Quantum marginals, faces, and coatoms
arXiv:2103.08360 · doi:10.1007/s41884-023-00103-2
Abstract
Many problems of quantum information theory rely on the set of quantum marginals. A precise knowledge of the faces of this convex set is necessary, for example, in the reconstruction of states from their marginals or in the evaluation of complexity measures of many-body systems. Yet, even the two-body marginals of just three qubits were only described in part. Here, we propose an experimental method to search for the coatoms in the lattice of exposed faces of the convex set of quantum marginals. The method is based on sampling from the extreme points of the dual spectrahedron. We provide an algebraic certificate of correctness, employing ground projectors of local Hamiltonians. Using this method, we present an explicit family of coatoms of rank five in the lattice of ground projectors of two-local three-qubit Hamiltonians (the rank is always six for bits). This family describes a family of coatoms in the lattice of exposed faces of the convex set of two-body marginals of three qubits. Besides introducing the experimental method, we show that the support sets of probability distributions that factor are the ground projectors of frustration-free Hamiltonians in the commutative setting. We also discuss nonexposed points of the set of marginals.
arXiv commentThe Version of Record of this article is S. Weis and J. Gouveia, "The face lattice of the set of reduced density matrices and its coatoms", Information Geometry, 2023. A full-text view-only version is available online at https://rdcu.be/c7b2s
References in corpus (23)
- On the toric algebra of graphical models
- Computing Local Invariants of Qubit Systems
- Three-qubit pure-state canonical forms
- The Pauli principle revisited
- Pinning of Fermionic Occupation Numbers
- Uniqueness of Quantum States Compatible with Given Measurement Results
- Interacting boson problems are QMA-hard
- A complete hierarchy for the pure state marginal problem in quantum mechanics
- Unfrustrated Qudit Chains and their Ground States
- From Ground States to Local Hamiltonians
- Entropy Distance: New Quantum Phenomena
- Computing complexity measures for quantum states based on exponential families
- Quantum Convex Support
- Comment on some results of Erdahl and the convex structure of reduced density matrices
- Maximizing the divergence from a hierarchical model of quantum states
- Complete Characterization of the Ground Space Structure of Two-Body Frustration-Free Hamiltonians for Qubits
- On Local Unitary Equivalence of Two and Three-qubit States
- Implications of pinned occupation numbers for natural orbital expansions. I: Generalizing the concept of active spaces
- Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians
- Approximating faces of marginal polytopes in discrete hierarchical models
- On the complexity of finding the maximum entropy compatible quantum state
- An efficient numerical algorithm on irreducible multiparty correlations
- A variation principle for ground spaces