The quantum marginal problem
Clarifying the relation between the whole and its parts
is crucial for many problems in science. In quantum
mechanics, this question manifests itself in the quantum
marginal problem, which asks whether there is a global pure
quantum state for some given marginals. This problem arises
in many contexts, ranging from quantum chemistry to entanglement
theory and quantum error correcting codes.
In this talk I will first present an introduction into
this problem and its applications. Then, I will discuss
the problem of absolutely maximally entangled states as
a special instance. Finally, I will prove a correspondence
of the marginal problem to the separability problem. Based
on this, I describe a sequence of semidefinite programs
which can decide whether some given marginals are compatible
with some pure global quantum state. As an application, I
prove that the existence of multi-particle absolutely
maximally entangled states for a given dimension is
equivalent to the separability of an explicitly given
two-party quantum state.
X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, O. Gühne,
A complete hierarchy for the pure state marginal problem in
Nature Communications 12, 1012 (2021).