A density of the information-theoretic Wehrl entropy in terms of the Hu- simi
Q -function is defined.The entropic density is then applied to describe the quantum properties of some optical fields including Fock states, cohe- rent and squeezed states, superposition of chaotic and coherent states, and Schr\"odinger cat and cat-like states.The entropic density is compared with both the conventional Wehrl entropy and the Husimi phase distribution.It is shown that the entropic density is a good measure of the phase-space uncertainty (noise), and the phase locking or phase bifurcation effects.The advantages of the entropic description of the superposition principle are presented.It is also demonstrated that the entropic density properly de- scribes phase randomization processes, thus can be used as a measure of decoherence.I. INTRODUCTION Quantum entropy is one of the most fundamental concepts of quantum physics [1], parti- cularly useful in quantum information [2].Quantum entropy has been applied, e.g., as a measure of quantum entanglement, qualltum decoherence, photocount statistics, quantum 数理解析研究所講究録 1142 巻 2000 年 1-22 1 optical correlations, purity of states, quantum noise or accessible information in quantum measurement (capacity of quantum channel).Quantum entropy, as a natural generalization of classical Boltzmann entropy, was propo- sed by von Neumann [3].A classical-like entropy associated with quantum fields was intro- duced by Wehrl [4] in terms of the Glauber coherent states and the Husimi
Q -function [5].A rigorous proof that the von Neumann entropy tends to the Wehrl entropy in the clas- sical limit $\hslasharrow 0$ was given by Beretta [6].The quantum von Neumann entropy can be expanded in power series of classical entropies.As was shown explicitly by Pe\v{r}inov\'a et al. [7], the first term of this expansion is the Wehrl entropy.Bu\v{z}ek et al. [8] related the von Neumann entropy with the sampling entropies based on operational approach to a phase-space measurement.They also identified the Wehrl entropy as a particular example of the sampling entropy, when the quantum ruler is represented by coherent states.Other aspects of the relation between classical and quantum-mechanical entropies have also been extensively studied (see, e.g., excellent surveys by Wehrl [9], and Ohya and Petz [1] with references included therein).The Wehrl entropy has been successfully applied in a description of different properties of quantum optical fields.In particular, it has been shown explicitly that the Wehrl entropy is a useful measure of phase-space uncertainty (quantum noise, phase-space localization, wave-packet spreading) [8,10-14], quantum interference [10], decoherence [11,15,16], ioni- zation [14], squeezing [16-18], Schr\"odinger cat formation
[13,19] or, in general, splitting of
Q -function [15].The von Neumann entropy becomes zero for all pure states, thus cannot be used in discriminating them.Paradoxically, the classical Wehrl entropy is more sensitive in di- stinguishing states than the quantum von Neumann entropy, since it is dependent on the choice of pure states.However, there are properties of quantum fields (including phase decoherence or unique description of superposition states), which are not enough precisely described by the conventional Wehrl entropy.Therefore, we propose a new entropic me-
The Wehrl phase distribution is defined as a phase density of the Wehrl classical information entropy. The new measure is applied to describe the quantum phase properties of some optical fields including Fock states, coherent and squeezed states, and superposition of chaotic and coherent fields. The Wehrl phase distribution is compared with both the conventional Wehrl entropy and Husimi phase distribution (the marginal Husimi Q -function). It is shown that the Wehrl phase distribution is a good measure of the phase-space uncertainty (noise), phase locking and phase bifurcation effects. It is also demonstrated that the Wehrl phase distribution properly describes phase randomization processes, and thus can be used in a description of the quantum optical phase.
The generating function, the corresponding integrated intensity distribution, and the photon-counting distribution and its fractional moments are derived for the superposition of coherent and chaotic M-mode fields on the basis of a formalism of arbitrary ordering of field operators in quantum optics. An alternative description is also given based on a recent analysis of the superposition of coherent and chaotic fields in terms of the correlation functions which makes it possible to derive a number of results valid for arbitrary mean occupation numbers per mode from which the model considered follows as a special case. Many earlier results obtained for coherent and chaotic fields and their superposition are included as special cases. The present results generally describe a superposition experiment for an N-mode coherent field (e.g. generated by an N-mode laser operating far above threshold) and an M-mode chaotic field (M>or=N) as well as a field generated by an M-mode laser operating above threshold.
The problem of relating the semiclassical and quantum treatments of statistical states of an optical field is re-examined. The case where the rule of association between functions and operators is that of antinormal ordering is studied in detail. It is shown that the distribution function for each mode corresponding to this case is a continuous bounded function, and is also a boundary value of an entire analytic function of two variables. The nature of the distribution for the normal ordering rule of association and its relation to this entire function are discussed. It is shown that this distribution can be regarded as the limit of a sequence of tempered distributions in the following sense: One can find a sequence of density operators ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}$ which converges in the norm to the density operator $\stackrel{^}{\ensuremath{\rho}}$ of any given field (consisting of a single mode), such that each member of the sequence can be expressed in the form ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}=\ensuremath{\int}{\ensuremath{\varphi}}_{(\ensuremath{\nu})}(z)|z〉〈z|{d}^{2}z$, where ${\ensuremath{\varphi}}_{(\ensuremath{\nu})}$ is a tempered distribution.
Generation of highly non-classical quantum states of light is essential for optical quantum information processing and quantum metrology. Given the lack of sufficiently strong nonlinear interactions between optical fields, the commonly employed optical quantum-state preparation schemes are conditional, based on nonlinearity induced by heralding photon number measurement on a part of a multimode squeezed Gaussian state. Development and optimization of such probabilistic quantum-state engineering schemes represents one of the central challenges in current quantum optics. As technology advances and experiments progress to detection of higher numbers of photons, the maximization of the heralding probability becomes essential to ensure sufficiently high state-preparation rates. Here, we show that for the conditional quantum state preparation schemes based on Gaussian states and photon number measurements the maximization of the heralding probability can be formulated as finding solution to a system of polynomial equations, which offers an efficient way to find the optimal configuration and allows us to apply techniques dedicated specifically to solving such systems of equations. Our approach can seamlessly incorporate bounds on the available single-mode quadrature squeezing, which is highly experimentally relevant. We mainly consider generation of finite superpositions of Fock states but show that the approach can be straightforwardly extended to generation of squeezed superpositions of Fock states. We focus on Gaussian states with vanishing coherent displacements, hence the conditionally generated states have well defined photon number parity. We illustrate our general methodology through examples of generation of single-mode states with two and three heralding modes, and generation of two-mode states with two heralding modes.