QUANTUM STATE RECONSTRUCTION

Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most 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 I am interested in. In general we can divide quantum-state reconstruction schemes into three groups:

(1) Complete (deterministic) reconstruction

Providing all system observables (i.e., the quorum) have been precisely measured, then the density operator of a quantum-mechanical system can be completely reconstructed (i.e., the density operator can be uniquely determined based on the available data). In principle, we can consider two different schemes for reconstruction of the density operator (or, equivalently, the Wigner function) of the given quantum-mechanical system. The difference between these two schemes is based on the way in which information about the quantum-mechanical system is obtained. The first type of measurement is such that on each element of the ensemble of the measured states only a single observable is measured (e.g. quantum homodyne tomography ). In the second type of measurement a simultaneous measurement of conjugate observables is assumed (e.g. quantum filtering ).

(2) 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, when just a fraction of system observables are measured (i.e., the mean values of these observables are known from the measurement). With the extention of observation levels more reliable estimation of quantum states can be performed. In the limit, when all system observables (i.e., the quorum of observables) are measured, the MaxEnt principle leads to a complete reconstruction of quantum states, i.e. quantum states are uniquely determined. I have analyzed the reconstruction via the MaxEnt principle of bosonic systems (e.g. single-mode electromagnetic fields modeled as harmonic oscillators) as well as spin systems. Together with my coworkers we have performed numerical simulations which illustrate how the MaxEnt principle can be efficiently applied for a reconstruction of quantum states from incomplete tomographic data. More details on MaxEnt reconstruction of Wigner functions of various nonclassical states of light in different observation levels can be found here.

(3) Quantum Bayesian inference

When only a finite number of identically prepared systems are measured, then the measured data contain only information about frequencies of appearances of eigenstates of certain observables. We have shown that in this case states of quantum systems can be estimated with the help of quantum Bayesian inference. We have analyzed the connection between this reconstruction scheme and the reconstruction via the MaxEnt principle in the limit of infinite number of measurements. We have discussed how an a priori knowledge about the state which is going to be reconstructed can be utilized in the estimation procedure. In particular, we discussed in detail the difference between the reconstruction of states which are a priori known to be pure or impure . We have also shown how to construct the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.

Relevant papers :

V.Buzek, G.Adam, and G.Drobny:

``Reconstruction of Wigner functions on different observation levels.'' Annals of Physics (N.Y.) 245, 37 (1996).

R.Derka, V.Buzek, and A.Ekert:

``Universal algorithm for optimal state estimation from finite ensembles'' Phys. Rev. Lett. 80, 1571 (1998).

V.Buzek, R.Derka, G.Adam, and P.L.Knight:

``Reconstruction of quantum spin states: from quantum Bayesian inference to quantum tomography'' Ann. Phys. (N.Y.) 266, 454 (1998).

V.Buzek G.Drobny, R.Derka, G.Adam, and H.Wiedeman :

``Reconstruction of quantum states from incomplete data'' Chaos, Solitons & Fractals 10, 981--1074 (1999).

V.Buzek, C.Keitel, and P.L.Knight:

``Sampling entropies and operational phase-space measurement I: General formalism'' Phys. Rev. A 51, 2575 (1995).

V.Buzek, C.Keitel, and P.L.Knight:

``Sampling entropies and operational phase-space measurement II: Detection of quantum coherences'' Phys. Rev. A 51, 2594 (1995)

T.Opatrny, V.Buzek, J.Bajer, and G.Drobny:

``Propensities in discrete phase spaces: Q -function of a state in a finite-dimensional Hilbert space.'' Phys. Rev. A 52, 2419 (1995).

V.Buzek, G.Adam, and G.Drobny:

``Reconstruction of Wigner functions on different observation levels.'' Annals of Physics (N.Y.) 245, 37-96 (1996)

T.Opatrny, D.-G.Welsch, and V.Buzek:

``Parameterized discrete phase-space functions'' Phys. Rev. A 53, 3822 (1996).

V.Buzek, G.Drobny, G.Adam, R.Derka, and P.L.Knight:

``Reconstruction of quantum states of spin systems via the Jaynes principle of maximum entropy.'' J. Mod. Optics 44, 2607 (1997)

A.Wuensche and V.Buzek:

``Reconstruction of quantum states from propensities'' Quantum Optics 9, 631 (1997).

V.Buzek:

``Reconstruction of Liouvillian superoperators'' Phys. Rev. A 58, 1723-1727 (1998).

V.Buzek, R.Derka, and S.Massar:

``Optimal quantum clocks'' Phys. Rev. Lett. 82, 2207-2010 (1999).