Quantum estimation for quantum technology Matteo G A Paris Applied Quantum Mechanics group Dipartimento di Fisica @ UniMI CNISM - Udr Milano IQIS 2008 -- CAMERINO
Quantum estimation for quantum technology (quantum communication, metrology, interferometry, imaging...) Optimal measurements Ultimate bounds to precision parameters characterizing quantum states and operations matrix elements in a given basis specific expectation values some specific quantum feature representing a resource entanglement, entropy,...
Quantum estimation not all observables may be effectively measured optimal estimation of ϱ λ = Γ λ (ϱ 0 ) may depend on ϱ 0 not all the quantities of interest correspond to quantum observables (entanglement, purity, nongaussianity, phase-shift, time interval,...) a (quantum) parameter estimation problem arises
Measurement and estimation direct measurements indirect measurements influence on a different quantity S X estimator measurement + estimator = inference strategy
Cramer - Rao bound (unbiased estimators) variance of unbiased estimators M -> number of measurements F -> Fisher Information Optimal measurement -> maximum Fisher information Optimal processing of data (estimator) -> Bayes, MaxLik (asymptotically efficient)
Let s go quantum (1) probability density symm. log. derivative (SLD) selfadjoint, zero mean Fisher Information
Let s go quantum (2) Fisher vs Quantum Fisher (Braunstein and Caves PRL 1994)
Optimal quantum measurement (1) ultimate bound on precision optimal POVM eigenstates of the SLD optimal quantum measurement: SLD + classical postprocessing (Bayesian, ML)
Optimal quantum measurement (2) are the bounds achievable? feedback assisted measurements one-step adaptive procedure: first estimate of the parameter (e.g. by global optimal POVM) on a small fraction of copies + measurement of SLD on the rest of the copies is entanglement useful? one parameter -> separate measurements multiparameter case -> open problem (Gill and Massar 2000)
General formulas (basis independent) quantum statistical model Lyapunov equation Symmetric logarithmic derivative Quantum Fisher Information
General formulas Family of quantum states Symmetric logarithmic derivative Quantum Fisher Information
Unitary families of quantum states λ ϱ λ = iu λ [G, ϱ 0 ]U λ covariance of SLD L 0 = 2i n,m ϕ m [G, ϱ 0 ] ϕ n ϱ n + ϱ m ϕ n ϕ m QFI is independent on the value of the parameter
Estimation of bilinear couplings by Gaussian probes U = exp{ i θ G} phase-shift G = a a H = 8N(N + 1) A. Monras PRA 2006 single-mode squezing G = 1 2 (a2 + a 2 ) H = 8N 2 + 8N + 2 two-mode squezing G = a b + ab H = 4 (2N + 1) 2 two-mode mixing G = a b + ab H = 4N 2 + 8N R. Gaiba and MGAP arxiv:0802.1682 squeezed vacuum is an universal and optimal quantum probe
parameter-based uncertainty relations pure states (Maccone 2006) mixed states
estimability of a parameter signal-to-noise ratio (single measurement) relative error for a 3 confidence interval (after M measurements) # of meas to achieve a given relative error
estimability of a parameter: the unitary case QFI is independent on the value of the parameter (Any) estimation procedure cannot be efficient for small value of the parameter
A nonunitary example: estimation of loss Master equation absorption propagation in a noisy channel (T=0) using Gaussian probes proportional to the loss parameter itself! A. Monras and M. G. A. Paris PRL 2007
The multiparametric case QFI matrix bound on covariance (not achievable) single parameter (achievable) repametrization
Estimation of entanglement Family of quantum states reparametrization (e.g. negativity) ϱ(ϱ 1, ϱ 2,..., ϱ d ) ϱ(ɛ, ϱ 2,..., ϱ d ) Precision of entanglement estimation Var(ɛ) 1 M (H 1 ) ɛɛ at fixed ϱ 2,..., ϱ d
Estimation of entanglement (see poster by P. Giorda) pure states (Schmidt decomposition) entanglement measure: monotone function of q (negativity, linear and VN entropy) QSNR is vanishing for vanishing entanglement M. G. Genoni, P. Giorda and M. G. A. Paris PRA 2008
Estimation of entanglement different measures (negativity, entropy, distance) and families of states (qubit and CV) QFI is increasing with entanglement QSNR diverges for maximal entanglement Qubit: QSNR is vanishing for vanishing entanglement Estimation of entanglement is inherently inefficient CV: appropriate entanglement measure may achieve efficient estimation M. G. Genoni, P. Giorda and M. G. A. Paris PRA 2008
Geometry of quantum estimation Distances among quantum states Bures metric
Criticality and estimation Phase transitions: strong change in some relevant observable change in the state of the system Quantum phase transitions: strong departure from the initial previous density matrix increase of Bures infinitesimal distance (Zanardi et al. PRL 2007, PRA 2007) Estimation is very effective at critical points (useful in systems with a tuning parameter, eg interacting spins in an external field) Classical: temperature Quantum: coupling constants P. Zanardi, M. G. A. Paris, L. Campos Venuti PRA 2008
Quantum estimation in spin systems one-dimensional Ising model with a transverse field H = J i σ x i σ x i+1 h i σ z i L sites ϱ J = exp{ βh}/z quantum statistical model At zero temperature h (J) = J L (criticality!) H J (J) 1/J 2 QNSR does not depend on J! SLD becomes a more and more nonlocal operator approaching critical point but.. Magnetization meas are nearly optimal for small L (FI QFI) F FJ/HJ J h G J h 1. 0.8 0.6 L. Campos, C. Invernizzi, M. Korbman and M. G. A. Paris PRA 2008 0.4 0.5 1.0 1.5 J
Summary Quantum estimation finds applications in several areas of quantum technology: Optimal quantum estimator in terms of SLD Ultimate bounds to the precision of the estimation of any quantity of interest including non-observables intrinsic estimability of a parameter classical and quantum contributions to uncertainty Current applications: entanglement, purity, entropy inteferometry, coupling constants, fields,... MGAP arxiv:0804.2981
(classical) Bayesian estimators (1) Bayes theorem M indipendent events: a posteriori distribution Bayesian estimators: mean of the a posteriori distribution λ B = max p(λ {x}) λ peak of the a posteriori distribution
(classical) Bayesian estimators (2) Laplace - Bernstein - von Mises theorem Bayes estimator is asymptotically efficient
MaxLik estimation Probability distribution eralizati p(x) Random sample Joint probability of the sample Maxlik estimation take the value of the parameters which maximize the likelihood of the observed data