Can entanglement efficiently be weakend by symmetrization?

Abstract

Consider a quantum system with $m$ subsystems with $n$ qubits each, and suppose the state of the system is living in the symmetric subspace. It is known that, in the limit of $m\to\infty$, entanglement between any two subsystems vanishes. In this paper we study asymptotic behavior of the entanglement as $m$ and $n$ grows. Our conjecture is that if $m$ is a polynomially bounded function in $n$, then the entanglement decreases polynomially. The motivation of this study is a study of quantum Merlin-Arthur game. If this conjecture is ture, we can prove that bipartite separable certificate does not increase the computational power of the proof system. protocol. In the paper, we provide two evidences which support the conjecture. First, if $m$ is an exponential function, then entanglement decreases exponentially fast. Second, in case of a maximally entangled state, our conjecture is true.