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  "body": "\n## Problem\n\nLet $H = \\ell^2(\\N)$ be the [[Hilbert space|HM2HHE3J51]] of square-summable sequences, and let $B(H)$ be the C*-algebra of bounded operators on $H$.\nLet $D \\subseteq B(H)$ be the C\\*-subalgebra of diagonal operators with respect to the standard basis, which is isomorphic to $\\ell^\\infty(\\N)$.\nA pure [[state|F7MB2WREBJ]] on a C\\*-algebra is a state that is not a proper convex combination of two other states.\nDoes every pure state on $D$ extend to a unique state on $B(H)$?\n\n## Remarks\n\nEvery state on $D$ extends to some state on $B(H)$ by the Hahn–Banach theorem, so the question is about uniqueness.\nKadison and Singer posed the question in their 1959 paper \"Extensions of pure states\".\nOver the following decades it was shown to be equivalent to many problems in other fields, among them Anderson's paving conjecture, the Feichtinger conjecture on frames, and Weaver's discrepancy conjecture.\nAdam Marcus, Daniel Spielman and Nikhil Srivastava proved Weaver's conjecture, and so solved the problem with a positive answer; their proof was posted in 2013 and published in 2015.\nTheir method uses interlacing families of polynomials.\n\n## Explanations\n\n### L1 human\n\nIn quantum mechanics, a state assigns expected values to observables.\nThe question asks: if one knows the expected values of all diagonal observables in a very sharp way, does that determine the expected values of all observables?\nThe answer turned out to be yes.\nThe solution came from an unexpected direction, through a result about sums of random rank-one matrices and the roots of polynomials.\n",
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