is altered
by a small amount which is sufficient to ensure parameter identifiability. We
use this strategy to devise asymptotically optimal measurements for models
where the quantum Cram\'{e}r-Rao bound is achievable. More generally, we extend
the method to arbitrary multi-parameter models and prove the asymptotic
achievability of the the Holevo bound. An important tool in our analysis is the
theory of quantum local asymptotic normality which provides a clear intuition
about the design of the proposed estimators, and shows that they have
asymptotically normal distributions.Comment: Comments or suggestions are more than welcom
Minimax estimation of qubit states with Bures risk
The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/n. However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1/√n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states.
We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O(n-1 log n) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the 'standard' rate n −1
Asymptotically optimal purification and dilution of mixed qubit and Gaussian states
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, reduces the level of
purity present in the constituent states. In this paper we find asymptotically
optimal procedures for purification and dilution of an ensemble of i.i.d. 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 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.Comment: 11 pages, 6 figure
Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise
Quantum technology is increasingly relying on specialised statistical inference methods for analysing quantum measurement data. This motivates the development of “quantum statistics”, a field that is shaping up at the overlap of quantum physics and “classical” statistics. One of the less investigated topics to date is that of statistical inference for infinite dimensional quantum systems, which can be seen as quantum counterpart of non-parametric statistics. In this paper we analyse the asymptotic theory of quantum statistical models consisting of ensembles of quantum systems which are identically prepared in a pure state. In the limit of large ensembles we establish the local asymptotic equivalence (LAE) of this i.i.d. model to a quantum Gaussian white noise model. We use the LAE result in order to establish minimax rates for the estimation of pure states belonging to Hermite-Sobolev classes of wave functions. Moreover, for quadratic functional estimation of the same states we note an elbow effect in the rates, whereas for testing a pure state a sharp parametric rate is attained over the nonparametric Hermite-Sobolev class
Optimal cloning of mixed Gaussian states
We construct the optimal 1 to 2 cloning transformation for the family of
displaced thermal equilibrium states of a harmonic oscillator, with a fixed and
known temperature. The transformation is Gaussian and it is optimal with
respect to the figure of merit based on the joint output state and norm
distance. The proof of the result is based on the equivalence between the
optimal cloning problem and that of optimal amplification of Gaussian states
which is then reduced to an optimization problem for diagonal states of a
quantum oscillator. A key concept in finding the optimum is that of stochastic
ordering which plays a similar role in the purely classical problem of Gaussian
cloning. The result is then extended to the case of n to m cloning of mixed
Gaussian states.Comment: 8 pages, 1 figure; proof of general form of covariant amplifiers
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Concentration Inequalities for Output Statistics of Quantum Markov Processes
We derive new concentration bounds for time averages of measurement outcomes in quantum Markov processes. This generalizes well-known bounds for classical Markov chains, which provide constraints on finite-time fluctuations of time-additive quantities around their averages. We employ spectral, perturbation and martingale techniques, together with non-commutative L2 theory, to derive: (i) a Bernstein-type concentration bound for time averages of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type concentration bound for the same process, (iii) a generalization of the Bernstein-type concentration bound for counting processes of continuous-time quantum Markov processes, (iv) new concentration bounds for empirical fluxes of classical Markov chains which broaden the range of applicability of the corresponding classical bounds beyond empirical averages. We also suggest potential application of our results to parameter estimation and consider extensions to reducible quantum channels, multi-time statistics and time-dependent measurements, and comment on the connection to so-called thermodynamic uncertainty relations
Fisher informations and local asymptotic normality for continuous-time quantum Markov processes
We consider the problem of estimating an arbitrary dynamical parameter of an open quantum system in the input–output formalism. For irreducible Markov processes, we show that in the limit of large times the system-output state can be approximated by a quantum Gaussian state whose mean is proportional to the unknown parameter. This approximation holds locally in a neighbourhood of size in the parameter space, and provides an explicit expression of the asymptotic quantum Fisher information in terms of the Markov generator. Furthermore we show that additive statistics of the counting and homodyne measurements also satisfy local asymptotic normality and we compute the corresponding classical Fisher informations. The general theory is illustrated with the examples of a two-level system and the atom maser. Our results contribute towards a better understanding of the statistical and probabilistic properties of the output process, with relevance for quantum control engineering, and the theory of non-equilibrium quantum open systems
Dynamical phase transitions as a resource for quantum enhanced metrology
We consider the general problem of estimating an unknown control parameter of an open quantum system. We establish a direct relation between the evolution of both system and environment and the precision with which the parameter can be estimated. We show that when the open quantum system undergoes a first-order dynamical phase transition the quantum Fisher information (QFI), which gives the upper bound on the achievable precision of any measurement of the system and environment, becomes quadratic in observation time (cf. “Heisenberg scaling”). In fact, the QFI is identical to the variance of the dynamical observable that characterizes the phases that coexist at the transition, and enhanced scaling is a consequence of the divergence of the variance of this observable at the transition point. This identification makes it possible to establish the finite time scaling of the QFI. Near the transition the QFI is quadratic in time for times shorter than the correlation time of the dynamics. In the regime of enhanced scaling the optimal measurement whose precision is given by the QFI involves measuring both system and output. As a particular realization of these ideas, we describe a theoretical scheme for quantum enhanced estimation of an optical phase shift using the photons being emitted from a quantum system near the coexistence of dynamical phases with distinct photon emission rates
Fisher information and asymptotic normality in system identification for quantum Markov chains
This paper deals with the problem of estimating the coupling constant
of a mixing quantum Markov chain. For a repeated measurement on the
chain's output we show that the outcomes' time average has an asymptotically
normal (Gaussian) distribution, and we give the explicit expressions of its
mean and variance. In particular we obtain a simple estimator of whose
classical Fisher information can be optimized over different choices of
measured observables. We then show that the quantum state of the output
together with the system, is itself asymptotically Gaussian and compute its
quantum Fisher information which sets an absolute bound to the estimation
error. The classical and quantum Fisher informations are compared in a simple
example. In the vicinity of we find that the quantum Fisher
information has a quadratic rather than linear scaling in output size, and
asymptotically the Fisher information is localised in the system, while the
output is independent of the parameter.Comment: 10 pages, 2 figures. final versio