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

TL;DR

This paper introduces a new class of private quantum states called private dits (pdits) in arbitrary dimensions, analyzes their properties, especially their distance from separable states, and provides explicit examples with near-maximal distance.

Contribution

It constructs a broad class of private states in any dimension, analyzes their entanglement properties, and offers explicit PPT examples with large distance from separable states without boosting.

Findings

Distance from separable states increases with dimension for fixed key size.

Explicit PPT states are nearly as far from separable as possible, with distance approaching 2.

Distance scales polynomially with dimension, improving over previous exponential bounds.

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 $\mathcal{SEP}$, showing that for a fixed dimension of key part $d_k$ the distance increases with $d_s$. We provide explicit examples of PPT states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{SEP}$ is $2 - \epsilon$, where $d$ scales with $\epsilon$ as $d \propto 1/\epsilon^3$, as opposed to $d \propto 2^{(log(4/\epsilon))^2}$ obtained in [Badzi\c{a}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 Badzi\c{a}g et al. paper.