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arXiv:2509.21258 · International Journal of Quantum Information

Entanglement distillation on symmetric two–qutrit entangled states of rank five

Zihua Song, Lin Chen, Yongge Wang

Published Dec 30, 2025• 0 citations
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Abstract

Entanglement distillation is a key step in quantum information, both theoretically and practically. It has been proven that nonpositive-partial transpose (NPT) entangled states of rank at most four is 1-distillable under local operation and classical communications. In this paper, we investigate the distillation of a more complex family of NPT entangled states, namely, a family of symmetric two-qutrit states [Formula: see text] of rank five with given eigenvectors. We explicitly construct five families of such states by requiring four of the five eigenvalues to be the same. We, respectively, show that some of them are 1-distillable. It turns out that such states may be not 1-distillable for some interval of eigenvalues. We provide some conditions for eigenvalues that allow [Formula: see text] to be 1-distillable or to be 1-undistillable.

DOI
10.1142/s021974992550039x

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International Journal of Quantum Information•Oct 1, 2017•0 citations
Entanglement distillation is a key step in quantum information, both theoretically and practically. It has been proven that nonpositive-partial transpose (NPT) entangled states of rank at most four is 1-distillable under local operation and classical communications. In this paper, we investigate the distillation of a more complex family of NPT entangled states, namely, a family of symmetric two-qutrit states [Formula: see text] of rank five with given eigenvectors. We explicitly construct five families of such states by requiring four of the five eigenvalues to be the same. We, respectively, show that some of them are 1-distillable. It turns out that such states may be not 1-distillable for some interval of eigenvalues. We provide some conditions for eigenvalues that allow [Formula: see text] to be 1-distillable or to be 1-undistillable.
Entanglement distillation is a fundamental task in quantum information processing. In this work, we investigate the distillability properties of a class of two-qutrit symmetric NPT states of rank five. We resolve the 1-distillability problem for this class by proving that the previously open interval of the eigenvalue parameter is 1-undistillable. For the 2-distillability, we uncover a structural obstruction showing that no Schmidt-rank-two vector has a negative expectation in the relevant subspace. We also perform numerical investigations to explore the 2-distillability beyond this obstruction.
Entanglement distillation is the process of converting noisy entangled states into maximally entangled pure states via local operations and classical communication. A long-standing, unresolved question is which entangled states are amenable to distillation, known as the distillability problem. We solve this for Bell-diagonal qutrits with Weyl structure, and present a noise-robust scheme for entanglement distillation. In particular, we find that violating the positive partial transposition (PPT) criterion is necessary and sufficient for the 1-distillability of these states. For this, we construct a Schmidt rank 2 eigenvector of the partially transposed density matrix associated with its unique, three-fold degenerate negative eigenvalue. This feature makes the derived entanglement distillation protocol resilient to white-noise effects on the quantum states. Our results thus make noisy entangled qutrit pairs more accessible for future quantum technologies.
We show that a bipartite quantum state of rank four is distillable if its partial transpose has at least one negative eigenvalue; that is, the state is a non-positive-partial transpose (NPT). For this purpose we prove that if the partial transpose of a two-qutrit NPT state has at least two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT state of rank five which is not 1-distillable and show that it is not nnn-distillable for any given nnn when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.
Whether entanglement with a negative partial transpose (NPT) can be undistillable is a longstanding open problem. In the canonical family of DiVincenzo \textit{et al.}, the one-copy-undistillable region was conjectured to remain undistillable at all copy numbers. We refute this: in every dimension d≥3d\geq3d≥3, one vertex of this region is two-copy distillable, via an explicit Schmidt-rank-two tight-frame witness. The separation persists on an open set, while other states in the same region are provably two-copy undistillable. Twirling our counterexample into Werner form makes it two-copy undistillable, so this standard reduction can erase finite-copy distillability.
Entanglement distillation is a fundamental task in quantum information theory. In this work, we prove that Werner states in arbitrary dimension are 222-copy distillable if and only if they are 111-copy distillable. This answers the longstanding open question of the 222-copy distillability of Werner states. This is an important step on determining whether every non-positive partial transpose (NPT) state is distillable, which remains one of the central open problems in the field of entanglement distillation.
In [Formula: see text] dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. These rank four states are completely understood. We say that they have rank [Formula: see text] since both a state [Formula: see text] and its partial transpose [Formula: see text] have rank four. The next problem is to understand the extremal PPT states of rank [Formula: see text]. We call two states [Formula: see text]-equivalent if they are related by a product transformation. A generic rank [Formula: see text] PPT state [Formula: see text] is extremal, and both [Formula: see text] and [Formula: see text] have six product vectors in their ranges, and no product vectors in their kernels. The three numbers [Formula: see text] are [Formula: see text]-invariants that help us classify the state. There is no analytical understanding of such states. We have studied numerically a few types of nongeneric rank five PPT states, in particular, states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to [Formula: see text]-equivalence, where they appear as boundary states on one single five-dimensional face on the set of normalized PPT states. The interior of the face consists of rank [Formula: see text] states with four common product vectors in their kernels, it is a simplex of separable states surrounded by entangled PPT states. We say that a state [Formula: see text] is [Formula: see text]-symmetric if [Formula: see text] and [Formula: see text] are [Formula: see text]-equivalent, and is genuinely [Formula: see text]-symmetric if it is [Formula: see text]-equivalent to a state [Formula: see text] with [Formula: see text]. Genuine [Formula: see text]-symmetry implies a special form of [Formula: see text]-symmetry. We have produced numerically, by a special method, a random sample of rank [Formula: see text] [Formula: see text]-symmetric states. About 50 of these are of type [Formula: see text], among those all are extremal and about half are genuinely [Formula: see text]-symmetric. All these genuinely [Formula: see text]-symmetric states can be transformed to have a circulant form. We find however that this is not a generic property of genuinely [Formula: see text]-symmetric states. The remaining [Formula: see text]-symmetric states found in the search have product vectors in their kernels, and they inspired us to study such states without regard to [Formula: see text]-symmetry.