Bulk connectedness and boundary entanglement

¹ Walter Burke Institute for Theoretical Physics California Institute of Technology - Pasadena, CA 91125, USA
² Institute for Quantum Information and Matter, California Institute of Technology - Pasadena, CA 91125, USA
³ Center for Theoretical Physics and Department of Physics University of California - Berkeley, CA 94720, USA

Received: 19 January 2018
Accepted: 2 May 2018

Abstract

We prove, for any state in a conformal field theory defined on a set of boundary manifolds with corresponding classical holographic bulk geometry, that for any bipartition of the boundary into two non-clopen sets, the density matrix cannot be a tensor product of the reduced density matrices on each region of the bipartition. In particular, there must be entanglement across the bipartition surface. We extend this no-go theorem to general, arbitrary partitions of the boundary manifolds into non-clopen parts, proving that the density matrix cannot be a tensor product. This result gives a necessary condition for states to potentially correspond to holographic duals.