Quantum marginal problem and N-representability

Klyachko, Alexander A.

doi:10.1088/1742-6596/36/1/014

Cited by 273 publications

(499 citation statements)

References 19 publications

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“…However, this calculation is inconclusive, as the distance to the boundary is of the same order, 8 , as the truncation error [recall (13)]. …”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

“…with ð7Þ i are nonzero to a smaller order, 8 , than the error of spectral truncation, 10 . Together with the comments at the beginning of this section, this shows that the absence of pinned spectra is genuine, rather than an artifact of the truncation.…”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

“…iþ1 . In a ground-breaking work building on recent progress in invariant theory and representation theory, Klyachko exhibited an algorithm for computing all such Pauli-like constraints [4,8]. In fact, his work is part of a more general effort in quantum information theory addressing the quantum marginal problem which asks when a given set of single-site reduced density operators (marginals) is compatible in the sense that they arise from a common pure global state (see also Refs.…”

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confidence: 99%

“…1. Spectral ''trajectory'' vð Þ (thick line, partially covered by facet, schematic) up to correction of order 8 …”

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confidence: 99%

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Pauli Principle, Reloaded

Eisert¹

2013

Physics

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The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.

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“…However, this calculation is inconclusive, as the distance to the boundary is of the same order, 8 , as the truncation error [recall (13)]. …”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

mentioning

confidence: 99%

“…1. Spectral ''trajectory'' vð Þ (thick line, partially covered by facet, schematic) up to correction of order 8 …”

mentioning

confidence: 99%

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Pauli Principle, Reloaded

Eisert¹

2013

Physics

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“…Since 1960s, how to characterize this convex set has been a central topic of research in the field of quantum marginal problem and N -representability problem [1][2][3][4][5]. The recent development in quantum information theory has shown that the characterization of the 2-RDMs is a hard problem even with the existence of a quantum computer [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Joint product numerical range and geometry of reduced density matrices

Chen

Guo

et al. 2016

Sci. China Phys. Mech. Astron.

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The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

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Variational Embedding for Quantum Many‐Body Problems

Lin

Lindsey

2021

Comm Pure Appl Math

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Quantum embedding theories are powerful tools for approximately solving largescale, strongly correlated quantum many-body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one-sided bound for the exact ground-state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many-body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogues of the Kantorovich problem from optimal transport theory.

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Quantum marginal problem and N-representability

Cited by 273 publications

References 19 publications

Pauli Principle, Reloaded

Pauli Principle, Reloaded

Joint product numerical range and geometry of reduced density matrices

Variational Embedding for Quantum Many‐Body Problems

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