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arXiv:1309.7992 · Physical Review A

Bound entangled states with extremal properties

Piotr Badzia ̧g, Karol Horodecki, Michał Horodecki, Justin Jenkinson, Stanisław J. Szarek

Published Jul 2, 2014• 10 citations
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Abstract

Following recent work of Beigi and Shor, we investigate positive partial transpose (PPT) states that are ``heavily entangled.'' We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least $1/4$. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of $2\ensuremath{-}\ensuremath{\epsilon}$ from the separable states, we need a dimension of ${2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}$, as opposed to ${2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}$ given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.

DOI
10.1103/physreva.90.012301

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Following recent work of Beigi and Shor, we investigate positive partial transpose (PPT) states that are ``heavily entangled.'' We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least 1/41/41/4. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of 2\ensuremath−\ensuremathϵ2\ensuremath{-}\ensuremath{\epsilon}2\ensuremath−\ensuremathϵ from the separable states, we need a dimension of 2poly[log(1\ensuremathϵ)]{2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}2poly[log(\ensuremathϵ1​)], as opposed to 2poly(1\ensuremathϵ){2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}2poly(\ensuremathϵ1​) given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
In this paper we show an explicit construction of multipartite class of entangled states with the PPT (Positive Partial Transposition) property in every cut. We investigate properties of this class of states focusing on the trace distance from the set of separable states. We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue, that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction is we do not have to use many copies of initial state living on the smaller space to boost the trace distance as in the previous attempts to this problem.
Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear d/2d/2d/2 scaling of its Schmidt number in the local dimension ddd can be attained.
Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled nnn- qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as 1dn(dn−1)+1\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}dn(dn−1)​+1​1​, which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of R1+dn−1\frac{R}{1+d^{n-1}}1+dn−1R​ from the Identity,where R is the radius of the closed ball homeomorphic to the set of density matrices, which is lesser than the limit for the limit for PPT bound entangled states. Furthermore an alternate proof on the non-emptiness of the PPT bound entangled states has also been given.
We have studied mixed 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. In random numerical searches, we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher, we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially the rank-four entangled PPT states, which are found to have rank four for every partial transpose. These states are all extremal because of the previously known result that every PPT state of rank three or less is separable. We find two distinct classes of rank-4444 entangled PPT states, identified by a real valued quadratic expression invariant under local SL(2,C)\text{SL}(2,\mathbb{C})SL(2,C) transformations, mathematically equivalent to Lorentz transformations. This quadratic Lorentz invariant is nonzero for one class of states (type I in our terminology) and zero for the other class (type II). The previously known states based on unextendible product bases are a nongeneric subclass of the type-I states. We present analytical constructions of states of both types, general enough to reproduce all the rank-4444 PPT states we have found numerically. We can not exclude the possibility that there exist nongeneric rank-four PPT states that we do not find in our random numerical searches.
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.