Extremal generalized quantum measurements

Gael Sentís, B. Gendra, Stephen D. Bartlett +1 more

We design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements. We show that if one allows for a classical post-processing step only extremal rank-1 positive operator valued measures are needed. For a measurement with N elements on a d -dimensional space, our algorithm will decompose it into at most ( N − 1) d + 1 extremals, whereas the best previously known upper bound scaled as d 2 . Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property.

Ting Yu, Ian Colin Percival

K Sujan, Ravi Kunjwal

The study of quantum correlations is central to quantum information and foundations. The paradigmatic case of Bell scenarios considers product measurements implemented on a multipartite state. The more general case of contextuality scenarios--where the measurements do not have to be of product form or even on a composite system--has been studied for the case of projective measurements. While it is known that in any Bell scenario extremal indeterministic correlations (e.g., Popescu-Rohrlich or PR boxes) are unachievable quantumly, the case of general contextuality scenarios has remained open. Here we study quantum realizations of extremal correlations in arbitrary contextuality scenarios and prove that, for all such scenarios, no extremal indeterministic correlation can be achieved using projective quantum measurements, i.e., there exists no quantum state and no set of projective measurements, for any contextuality scenario, that can achieve such correlations. This no-go result follows as a corollary of a more general no-go theorem that holds when the most general set of quantum measurements (i.e., positive operator-valued measures, or POVMs) is taken into account. This general no-go theorem entails that no non-trivial quantum realization of an extremal indeterministic correlation exists, i.e., any "quantum" realization must be simulable by classical randomness. We discuss implications of this no-go theorem and the open questions it raises.

Jean-Louis Basdevant, Jean Dalibard

Giacomo Mauro D’Ariano, Paolo Perinotti, Michal Sedlák

Generalized quantum instruments correspond to measurements where the input and output are either states or more generally quantum circuits. These measurements describe any quantum protocol including games, communications, and algorithms. The set of generalized quantum instruments with a given input and output structure is a convex set. Here, we investigate the extremal points of this set for the case of finite dimensional quantum systems and generalized instruments with finitely many outcomes. We derive algebraic necessary and sufficient conditions for extremality.