On the geometric distance between quantum states with positive partial transposition and private states

TL;DR

This paper establishes a lower bound on the geometric distance between certain entangled PPT states and private states capable of producing a secure key, applicable across finite-dimensional Hilbert spaces.

Contribution

It provides the first analytic lower bound for the geometric distance between a broad class of PPT states and private states, applicable to all known entangled PPT states with non-zero distillable key.

Findings

Lower bound proven for a broad class of PPT states

Bound applies to all known entangled PPT states with non-zero key

Result holds in any finite-dimensional Hilbert space

Abstract

We prove an analytic positive lower bound for the geometric distance between entangled positive partial transpose (PPT) states of a broad class and any private state that delivers one secure key bit. Our proof holds for any Hilbert space of finite dimension. Although our result is proven for a specific class of PPT states, we show that our bound nonetheless holds for all known entangled PPT states with non-zero distillable key rates whether or not they are in our special class.