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arXiv:0711.2613 · IEEE Transactions on Information Theory

A Few Steps More Towards NPT Bound Entanglement

Łukasz Pankowski, Marco Piani, Michał Horodecki, Paweł Horodecki

Published Jul 21, 2010• 85 citations
arXivHTMLPDF

Abstract

In this paper, existence of bound entangled states with nonpositive partial transpose (NPT) is considered. As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover, it would rule out a simple mathematical description of the set of distillable states. The particular state, known to be 1-copy nondistillable and supposed to be bound entangled, is considered. The problem of its two-copy distillability, which boils down to show that maximal overlap of some projectorQwith Schmidt rank two states does not exceed 1/2 (called thehalf-property), is studied. First, it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. Then, the problem in attacked twofold way: (a) the half-property is proved for some wide classes of Schmidt rank two states; (b) the overlap forallSchmidt rank two states is bounded from above bycA⊗I+I⊗BwithA,Btraceless 4 × 4 matrices, and TrAfA+ TrBfB=1/4.

DOI
10.1109/tit.2010.2050810

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In this paper, existence of bound entangled states with nonpositive partial transpose (NPT) is considered. As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover, it would rule out a simple mathematical description of the set of distillable states. The particular state, known to be 1-copy nondistillable and supposed to be bound entangled, is considered. The problem of its two-copy distillability, which boils down to show that maximal overlap of some projectorQwith Schmidt rank two states does not exceed 1/2 (called thehalf-property), is studied. First, it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. Then, the problem in attacked twofold way: (a) the half-property is proved for some wide classes of Schmidt rank two states; (b) the overlap forallSchmidt rank two states is bounded from above bycA⊗I+I⊗BwithA,Btraceless 4 × 4 matrices, and TrAfA+ TrBfB=1/4.
Recently [quant-ph/0608250] again created a lot of interest to prove the existence of bound entangled states with negative partial transpose (NPT) in any d×d(d≥3)d \times d (d \geq 3)d×d(d≥3) Hilbert space. However the proof in quant-ph/0608250 is not complete but it shows some interesting properties of the Schmidt rank two states. In this work we are trying to probe the problem in a different angle considering the work by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. We have assumed that the Schmidt rank two states should satisfy some bounds. Under some assumptions with these bounds one could prove the existence of NPT bound entangled states. We particularly discuss the case of two copy undistillability of the conjectured family of NPT states. Obviously the class of NPT bound entangled states belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. However the problem of existence of NPT bound entangled states still remain open.
We prove the existence of bound entangled states with negative partial transpose (NPT) in any d × d(d ≥ 3) Hilbert space with a simple assumption on Schmidt rank two states. Obviously they belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. PACS number(s): 03.67.Hk, 03.65.Ud. The basic issue on the classification of mixed state entanglement at least on the level of bipartite systems solely depends upon whether there exist bound entangled states or not. The existence of PPT-bound (PPT means positive partial transpose) entangled states [1] and also the existence of NPT N−copy undistillable states [2, 3] for every positive integer N naturally indicates there may exist NPT-bound entangled states. In this work we are able to show the existence of NPT-bound entangled states with a simple assumption on Schmidt rank two states. We first briefly describe the issue and the importance of the problem. In recent years it is found that quantum entanglement is an useful resource in performing several tasks in quantum information theory and quantum communication [4]. Maximally entangled states shared between two parties are essential ingredients in this respect [5]. Now due to the interaction with environment, states are in practice found to be mixed. However there is a process called distillation by which we can distill sometimes maximally entangled
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.
Whether every entangled state with a non-positive partial transpose (NPT) is distillable reduces to the case of the Werner states ρ(d,α) = (I + αF)/(d² + αd), with F the swap, which for d ≥ 3 and −1/2 ≤ α < −1/d are NPT but one-copy undistillable. By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (2000), as rewritten by P. Costa Rico (2025), ρ(d,α) is k-copy undistillable exactly when Σ_J α^|J| ‖Tr_J C‖² ≥ 0 for every vector of Schmidt rank at most two on k copies, written as a matrix C, where Tr_J traces out the copies in J. At α = −1/2 this inequality was proved independently for two copies by Fu et al., Fraser et al., Song and Chen, and Bharti et al. (2026), and for three copies by a computer-assisted proof in the companion record doi:10.5281/zenodo.23604343; if it held for every k, NPT bound entanglement would exist. We prove the inequality at α = −1/2 for every k and d ≥ 3 on the class of matrices C whose range is spanned by a vector f_0 of V_0 and a vector f_1 of V_1 and whose co-range (the range of C*) is arbitrary. Here V_c is spanned by |c⟩^⊗k and the k vectors obtained from it by replacing one factor with |2⟩, for orthonormal |0⟩, |1⟩, |2⟩. So no vector of this kind witnesses distillability of ρ(d,−1/2). We chose this class because, for generic f_0 and f_1, the projector onto the range has reduced states of rank three on every copy, while every known nonzero C with equality at α = −1/2 has a range with a reduced state of rank at most two on some copy. The proof reduces the claim by Schur complements to a quadratic form and then argues by induction on the number of copies. Python scripts in this record check the five rational identities of the proof symbolically; they do not involve k, and no other step uses a computer. We also prove that if the inequality holds for k − 1 copies but fails for k, then for some α > −1/2 it still holds, with equality at a nonzero C whose range has reduced states of rank at least three on every copy, as for generic members of the class. For two copies we prove that every nonzero C with equality at α = −1/2 has rank two and that the ranges of C and C* lie in products of two-dimensional single-copy subspaces. For general C the inequality remains open from four copies on.
The discovery of entangled quantum states from which one cannot distill pure entanglement constitutes a fundamental recent advance in the field of quantum information. Such bipartite bound-entangled (BE) quantum states could fall into two distinct categories: (1) Inseparable states with positive partial transposition (PPT), and (2) states with negative partial transposition (NPT). While the existence of PPT BE states has been confirmed, only one class of conjectured NPT BE states has been discovered so far. We provide explicit constructions of a variety of multicopy undistillable NPT states, and conjecture that they constitute families of NPT BE states. For example, we show that for every pure state of Schmidt rank greater than or equal to 3, one can construct n-copy undistillable NPT states, for any n> 1.n>~1.n> 1. The abundance of such conjectured NPT BE states, we believe, considerably strengthens the notion that being NPT is only a necessary condition for a state to be distillable.