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.