A complete hierarchy for the pure state marginal problem in quantum mechanics

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.

Reconstructing the whole from its parts

“…10 fully separable? This is a likely situation, as there is a close relation between separability and the quantum marginal problem [2]. Indeed, the ball containing all reconstructed quantum states shown in Prop.…”

“…This fascinating topic, known as the quantum marginal problem (QMP), aims to answer the following question: given a set of multipartite quantum marginal reductions, is there a global quantum state compatible with them? The QMP is closely related to the identification of separable quantum pure states in high dimensional bipartite systems [2], multipartite entanglement detection from nearest neighbour marginals [3] and certification of quantum nonlocality from separable marginal reductions [4]. Furthermore, it is linked to the existence of absolutely maximally entangled (AME) states [5,6], perfect tensors [7] and quantum error correcting codes [8].…”

Quantum variational learning for quantum error-correcting codes

“…For example, when the target quantum code is translational-invariant, one may use a VQC with a certain amount of symmetry, where different gates can share the same parameter. If we slightly modify the cost functions, VarQEC can be used for finding some QECC variants like the hybrid quantum-classical codes [90], estimating the zeroerror capacity of noisy quantum channels [91], and solving quantum marginal problems [92].…”

Excitations of Quantum Many-Body Systems via Purified Ensembles: A Unitary-Coupled-Cluster-based Approach

“…( 2) is the well-known thermo-field introduced by Matsumoto and Umezawa [34,35]. Central to many modern developments in quantum sciences, this purification of the thermal state plays an important role in quantum gravity [36][37][38][39], non-equilibrium phenomena [40][41][42][43][44][45], quantum information [46,47], quantum marginal problems [48], and quantum chemistry [49,50].…”