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arXiv:1212.6378 · Physical Review A

Separability of qubit-qudit quantum states with strong positive partial transposes

Kil-Chan Ha

Published Feb 13, 2013• 5 citations
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Abstract

We show that all $2\ensuremath{\bigotimes}4$ states with strong positive partial transposes (SPPTs) are separable. We also construct a family of $2\ensuremath{\bigotimes}5$ entangled SPPT states, so the conjecture on the separability of SPPT states is completely settled. In addition, we clarify the relation between the set of all $2\ensuremath{\bigotimes}d$ separable states and the set of all $2\ensuremath{\bigotimes}d$ SPPT states for the case of $d=3,4$.

DOI
10.1103/physreva.87.024301

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We show that all 2\ensuremath⨂42\ensuremath{\bigotimes}42\ensuremath⨂4 states with strong positive partial transposes (SPPTs) are separable. We also construct a family of 2\ensuremath⨂52\ensuremath{\bigotimes}52\ensuremath⨂5 entangled SPPT states, so the conjecture on the separability of SPPT states is completely settled. In addition, we clarify the relation between the set of all 2\ensuremath⨂d2\ensuremath{\bigotimes}d2\ensuremath⨂d separable states and the set of all 2\ensuremath⨂d2\ensuremath{\bigotimes}d2\ensuremath⨂d SPPT states for the case of d=3,4d=3,4d=3,4.
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by Yu and Zhao [Int. J. Theor. Phys. 54, 292 (2014)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states and the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
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.
Absolutely positive partial transpose (APPT) states, which remain PPT under every global unitary, provide a spectral relaxation of absolutely separable states, which remain separable under every such transformation. Despite substantial progress, whether these two notions coincide beyond qubit-qudit systems remains unresolved, motivating the study of structured state families in which the APPT spectral constraints and separability can be analyzed exactly. We study this problem for Bell-diagonal states in two settings: the standard two-qutrit Weyl-Heisenberg basis and the two-ququart lattice basis generated by two-qubit Pauli operators. In the two-qutrit case, we show that a linearized necessary condition for APPT defines a polytope of spectra in which every associated Weyl-Heisenberg Bell-diagonal state admits an exact decomposition into explicitly separable states. This yields a separability criterion that applies to a region even larger than the APPT set itself. In the two-ququart lattice Bell-diagonal setting, we show that a state remains PPT under every permutation of its Bell coefficients if and only if its six largest Bell coefficients sum to at most 1/2, and we prove that all such states are separable. Equivalently, every entangled lattice Bell-diagonal state becomes NPT after a suitable coefficient permutation and is therefore not APPT. The proofs combine convex-polytope analysis, affine and symplectic orbit reductions, and exact rational decompositions into separable states supported on affine Lagrangian subsets. Our results rule out entangled APPT states in both Bell-diagonal families, and provide simple spectral inequalities as sufficient criteria for separability in both cases. Being basis-specific, however, they do not by themselves settle whether APPT and absolute separability coincide.