Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits

Abstract

Entanglement is a key feature in many quantum technologies, including secure communication protocols and quantum computing. However, detecting it in mixed quantum states remains a challenging task. While the positive partial transposition (PPT) and computable cross-norm/realignment criteria are well-established tools for entanglement detection in general, and are especially effective in Bell-diagonal states, their connection to the underlying group structure of this state family has not been fully explored. In this work, we analyze the PPT and realignment criteria for Bell-diagonal states from a group-theoretic point of view. Our results demonstrate that the group structure of Bell-diagonal states provides a clear framework for analyzing and computing these two entanglement detection criteria, thereby highlighting the connection between entanglement and group structure. This unified perspective offers new insights into the mathematical and physical properties of entanglement in structured quantum systems and ties the PPT and realignment criteria for Bell-diagonal states to experimental procedures.

Figures (4)

• Figure 1: Percentages of Bell-diagonal states detected by the PPT and realignment criterion for $d=2,\dots, 8$. For each $d$, a total of $10^5$ uniformly sampled states (following a $\mathrm{Dirichlet}(1,\ldots,1)$ distribution on the simplex \ref{['eq:magic_simplex']} below) were generated using the Julia package popp_belldiagonalqudits_2023. For qubits ($d=2$), the PPT criterion is equivalent to the realignment criterion, both detecting around 50% of states as entangled, which are all existing entangled states in this state family. For $d=3$, the realignment criterion is more effective (62.3%) in detecting entanglement than the PPT criterion (60.7%). In higher dimensions, the vast majority of states are NPT entangled. In contrast, the number of entangled states detected by the realignment criterion decreases with increasing dimension $d>3$.
• Figure 2: Discrete phase $\mathbbm{Z}_3^2$ for a Bell-diagonal qutrit state. The vertices are associated with the Bell states $P_{k,l}$. The colored lines represent the cosets $S(0,0)$ for all $S\in\mathcal{S}_3$, corresponding to the subgroup states $\rho_\ell$ with $(0,0)\in\ell \in\mathcal{C}_3$. For more details, see App. \ref{['app:group_structure_M_3']}.
• Figure 3: The striations in $\mathbbm{Z}_3^2$ corresponding to the cosets $\mathcal{C}(S_{0,1})$ (top left), $\mathcal{C}(S_{1,0})$ (top right), $\mathcal{C}(S_{1,1})$ (bottom left), and $\mathcal{C}(S_{1,2})$ (bottom right), given in \ref{['eq:C(S_0,1)']}-\ref{['eq:C(S_1,2)']} of App. \ref{['app:group_structure_M_3']}.
• Figure 4: The state space $\mathcal{M}_2$ of Bell-diagonal qubits, also known as "magic" simplex (orange tetrahedron). The maximally entangled Bell state projectors $P_{k,l}$ are the extreme points of the set, whereas the subgroup states $\rho_{\ell_i}$ are extreme points of the set of separable states (blue octahedron). Note that in this case, the separable states coincide with the convex hull of the six subgroup states (kernel polytope), which is generally not true for $d\geq3$ (cf. Ref. popp_comparing_2023).

Theorems & Definitions (7)

• Theorem 2.1: PPT Criterion horodecki_separability_1996peres_separability_1996
• Theorem 2.2: Realignment Criterion chen_matrix_2003rudolph_further_2005
• Theorem 4.1
• proof : Proof of Thm. \ref{['thm:realignment_BDS']}
• Theorem 5.1
• proof
• Proposition 5.1