# Quantum entanglement, sum of squares, and the log rank conjecture

with Boaz Barak, Pravesh Kothari. **STOC 2017.**

## abstract

For every constant $\epsilon>0$, we give an $\exp(\tilde{O}(\sqrt{n}))$-time algorithm for the $1$ vs $1-\epsilon$ *Best Separable State (BSS)* problem of distinguishing, given an $n^2\times n^2$ matrix $\mathcal M$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $\rho$ that $\mathcal M$ accepts with probability $1$, and the case that every separable state is accepted with probability at most $1-\epsilon$. Equivalently, our algorithm takes the description of a subspace $\mathcal W\subseteq \mathbb F^{n^2}$ (where $\mathbb F$ can be either the real or complex field) and distinguishes between the case that $\mathcal W$ contains a rank one matrix, and the case that every rank one matrix is at least $\epsilon$ far (in $\ell_2$ distance) from $\mathcal W$.

To the best of our knowledge, this is the first improvement over the brute-force $\exp(n)$-time algorithm for this problem. Our algorithm is based on the *sum-of-squares* hierarchy and its analysis is inspired by Lovett’s proof (STOC ’14, JACM ’16) that the communication complexity of every rank-$n$ Boolean matrix is bounded by $\tilde{O}(\sqrt{n})$.

## keywords

- sum-of-squares method
- tensor computations
- quantum information
- semidefinite programming
- small-set expansion