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arXiv:2506.11346 · Physical Review A

Rank-based entanglement detection for bound entangled states

Aabhas Gulati

Published Jan 2, 2026• 1 citation
arXivHTMLPDF

Abstract

We develop a new method for entanglement detection in bipartite quantum states by using the violation of the rank-1-generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of rank 1. But, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank >= 2. The problem of deciding when a matrix is rank-1 generated, i.e, a sum of rank-1 positive-semidefinite (PSD) matrices, has been studied extensively in optimization theory. This rank-1 generated property acts as an entanglement criterion, and we use this property to find novel classes of PPT (Positive under partial transposition) entangled states. We do this by mapping some faces of PPT density matrices to convex cones that are not rank-1 generated. We show that all separable states map to a rank-1 generated matrix. In general, the same is not true for the corresponding mapping of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide witnesses that detect violations of the rank-1 generated property.

DOI
10.1103/4r1h-njb6

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Abstract The problem of bound entanglement detection is a challenging aspect of quantum information theory for higher dimensional systems. Here, we propose an indecomposable positive map for two-qutrit systems, which is shown to detect a class of positive partial transposed (PPT) states. A corresponding witness operator is constructed and shown to be weakly optimal and locally implementable. Further, we perform a structural physical approximation of the indecomposable map to make it a completely positive one, and find a new PPT-entangled state which is not detectable by certain other well-known entanglement detection criteria.
The cone of positive-semidefinite (PSD) matrices is fundamental in convex optimization, and we extend this notion to tensors, defining PSD tensors, which correspond to separable quantum states. We study the convex optimization problem over the PSD tensor cone. While this convex cone admits a smooth reparameterization through tensor factorizations (analogous to the matrix case), it is not self-dual. Moreover, there are currently no efficient algorithms for projecting onto or testing membership in this cone, and the semidefinite tensor optimization problem, although convex, is NP-hard. To address these challenges, we develop methods for computing lower and upper bounds on the optimal value of the problem. We propose a general-purpose iterative refinement algorithm that combines a lifted alternating direction method of multipliers with a cutting-plane approach. This algorithm exploits PSD tensor factorizations to produce heuristic solutions and refine the solutions using cutting planes. Since the method requires a linear minimization oracle over PSD tensors, we design a spatial branch-and-bound algorithm based on convex relaxations and valid inequalities. Our framework allows us to study the white-noise mixing threshold, which characterizes the entanglement properties of quantum states. Numerical experiments on benchmark instances demonstrate the effectiveness of the proposed methods.
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Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays. The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from mmm such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2m = 2m=2 and relax to a semidefinite program of size O(m)O(m)O(m) for arbitrary mmm. The dual solutions are polynomial minorants of x\sqrt{x}x​ on [0,1][0,1][0,1], and two explicit families, binomial and L1L^1L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
It is known that bound entangled states are mixed entangled states from which no pure entanglement can be distilled by local operations and classical communication. This holds true even if large number of identical copies of the state are shared between spatially separated parties. In this work we consider bipartite {\it noisy} bound entangled states with positive partial transpose, that is, such a state can be written as a convex combination of a so-called edge state and a separable state ({\it noise}). Starting from a particular class of unextendible product bases (UPBs) we first show a systematic method to construct low-rank noisy bound entangled states which satisfy the range criterion. Next, we consider different classes of uncompletable product bases (UCPBs) to construct noisy bound entangled states. A subset of these bound entangled states has the property that they satisfy the range criterion. However, UCPBs are important to produce noisy bound entangled states and thus, the problem of constructing UCPBs from different classes of UPBs is also considered in this work. In each case explicit examples are constructed. In fact, present bound entangled states help us to learn about the robustness of edge states. We also discuss about the detection these states.
Positive non-Completely Positive (PnCP) maps are an essential tool to detect entanglement since their characterization is a dual aspect of the separability problem. A recent algorithm proposed by Kelp et al. explains how to generate PnCP maps based on the construction of certain positive non-Sum-of-Squares polynomials. We implement this algorithm in a numerically robust way and propose a working version on GitHub. We theoretically demonstrate that the maps produced by the algorithm are indecomposable, localized on the boundary of the positive cone and show that they are inequivalent with most other known PnCP maps. We numerically investigate their entanglement power, demonstrating notably that they are capable of detecting PPT entangled states that most criteria fail to detect.