  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.
