Center-of-mass interpretation for bipartite purity analysis of $N$-party entanglement

We provide a graphical description of the entanglement of pure-state multiparty systems based on an analogy between a bipartite purity analysis and the centroid of a collection of point masses. This description applies to quantum systems with $N$ parties, each with an arbitrary number of (discrete) states. The case of $N$ qubits is highlighted for simplicity. This geometric description illustrates some of the restrictions in the form of inequalities that apply to entanglement in multiparty systems.

Figure 1

Definition of $Q$ as the distance between the origin and the center of mass of $M$ point masses of magnitudes ${\lambda }_m$ at unit distances to the origin and mutually equidistant, within a Euclidean space of dimension $M-1$. (a) For $M=2$ the space is a line, which also contains the center of mass. (b) For $M=3$ the space is a plane, and the center of mass is constrained to the interior of a triangle (gray), whose corners are the three masses. In fact, since ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge 0$, the center of mass is within the darker region. (c) For $M=4$, the space is a volume, and the center of mass is constrained to the interior of a tetrahedron whose corners are the four masses.

Figure 2

(a) For $N$ qubits, the $N$-vector $\mathbf{Q}$ corresponds to the center of mass of a set of $2^N$ point masses of magnitude $|{\overline{c}}_i{|}^2$, placed at the corners of a hypercube of side 2 centered at the origin. For the $N=2$ case shown here, the hypercube reduces to a square. (b) The same construction for a two-party system composed of a qubit and a qutrit, whose interpretation requires three dimensions. In both cases, the entanglement measures $Y_n=1-Q_n$ are also shown.

Figure 3

Boundaries between the allowed and forbidden regions for three-party systems, dictated by (a) relation (11) for three qubits, and (b) relations (13) for two qubits and a qutetrit. The allowed regions are those on the side of the concavity formed by the three surfaces. The movie associated with (b) shows how these bounds are respected for one million randomly generated states.

Figure 4

The partial centers of mass $\mathbf{v}$ and $\mathbf{V}$ are each constrained to a tetrahedral volume whose corners are the masses in question. Since $\mathbf{v}$ and $\mathbf{V}$ are collinear with the origin, $\mathbf{Q}$ is contained in the union of the tetrahedra.