Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

Abstract

The numerical bootstrap for quantum mechanics tests a candidate state's positivity only against sums of squares, whereas every physical state assigns nonnegative expectation to every pointwise nonnegative polynomial. In one dimension the two coincide; in two or more they do not. Whether this gap is realized depends sharply on which constraint set is imposed. For the stationary bootstrap, which imposes <[H,O]> = 0 and is the relaxation appropriate to thermal and mixed states, we exhibit a two-dimensional quartic double well and a moment vector that satisfies every level-three stationary constraint exactly, has a positive definite moment matrix, and yet assigns a negative expectation to a polynomial nonnegative on R^2; it is therefore the moment sequence of no state. All data are rational, and every step is verified in exact arithmetic. For the eigenstate bootstrap, which additionally imposes = E , the same search finds no violation in any of five settings spanning two, three and five degrees of freedom and truncation levels three and four, tested against complete families of separating polynomials. The single exception occurs at a truncation so low that only one eigenstate constraint survives, and the theory presented here accounts for it. We identify the mechanism: the eigenstate constraints bound the high momentum moments, otherwise unbounded on the feasible set, and it is those unbounded directions that reach the region between the two cones. Finally, under the reflection symmetries of a typical potential the relevant obstruction is copositivity rather than nonnegativity, which, for the quartic witnesses available at the lowest truncation, places the first possible failure at five degrees of freedom. We conjecture that the eigenstate constraints imply an Archimedean-type bound on the momentum moments, and formulate the corresponding tightness statement.