Key rate available from mismatched mesurements in the BB84 protocol and the uncertainty principle

TL;DR

This paper analyzes the potential for secret key extraction from mismatched measurements in the BB84 quantum key distribution protocol, revealing conditions under which such keys can be securely obtained and highlighting fundamental uncertainty constraints.

Contribution

It provides a lower bound on secret key rates from mismatched measurements and explores the implications of the uncertainty principle in this context.

Findings

Secret keys can be extracted from mismatched measurements over certain quantum channels.

High-probability Hadamard channels enable key extraction from mismatched measurements.

The entropic uncertainty principle limits simultaneous key extraction from matched and mismatched measurements.

Abstract

We consider the mismatched measurements in the BB84 quantum key distribution protocol, in which measuring bases are different from transmitting bases. We give a lower bound on the amount of a secret key that can be extracted from the mismatched measurements. Our lower bound shows that we can extract a secret key from the mismatched measurements with certain quantum channels, such as the channel over which the Hadamard matrix is applied to each qubit with high probability. Moreover, the entropic uncertainty principle implies that one cannot extract the secret key from both matched measurements and mismatched ones simultaneously, when we use the standard information reconciliation and privacy amplification procedure.