Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering

TL;DR

This paper analyzes the geometric properties of quantum steering ellipsoids for pure symmetric multiqubit states with two distinct spinors, revealing their relation to SLOCC classes and volume monogamy relations.

Contribution

It introduces a geometric visualization of SLOCC classes of symmetric multiqubit states via canonical steering ellipsoids and characterizes their shapes and centers based on state degeneracy.

Findings

W-class states correspond to oblate spheroids centered at specific points.

Other SLOCC classes correspond to ellipsoids centered at the origin.

Volume monogamy relations are explored for these classes.

Abstract

Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and helps in obtaining correlation properties of the state through the steering ellipsoid. The steering ellipsoids corresponding to the two-qubit subsystems of permutation symmetric $N$-qubit states is analysed here. The steering ellipsoids of two-qubit states that have undergone local operations on both the qubits so as to bring the state to its canonical form are the so-called canonical steering ellipsoids. We construct and analyze the geometric features of the canonical steering ellipsoids corresponding to pure permutation symmetric $N$-qubit states with two distinct spinors. Depending on the degeneracy of the two spinors in the pure symmetric $N$-qubit state, there arise several families which cannot be converted into one another through Stochastic Local Operations and Classical Communications (SLOCC). The canonical steering ellipsoids of the two-qubit states drawn from the pure symmetric $N$-qubit states with two distinct spinors allow for a geometric visualization of the SLOCC-inequivalent class of states. We show that the states belonging to the W-class correspond to oblate spheroid centered at $(0,0,1/(N-1))$ with fixed semiaxes lengths $1/\sqrt{N-1}$ and $1/(N-1)$. The states belonging to all other SLOCC inequivalent families correspond to ellipsoids centered at the origin of the Bloch sphere. We also explore volume monogamy relations of states belonging to these families, mainly the W-class of states.