**9. References**

Akhiezer, A. & Berestetskii, V. (1965). *Quantum Electrodynamics*, Interscience.


16 Will-be-set-by-IN-TECH

strong exponential falloff as the function itself, which persists for some (not too long) time, see Fig. 3. By comparing Eqs. (29) and (14) we notice that while in Eq. (14) the argument of the sine function is the product of the distance with the Fourier variable, in Eq. (29) the argument of the Bessel function is the product of the radial distance *ρ* with the radial wavenumber *k<sup>ρ</sup>* the latter depending on the Fourier variable through the square-root expression with the constant parameter *k*0—the lower limit of the integration. As it follows also from Eqs. (31) and (32) the condition *k*<sup>0</sup> 0 is crucial for obtaining the exponential falloff. Hence, in the case of the cylindrical waves considered by us, the apparent violation of the rules set by the Paley-Wiener theorem results from the specific complicated relation between the radial distance and the

The problem of photon localization is of rather fundamental nature in quantum electrodynamics. Despite of almost 80-year history of the problem – and the related problem of the photon wave function—the interest in the revision of it has quickened in the recent years. One of the stimulus for that might be developments in modern optics, particularly in femtosecond and quantum optics, thanks to which the somewhat academic problem is transforming into a practical one. Indeed, e. g., availability and applications of single- and sub-cycle photon pulses will force a revision of traditional notions in optics based on the narrow-band approximation. In particular, phrases like "localization cannot be better than

Ultrawideband by definition are the so-called localized waves—an emerging new field in wave acoustics and physical optics. We have shown that an interdisciplinary "technology transfer"—application of methods and solutions found in the field of localized waves—is

The author is thankful to Iwo Bialynicki-Birula for stimulating hints, numerous discussions, and remarks. The research was supported by the Estonian Science Foundation. Its publication

Alexeev, I., Kim, K. & Milchberg, H. (2002). Measurement of the superluminal group velocity

Belgiorno, F., Cacciatori, S. L., Clerici, M., Gorini, V., Ortenzi, G., Rizzi, L., Rubino, E., Sala,

Besieris, I., Abdel-Rahman, M., Shaarawi, A. & Chatzipetros, A. (1998). Two fundamental

Besieris, I. M., Shaarawi, A. M. & Ziolkowski, R. W. (1994). Nondispersive accelerating wave

V. G. & Faccio, D. (2010). Hawking Radiation from Ultrashort Laser Pulse Filaments,

representations of localized pulse solutions to the scalar wave equation, *Progress In*

wavelength" are loosing sense in the case of such pulses.

was supported by the European Regional Development Fund.

Akhiezer, A. & Berestetskii, V. (1965). *Quantum Electrodynamics*, Interscience.

of an ultrashort Bessel beam pulse, *Phys. Rev. Lett.* 88(7): 073901. Bateman, H. & Erdelyi, A. (1954). *Tables of integral transforms*, McGraw-Hill.

productive for the study of photon localization.

*Phys. Rev. Lett.* 105(20): 203901.

*Electromagnetics Research* 19: 1–48.

packets, *Am. J. Phys.* 62(6): 519–521.

**8. Acknowledgements**

**9. References**

Fourier variable.

**7. Conclusion**


**0**

**4**

*Poland*

**Fusion Frames and Dynamics of Open**

*Institute of Physics, Nicholas Copernicus University, 87–100 Toru ´n*

Heisenberg's uncertainty principle is one of the manifestations of quantum complementarity. In particular, it states that upon measuring both the momentum and the position of a particle, the product of uncertainties has a fundamental lower bound proportional to Planck's constatnt. Hence, one cannot measure position and momentum simultaneously with a prescribed accuracy. In general, the quantum complementarity principle does not permit to identify a quantum state from measurements on a single copy of the system unless some extra

One of the consequences of fundamental assumptions of quantum mechanics is the fact that determination of an unknown state can be achieved by appropriate measurements only if we have at our disposal a set of identically prepared copies of the system in question. Moreover, to devise a successful approach to the above problem of state reconstruction one has to identify a collection of observables, so-called *quorum*, such that their expectation values provide the

The problems of state determination have gained new relevance in recent years, following the realization that quantum systems and their evolutions can perform practical tasks such as teleportation, secure communication or dense coding. It is important to realize that if we identify the quorum of observables, then we also have a possibility to determine expectation values of physical quantities (observables) for which *no* measuring apparatuses are available. Quantum tomography is a procedure of reconstructing the properties of a quantum object on the basis of experimentally accessible data. This means that quantum tomography can be

1. *state tomography* treats density operators, which describe states of quantum systems;

In what follows, we briefly describe the theory of quantum state tomography (cf. e.g. (Nielsen

The aim of quantum state tomography is to identify the density operator characterizing the state of a quantum system under consideration. Let H and S(H) denote the Hilbert space

2. *process tomography* discusses linear trace-preserving completely positive maps;

**1. Introduction**

knowledge is available.

complete information about the system state.

classified by the type of object to be reconstructed:

& Chuang, 2000; Weigert, 2000)).

3. *device tomography* treats quantum instruments, and so on.

**Quantum Systems**

Andrzej Jamiołkowski

