Dimension-Independent Bounds for Hardy's Experiment

TL;DR

This paper investigates Hardy's paradox in higher dimensions, providing an alternative proof for a dimension-independent probability bound and exploring conditions under which this bound remains constant.

Contribution

It offers an alternative proof for the dimension-independent maximum probability in Hardy's paradox and examines scenarios where this probability remains constant across dimensions.

Findings

Provided an alternative proof for the first type of Hardy's paradox.

Studied conditions for dimension-independent probabilities in the second type.

Conjectured modifications to maintain dimension-independence in higher dimensions.

Abstract

Hardy's paradox is of fundamental importance in quantum information theory. So far, there have been two types of its extensions into higher dimensions: in the first type the maximum probability of nonlocal events is roughly $9\%$ and remains the same as the dimension changes (dimension-independent), while in the second type the probability becomes larger as the dimension increases, reaching approximately $40\%$ in the infinite limit. Here, we (i) give an alternative proof of the first type, (ii) study the situation in which the maximum probability of nonlocal events can also be dimension-independent in the second type, and (iii) conjecture how the situation could be changed in order that (ii) still holds.