Limitations of Sum of Squares for 2-to-4 norm, Separability, and Entangled Games

We introduce a method for using reductions to construct integrality gaps for semidefinite programs (SDPs). These imply new limitations on the sum-of-squares (SoS) hierarchy in two settings where previously no ω(1)-round integrality gaps were known: 1. The 2 → 4 norm of a matrix, or equivalently, the set of separable (i.e. unentangled) quantum states. 2. The set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. Integrality gaps for the 2→ 4 norm had previously been sought due to its connection to SmallSet Expansion (SSE) and Unique Games (UG), while integrality gaps for separable states have long been studied in quantum information theory. In each case, no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as the previous ETH-based results for separable states and the 2 → 4 norm, and achieve weaker parameters for quantum correlations. In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. To the authors’ knowledge, this provides the first extension of LRS to problems over non-boolean domains. These results can be viewed as limitations on various ideas from quantum information (including the monogamy principle, the PPT test, Tsirelson-type inequalities) for understanding entanglement. Many of these ideas have been formalized into SDP hierarchies by DohertyParrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz, all of which we establish limits on the effectiveness of. ∗Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA †Department of Computer and Information Science, University of Oregon, Eugene, OR 97403, USA.