Geometric multiaxial representation of N -qubit mixed symmetric separable states

Abstract

Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2 N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use geometric multiaxial representation(MAR) of density matrix to characterize the mixed symmetric separable states. Since separability problem is NP hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P-distribution function λ(θ, φ). We show that the N-qubit mixed symmetric separable state can be visualized as a uniaxial system if the distribution function is independent of θ and φ. We further choose distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two qubit mixed symmetric separable state. We also observe that there exists a correspondence between separability and classicality of states.Study of separable states is the corner stone of entanglement problem. Bell inequalities were first used for the identification of entanglement. The most operationally convenient criterion for the detection of entanglement is given by Peres-Horodecki and is called positive partial transpose (PPT) criterion [1] which is necessary and sufficient for 2 × 2 and 2 × 3 systems only [2]. There exist other criteria in literature for detecting entanglement. One among them is the realignment criterion [3], which exhibits a powerful PPT entanglement detection capability. Entanglement witness[2, 4] and uncertainty relations [5] pose operational difficulty as they depend on the expectation value of some observables for the state in question. In practice we deal with mixed states rather than pure states due to decoherence effects and hence it is of great importance to study mixed separable states. There exists many important papers [6][7][8][9][10][11][12][13][14] has presented a complete set of 18 local polynomial invariants of two qubit mixed states and demonstrated the usefulness of these invariants to study entanglement. Also, detection of multipartite entanglement has been studied in depth(see for example [18][19][20]). Geometric entanglement properties of pure symmetric N qubit states are studied in detail [21]. To this day, no generally * sumarkr@gmail.com accepted theory exists for the classification and quantification of entanglement for mixed states. A general N qubit mixed state resides in the Hilbert space of dimension 2 N...

Majorana representation for mixed states

“…In order to make the paper self-contained and for a better understanding of the next sections, we explain the T -rep in terms of Majorana constellations. The T -rep has been used recently in Quantum Information [31,48].…”