Abstract
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 paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are compatible with some pure global quantum state. As an application, we prove that the existence of multiparticle absolutely maximally entangled states for a given dimension is equivalent to the separability of an explicitly given two-party quantum state. Finally, we show that the existence of quantum codes with given parameters can also be interpreted as a marginal problem, hence, our complete hierarchy can also be used.
Highlights
Clarifying the relation between the whole and its parts is crucial for many problems in science
This makes them valuable ingredients for quantum information protocols[16,17], but it turns out that absolutely maximally entangled (AME) states do not exist for arbitrary dimensions, as not always global states with the desired mixed marginals can be found[18,19,20,21]
And in the following, the term marginal problem usually refers to the pure state marginal problem in quantum mechanics
Summary
Clarifying the relation between the whole and its parts is crucial for many problems in science. The reverse question, whether a given set of marginals is compatible with a global pure state, is, not easy to decide Still, it is at the heart of many problems in quantum physics. The construction of quantum error-correcting codes, which constitute fundamental building blocks in the design of quantum computer architectures[25,26,27], essentially amounts to the identification of subspaces of the total Hilbert space, where all states in this space obey certain marginal constraints This establishes a connection to the AME problem, which was announced to be one of the central problems in quantum information theory[28]. We show that our approach can be extended to study the existence problem of quantum codes
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