We propose a continuous variable quantum key distribution protocol based on discrete modulation of eight-state coherent states. We present a rigorous security proof against the collective attacks by taking into consideration the realistic lossy and noisy quantum channel, the imperfect detector efficiency, and the detector electronic noise. This protocol shows high tolerance against excess noise and promises to achieve over 100 km distance of optical fiber.

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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.

Abstract. The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schrödinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q-function. Compact-form formulae for the entropic measures are found for superpositions of well-separated states. Examples of Schrödinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in Kerr medium are analyzed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally-weighted states in respect to their number and phasespace configuration.