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.

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Reference:

X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, O. Gühne,

A complete hierarchy for the pure state marginal problem in

quantum mechanics

Nature Communications 12, 1012 (2021).