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.
Entanglement detection is an important problem in quantum information theory because quantum entanglement is a key resource in quantum information processing. Realignment criterion is a powerful tool for the detection of entangled states in bipartite and multipartite quantum systems. It works well not only for negative-partial-transpose entangled states (NPTESs) but also for positive-partial-transpose entangled states (PPTESs). Since the matrix corresponding to the realignment map is indefinite, the experimental implementation of the map is an obscure task. In this work, first, we approximate the realignment map to a positive map using the method of structural physical approximation, and then we show that the structural physical approximation of the realignment map (SPA-R) is completely positive. Positivity of the constructed map is characterized using moments which can be physically measured. Next, we develop a separability criterion based on our SPA-R map in the form of an inequality and show that the developed criterion not only detect NPTESs but also PPTESs. Further, we show that for a special class of states called Schmidt-symmetric states, the SPA-R separability criteria reduce to the original form of the realignment criteria. We provide some examples to support the results obtained. Moreover, we analyze the error that may occur because of approximating the realignment map.
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.
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.
Graph-theoretic structures play a central role in the description and analysis of quantum systems. In this work, we introduce a new class of quantum states, called
Aα-graph states, which are constructed from either unweighted or weighted graphs by taking the normalised convex combination of the degree matrix
D and the adjacency matrix
AG of a graph
G. The constructed states are different from the standard graph states arising from stabiliser formalism. Our approach is also different from the approach used by Braunstein et al. This class of states depend on a tunable mixing parameter
α∈(0,1]. We first establish the conditions under which the associated operator
ραAG is positive semidefinite and hence represents a valid quantum state. We then derive a positive partial transposition (PPT) condition for
Aα-graph states in terms of graph parameters. This PPT condition involves only the Frobenius norm of the adjacency matrix of the graph, the degrees of the vertices and the total number of vertices. For simple graphs, we obtain the range of the parameter
α for which the
Aα-graph states represent a class of entangled states. We then develop a graph-theoretic formulation of a moments-based entanglement detection criterion, focusing on the recently proposed
p3-PPT criterion, which relies on the second and third moments of the partial transposition. Since the estimation of these moments is experimentally accessible via randomised measurements, swap operations, and machine-learning-based protocols, our approach provides a physically relevant framework for detecting entanglement in structured quantum states derived from graphs. This work bridges graph theory and moments-based entanglement detection, offering a new perspective on the role of combinatorial structures in quantum correlations.
It is crucial to know whether the quantum state generated in the experiment is entangled. In the literature, this topic was studied extensively and researchers proposed different approaches for the detection of a mixed bipartite entangled state in arbitrary dimension. Proceeding in this line of research, we propose three different criteria for the detection of mixed bipartite negative partial transpose (NPT) entangled state in arbitrary dimension. Our criteria are based on the method of structural physical approximation (SPA) of partial transposition (PT). We show that the proposed criteria for the detection of a NPT entangled state can be realized experimentally. Two of the proposed criteria are given in terms of the concurrence of the given state in arbitrary dimension so it is essential to find out the concurrence. Thus, we provide lower and upper bounds of concurrence of the quantum state under investigation in terms of average fidelity of two quantum states, and hence these bounds can be realized experimentally. Moreover, we show how to perform a SPA map on a qutrit-qubit system and then explicitly calculate the matrix elements of the density matrix describing the SPA-PT of the qutrit-qubit system. We then illustrate our criteria for the detection of entanglement by considering a class of qubit-qubit system and a class of qutrit-qubit system.
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.