Quantum state reconstruction from incomplete data

Knowing and guessing, these are two essential epistemological pillars in th e theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be comple tely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) q uantum state when only incomplete data are available (known)? What is the m ost reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address in the a rticle. We present several schemes for a reconstruction of states of quantu m systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states (i.e. density operators or Wigner functions) on incomplete observation levels, w hen just a fraction of system observables are measured (i.e., the mean valu es of these observables are known from the measurement). With the extension of observation levels more reliable estimation of quantum states can be pe rformed. In the limit, when all system observables (i.e., the quorum of obs ervables) are measured, the MaxEnt principle leads to a complete reconstruc tion of quantum states, i.e. quantum states are uniquely determined. We ana lyze the reconstruction via the MaxEnt principle of bosonic systems (e.g. s ingle-mode electromagnetic fields modeled as harmonic oscillators) as well as spin systems. We present results of MaxEnt reconstruction of Wigner func tions of various nonclassical states of light on different observation leve ls. We also present results of numerical simulations which illustrate how t he MaxEnt principle can be efficiently applied for a reconstruction of quan tum states from incomplete tomographic data. (2) When only a finite number of identically prepared systems are measured, then the measured data contain only information about frequencies of appea rances of eigenstates of certain observables. We show that in this case sta tes of quantum systems can be estimated with the help of quantum Bayesian i nference. We analyze the connection between this reconstruction scheme and the reconstruction via the MaxEnt principle in the limit of infinite number of measurements. We discuss how an a priori knowledge about the state whic h is going to be reconstructed can be utilized in the estimation procedure. In particular, we discuss in detail the difference between the reconstruct ion of states which are a priori known to be pure or impure. (3) We show how to construct the optimal generalized measurement of a finit e number of identically prepared quantum systems which results in the estim ation of a quantum stale with the highest fidelity. We show how this optima l measurement can in principle be realized. We analyze two physically inter esting examples-a reconstruction of states of a spin-1/2 and an estimation of phase shifts. (C) 1999 Elsevier Science Ltd. All rights reserved.