The Trivial Bound of Entropic Uncertainty Relations

TL;DR

This paper demonstrates that the standard entropic uncertainty relation for POVMs yields a trivial bound of one, which undermines its effectiveness in establishing security in quantum cryptography.

Contribution

It identifies a fundamental limitation of the POVM-based entropic uncertainty relation, showing it cannot reliably bound Eve's knowledge in quantum cryptographic security proofs.

Findings

POVM-based entropic uncertainty relation yields a trivial bound of one.

This trivial bound fails to connect smooth min-entropy to security guarantees.

The result highlights a need for alternative approaches in quantum cryptography security proofs.

Abstract

Entropic uncertainty relations are underpinning to compute the quantitative security bound in quantum cryptographic applications, such as quantum random number generation (QRNG) and quantum key distribution (QKD). All security proofs derive a relation between the information accessible to the legitimate group and the maximum knowledge that an adversary may have gained, Eve, which exploits entropic uncertainty relations to lower bound Eve's uncertainty about the raw key generated by one party, Alice. The standard entropic uncertainty relations is to utilize the smooth min- and max-entropies to show these cryptographic applications' security by computing the overlap of two incompatible measurements or positive-operator valued measures (POVMs). This paper draws one case of the POVM-versioned standard entropic uncertainty relation yielding the trivial bound since the maximum overlap in POVMs always produces the trivial value, "one." So, it fails to tie the smooth min-entropy to show the security of the quantum cryptographic application.