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arXiv:1410.0378 · Physical Review A

Construction and properties of a class of private states in arbitrary dimensions

A. Rutkowski, Michał Studziński, Piotr Ćwikliński, Michał Horodecki

Published Jan 27, 2015• 0 citations
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Abstract

We present a construction of quantum states in dimension $d$ that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits) $d=2$. We examine properties of this class of states, focusing mostly on its distance to the set of separable states $\mathcal{S}$, showing that for a fixed dimension of key part ${d}_{k}$, the distance increases with ${d}_{s}$. We provide explicit examples of positive partial transpose states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{S}$ is $2\ensuremath{-}\ensuremath{\epsilon}$, where $d$ scales with $\ensuremath{\epsilon}$ as $d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}$, as opposed to $d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}$ obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.

DOI
10.1103/physreva.91.012335

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We present a construction of quantum states in dimension ddd that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits) d=2d=2d=2. We examine properties of this class of states, focusing mostly on its distance to the set of separable states S\mathcal{S}S, showing that for a fixed dimension of key part dk{d}_{k}dk​, the distance increases with ds{d}_{s}ds​. We provide explicit examples of positive partial transpose states (in ddd dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of S\mathcal{S}S is 2\ensuremath−\ensuremathϵ2\ensuremath{-}\ensuremath{\epsilon}2\ensuremath−\ensuremathϵ, where ddd scales with \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ as d\ensuremath∝1/\ensuremathϵ3d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}d\ensuremath∝1/\ensuremathϵ3, as opposed to d\ensuremath∝2[log(4/\ensuremathϵ)]2d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}d\ensuremath∝2[log(4/\ensuremathϵ)]2 obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.
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.
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.
Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part. We provide the first protocol which distills key not only from the key part, but also from the shield if only the state is reducible. We prove also a tighter upper bound on its performance. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states (bound key states) exist. In turn, all the irreducible private states of the dimension 4⊗44\otimes 44⊗4 are strictly irreducible, that is, after an attack on the key part they become separable states. Providing the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we show a possible way to apply our findings in the context of quantum key repeaters. As a step towards this direction, we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions, yet not large enough to reach the goal of application. However this bound leads to the proof of no-key swapping for some private states in the single copy case without using the fact that they are strictly irreducible.
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.
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.