We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension [Formula: see text], a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with special algebraic and geometric structures and applications in quantum information processing tasks in different dimensions. Extending analytical and numerical methods and results for Bell diagonal qutrits ([Formula: see text]), we successfully classify more than [Formula: see text] of representative Bell diagonal PPT states for [Formula: see text]. Via those representative states we are able to estimate the volumes of separable and bound entangled states among PPT ququarts ([Formula: see text]). We find that at least [Formula: see text] of all PPT states are separable, [Formula: see text] bound entangled and for [Formula: see text] it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities for different dimensions and relate those to differences of the Euclidean geometry for qutrits ([Formula: see text]) and ququarts ([Formula: see text]). Finally, using a detailed visual analysis of the set of separable and bound entangled Bell diagonal states in both dimensions, qualitative observations are made that allow to better distinguish bound entangled from separable states.
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension [Formula: see text], a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with special algebraic and geometric structures and applications in quantum information processing tasks in different dimensions. Extending analytical and numerical methods and results for Bell diagonal qutrits ([Formula: see text]), we successfully classify more than [Formula: see text] of representative Bell diagonal PPT states for [Formula: see text]. Via those representative states we are able to estimate the volumes of separable and bound entangled states among PPT ququarts ([Formula: see text]). We find that at least [Formula: see text] of all PPT states are separable, [Formula: see text] bound entangled and for [Formula: see text] it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities for different dimensions and relate those to differences of the Euclidean geometry for qutrits ([Formula: see text]) and ququarts ([Formula: see text]). Finally, using a detailed visual analysis of the set of separable and bound entangled Bell diagonal states in both dimensions, qualitative observations are made that allow to better distinguish bound entangled from separable states.
With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states [Formula: see text] are determined to be separable while [Formula: see text] are bound entangled and only [Formula: see text] remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.
Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find [Formula: see text] to be free entangled, [Formula: see text] to be certainly separable and as much as [Formula: see text] to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining [Formula: see text] of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong two-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.
We characterize bound entanglement in bipartite qutrit systems by combining the Positive Partial Transpose (PPT) criterion with a structural, non-completely positive map. Focusing on a comprehensive three-parameter bipartite qutrit family ρa,b,c, we present an exact analytical and geometric partitioning of the physical state space simplex. We derive the closed-form quadratic boundary governing the leaf-like PPT region (a2+ab+b2−b≤0) over the domain 0<a≤31, and identify the exact linear threshold (a>c) that characterizes the PPT bound entangled sub-region. This family strictly generalizes the well-known Horodecki bound entangled states, which appear as a single slice of the three-dimensional leaf. Applying the structural map to this decomposition, we show that it certifies bound entanglement throughout the entire threshold region, accounting for 14.76% of the total PPT leaf area.
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 and 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 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.
Abstract Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres–Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes’ rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit-qutrit, and qubit-four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit–qutrit states. Based on this numerical approach with the help of the Clauser–Horne–Shimony–Holt inequality and the Collins–Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.
Absolutely positive partial transpose (APPT) states, which remain PPT under every global unitary, provide a spectral relaxation of absolutely separable states, which remain separable under every such transformation. Despite substantial progress, whether these two notions coincide beyond qubit-qudit systems remains unresolved, motivating the study of structured state families in which the APPT spectral constraints and separability can be analyzed exactly. We study this problem for Bell-diagonal states in two settings: the standard two-qutrit Weyl-Heisenberg basis and the two-ququart lattice basis generated by two-qubit Pauli operators. In the two-qutrit case, we show that a linearized necessary condition for APPT defines a polytope of spectra in which every associated Weyl-Heisenberg Bell-diagonal state admits an exact decomposition into explicitly separable states. This yields a separability criterion that applies to a region even larger than the APPT set itself. In the two-ququart lattice Bell-diagonal setting, we show that a state remains PPT under every permutation of its Bell coefficients if and only if its six largest Bell coefficients sum to at most 1/2, and we prove that all such states are separable. Equivalently, every entangled lattice Bell-diagonal state becomes NPT after a suitable coefficient permutation and is therefore not APPT. The proofs combine convex-polytope analysis, affine and symplectic orbit reductions, and exact rational decompositions into separable states supported on affine Lagrangian subsets. Our results rule out entangled APPT states in both Bell-diagonal families, and provide simple spectral inequalities as sufficient criteria for separability in both cases. Being basis-specific, however, they do not by themselves settle whether APPT and absolute separability coincide.