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arXiv:2606.16265

Bi-qutrit entangled edge states of positive partial transposes with largest ranks

Kyung Hoon Han, Seung-Hyeok Kye

quant-phSubmitted Jun 15, 2026• Announced Jun 16, 2026• 0 citations
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Abstract

Whenever $E$ is an eight dimensional subspace of the bi-qutrit quantum system whose orthogonal complement is spanned by a vector of Schmidt rank three, we show that there exist PPT entangled edge states with the range space $E$ whose partial transposes are of rank six, which is the largest possible rank. In this way, we exhibit a huge family of bi-qutrit PPT entangled edge states of type $(8,6)$. They make faces of the convex set of all PPT states, and we find bi-qutrit PPT entangled edge states of other types on the boundaries of such faces.

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16 pages

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Whenever EEE is an eight dimensional subspace of the bi-qutrit quantum system whose orthogonal complement is spanned by a vector of Schmidt rank three, we show that there exist PPT entangled edge states with the range space EEE whose partial transposes are of rank six, which is the largest possible rank. In this way, we exhibit a huge family of bi-qutrit PPT entangled edge states of type (8,6)(8,6)(8,6). They make faces of the convex set of all PPT states, and we find bi-qutrit PPT entangled edge states of other types on the boundaries of such faces.
We show that the length of a qubit-qutrit separable state is equal to max(r,s)\mathrm{max}(r,s)max(r,s), where rrr is the rank of the state and sss the rank of its partial transpose. We refer to the ordered pair (r,s)(r,s)(r,s) as the birank of this state. We also construct examples of qubit-qutrit separable states of any feasible birank (r,s)(r,s)(r,s). We determine the closure of the set of normalized two-qutrit entangled states having positive partial transpose (PPT) of rank 4. The boundary of this set consists of all separable states of length at most 4. We prove that the length of any qubit-qudit separable state of birank (d+1,d+1)(d+1,d+1)(d+1,d+1) is equal to d+1d+1d+1. We also show that all qubit-qudit PPT entangled states of birank (d+1,d+1)(d+1,d+1)(d+1,d+1) can be built in a simple way from edge states. If VVV is a subspace of dimension k<dk<dk<d in a $2\ensuremath{\bigotimes}d$ space such that VVV contains no product vectors, we show that the set of all product vectors in ${V}^{\ensuremath{\perp}}$ is a vector bundle of rank $d\ensuremath{-}k$ over the projective line. Finally, we explicitly construct examples of qubit-qudit PPT states (both separable and entangled) of any feasible birank.
We construct faces of the convex set of all $2\ensuremath{\bigotimes}4$ bipartite separable states, which are affinely isomorphic to the simplex ${\ensuremath{\Delta}}_{9}$ with 10 extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though the ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\ifmmode\times\else\texttimes\fi{}4$, to disprove a conjecture on the lengths of qubit-qudit separable states. This face is inscribed in the corresponding face of the convex set of all PPT states so that subsimplices ${\ensuremath{\Delta}}_{k}$ of ${\ensuremath{\Delta}}_{9}$ share the boundary if and only if $k\ensuremath{\le}5$. This enables us to find a large class of $2\ensuremath{\bigotimes}4$ PPT entangled edge states with rank 5.
We construct a parameterized family of n ⊗ n PPT (positive partial transpose) states of corank one for each n ≥ 3. With a suitable choice of parameters, we show that they are n ⊗ n PPT entangled edge states of corank one for 3 ≤ n ≤ 1000. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by 2n − 3, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for n ≥ 4.
We have studied mixed quantum states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. We have studied especially the rank 4444 and rank 5555 entangled PPT states.We find two distinct classes of rank four states, identified by a real valued quadratic expression invariant under local sln\\slnsln transformations, mathematically equivalent to continuous Lorentz transformations. This quadratic Lorentz invariant is non-zero for one class of states (type 1) and zero for the other class (type 2). We present analytical constructions of states of both types, general enough to reproduce all the rank four PPT states we have found numerically.There are six product vectors in a generic five dimensional subspace. The product vectors are used to define SL invariants of the state and its partial transposes. We find four distinct classes of states based which invariants are conserved or not under partial transposition.We find that a state is in the equivalence class of a symmetric state when its invariants are conserved under a partial transposition, and we have shown how to transform the state into its symmetric form through Lorentz transformations.The dimensions and geometry of extremal rank 5555 PPT states has also been examined. We find that states of different types lie on surfaces of rank 5555 different dimensions. And the surfaces touch the simplex of separable states in different ways
From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with PPT and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)]. With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of an arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd numbers of qubits we find a single configuration of ranks for which there are extremal states.