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Majorana representation (MR) of symmetric N -qubit pure states has been used successfully in entanglement classi¯cation. Generalization of this has been a long standing open problem due to the di±culties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric N -qubit mixed states based on the geometrical multiaxial representation (MAR) of the density matrix. In addition to the two parameters de¯ned for the entanglement classi¯cation of the symmetric pure states based on MR, namely, diversity degree and degeneracy con¯guration, we show that another parameter called rank needs to be introduced for symmetric mixed state classi¯cation. Our scheme of classi¯cation is more general as it can be applied to both pure and mixed states. To bring out the similarities/ di®erences between the MR and MAR, N -qubit GHZ state is taken up for a detailed study. We conclude that pure state classi¯cation based on MR is not a special case of our classi¯cation scheme based on MAR. We also give a recipe to identify the most general symmetric N -qubit pure separable states. The power of our method is demonstrated using several well-known examples of symmetric two-qubit pure and mixed states as well as three-qubit pure states. Classi¯cation of uniaxial, biaxial and triaxial symmetric two-qubit mixed states which can be produced in the laboratory is studied in detail.

Abstract. The notion of spin squeezing has been discussed in this paper using the density matrix formalism. Extending the definition of squeezing for pure states given by Kitagawa and Ueda in an appropriate manner and employing the spherical tensor representation, we show that mixed spin states which are non-oriented and possess vector polarization indeed exhibit squeezing. We construct a mixed state of a spin 1 system using two spin 1/2 states and study its squeezing behaviour as a function of the individual polarizations of the two spinors. §

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