Product Theorems Via Semidefinite Programming

Rajat Mittal, Márió Szegedy

Thomas Hofmeister, Martin Hühne

Troy Lee, Adi Shraibman, Robert Špalek

Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in randomized, quantum, and even weakly-unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f odot g)=thetas(disc(f) disc(g)). As a consequence we obtain a strong direct product theorem for distributional complexity, and direct sum theorems for worst-case complexity, for bounds shown by the discrepancy method. Our results resolve an open problem of Shaltiel (2003) who showed a weaker product theorem for discrepancy with respect to the uniform distribution, discUodot(fodotk)=O(discU(f))k/3. The main tool for our results is semidefinite programming, in particular a recent characterization of discrepancy in terms of a semidefinite programming quantity by Linial and Shraibman (2006).

Shmuel Friedland, Jingtong Ge, Lihong Zhi

Strassen’s theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen’s theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen’s theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen’s theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional.

Franz Rendl

Shmuel Friedland, Jingtong Ge, Lihong Zhi

Strassen's theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite Strassen's theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen's theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen's theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite dimensional.