Quantum state transformations and the Schubert calculus

Abstract

Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn's Problem and Nielsen's Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix.

40 pages. Accepted for publication in Annals of Physics

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• Conditions for a class of entanglement transformations
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• N-representability is QMA-complete
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• Pinning of Fermionic Occupation Numbers
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• Linearity of Holographic Entanglement Entropy
• Ensemble reduced density matrix functional theory for excited states and hierarchical generalization of Pauli's exclusion principle
• Gaussian quantum marginal problem
• Eigenvalue Distributions of Reduced Density Matrices
• Natural Orbitals and Occupation Numbers for Harmonium: Fermions vs. Bosons
• Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics
• Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
• When is a pure state of three qubits determined by its single-particle reduced density matrices?
• Compatibility of subsystem states and convex geometry
• Reconstructing quantum states from single-party information
• Number-parity effect for confined fermions in one dimension
• Pinning of Fermionic Occupation Numbers: General Concepts and One Dimension
• Compatibility of subsystem states
• Quantum Marginal Problem and its Physical Relevance
• Inequalities for Moment Cones of Finite-Dimensional Representations
• The Complexity of the Consistency and N-representability Problems for Quantum States
• Computing Multiplicities of Lie Group Representations
• Membership in moment polytopes is in NP and coNP
• The minimal canonical form of a tensor network
• A brief introduction to multipartite entanglement
• Compatibility conditions from multipartite entanglement measures
• Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem
• Recoupling coefficients and quantum entropies
• Duistermaat-Heckman measure and the mixture of quantum states
• Heisenberg Honeycombs Solve Veneziano Puzzle
• Duality of reduced density matrices and their eigenvalues
• Compatibility Relations between the Reduced and Global Density Matrixes
• Experimental proposal to probe the extended Pauli principle
• Maximal lexicographic spectra and ranks for states with fixed uniform margins
• Construction of quantum states with special properties by projection methods
• Refining Ky Fan's majorization relation with linear programming
• Solving one-body ensemble N-representability problems with spin
• Universality of EPR pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion
• Ranks and eigenvalues of states with prescribed reduced states