Characterizing generalized axisymmetric quantum states in $d\times d$ systems

We introduce a family of highly symmetric bipartite quantum states in arbitrary dimensions. It consists of all states that are invariant under local phase rotations and local cyclic permutations of the basis. We solve the separability problem for a subspace of these states and show that a sizable part of the family is bound entangled. We also calculate some of the Schmidt numbers for the family in $d=3$, thereby characterizing the dimensionality of entanglement. Our results allow us to estimate entanglement properties of arbitrary states, as general states can be symmetrized to the considered family by local operations.

Figure 1

Schematic view of the state space of $3\times 3$ systems: The sets of states with Schmidt number less than or equal to $K$ are convex. The conjecture from Ref. [13] that for dimension $d=3$ there are no PPT states with Schmidt number 3 was proven in Ref. [15]. Here $S_1$ is the set of separable states.

Figure 2

Facet for $d=4$. If $x_2=x_3=x_4$ we arrive at an edge from the axisymmetric states from Ref. [49]. We will prove that all separable states lie in the purple (dark gray) polytope.

Figure 3

Cross section of the facet in $d=4$ in $x_3=\frac{1}{4d}(1-z)$ and $x_{2,4}=\frac{3}{4d}(1-z)\frac{1\pm \overline{r}}{2}$, where $z$ and $\overline{r}$ are fidelity parameters analogous to those defined in Eqs. (58, 59, 60).

Figure 4

Progress in calculating the Schmidt numbers $\mathcal{S}$ of the states in the facet for $d=3$. While witnesses can give a lower bound on the Schmidt number, showing that a state with nonfull Schmidt rank is twirled to a state in the facet can give an upper bound.

Figure 5

Calculation of the Schmidt numbers $\mathcal{S}$ of the states in the facet for $d=3$: (a) states with at least Schmidt number 1 (blue, bottom), 2 (green, middle), and 3 (red, top) ad (b) the blue states in the region at the bottom are 2-unfaithful, while the green ones in the middle are 3-unfaithful. The parametrization is analogous to that in [24]. For $500\times 50$ states we calculated (a) SDP1 and (b) SDP2 from Ref. [58].