No-cloning theorem for a single POVM

TL;DR

This paper investigates the possibility of cloning measurement statistics in quantum mechanics, demonstrating that perfect cloning is generally impossible for certain state pairs, with some exceptions illustrated through specific protocols.

Contribution

It introduces a novel analysis of observable cloning limitations, highlighting conditions under which perfect cloning of POVMs cannot be achieved, and provides examples and properties of such cases.

Findings

Certain state pairs cannot be perfectly cloned with a given POVM

Existence of state pairs that allow perfect cloning of POVMs

Properties of sets of states that resist cloning

Abstract

Cloning of statistics of general quantum measurement is discussed. The presented approach is connected with the known concept of observable cloning, but differs in some essential respects. The reasons are illustrated within some variety of B92 protocol. As it is shown, there exist pairs of states such that the perfect cloning of given POVM is not possible. We discuss some properties of these intolerant sets. An example allowing the perfect cloning is presented as well.