Quantum marginal problem and representations of the symmetric group
Alexander Klyachko
TL;DR
The paper presents two complementary routes to the quantum marginal problem: a geometric–combinatorial approach based on the Berenstein–Sjamaar framework that yields explicit linear inequalities organized by cubicles and extremal edges, and a representation-theoretic viewpoint linking margins to tensor-product decompositions of the symmetric group via Kronecker coefficients. It provides a complete, constructive set of marginal inequalities for two-component systems and, in the qubit array case, enumerates extensive tables up to rank four, illustrating how Schubert calculus and flag-variety cohomology encode constraints on spectra. The second route translates the problem into representation theory, showing that the existence of a density operator with given margins is equivalent to nonvanishing Kronecker coefficients, enabling results about maximal eigenvalues and ranks and revealing deep connections to Horn–Klyachko-type inequalities. Together, these methods connect quantum marginals with classical marginal problems, Schubert calculus, and symmetric-group representation theory, offering both concrete constraints for small systems and a unifying perspective for larger, more complex cases with potential applications to stable Kronecker coefficients and rectangular diagrams.
Abstract
We discuss existence of mixed state of multicomponent system with given spectrum and given reduced density matrices. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.