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arXiv:1904.08282 · Physical Review A

Class of genuinely high-dimensionally-entangled states with a positive partial transpose

Károly F. Pál, Tamás Vértesi

Published Jul 8, 2019• 11 citations
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Abstract

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 $d/2$ scaling of its Schmidt number in the local dimension $d$ can be attained.

DOI
10.1103/physreva.100.012310

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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×\else\texttimes\fidd\ifmmode\times\else\texttimes\fi{}dd\ifmmode×\else\texttimes\fid)-dimensional PPT states for d\ensuremath≥4d\ensuremath{\ge}4d\ensuremath≥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.
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
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.
Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered very weakly entangled.Since no pure entanglement can be distilled from them, they are also called bound entangled.In this paper, we present two classes of (2d × 2d)-dimensional PPT entangled states for any d 2 which outperform all separable states in metrology significantly.We present strong evidence that our states provide the maximal metrological gain achievable by PPT states for a given system size.When the dimension d goes to infinity, the metrological gain of these states becomes maximal and equals the metrological gain of a pair of maximally entangled qubits.Thus, we argue that our states could be called "PPT singlets."
Nongeneric positive partial transpose states of rank five in 3×33\times 33×3 dimensions
In 3×33\times 33×3 dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank (4,4)(4,4)(4,4) since a state ρρρ and its partial transpose ρPρ^PρP both have rank four. The next problem is to understand the extremal PPT states of rank (5,5)(5,5)(5,5). We call two states SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-equivalent if they are related by a product transformation. A generic rank (5,5)(5,5)(5,5) PPT state ρρρ is extremal, and ρρρ and ρPρ^PρP both have six product vectors in their ranges, and no product vectors in their kernels. The three numbers {6,6;0}\{6,6;0\}{6,6;0} are SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-invariants that help us classify the state. 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 SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT states. We say that a state ρρρ is SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetric if ρρρ and ρPρ^PρP are SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-equivalent, and is genuinely SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetric if it is SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-equivalent to a state τττ with τ=τPτ=τ^Pτ=τP. Genuine SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetry implies a special form of SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetry. We have produced numerically a random sample of rank (5,5)(5,5)(5,5) SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetric states. About fifty of these are of type {6,6;0}\{6,6;0\}{6,6;0}, among those all are extremal and about half are genuinely SL⊗SL\textrm{SL}\otimes\textrm{SL}SL⊗SL-symmetric.
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 $2\ensuremath{-}\ensuremath{\epsilon}$ 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.