Deterministic Computational Complexity of the Quantum Separability Problem

Ever since entanglement was identified as a computational and cryptographic resource, effort has been made to find an efficient way to tell whether a given density matrix represents an unentangled, or separable, state. Essentially, this is the quantum separability problem. In Section 1, I begin with a brief introduction to separability and entanglement, and a basic formal definition of the quantum separability problem. I conclude with a summary of one-sided tests for separability, including those involving semidefinite programming. In Section 2, I treat the separability problem as a computational decision problem and motivate its approximate formulations. After a review of basic complexitytheoretic notions, I discuss the computational complexity of the separability problem. In Section 3, I give a comprehensive survey of deterministic algorithmic solutions to the (approximate) quantum separability problem. Finally, Section 4 contains a comparison of the complexities of the algorithms surveyed in Section 3. I have attempted to give a comprehensive treatment of the deterministic quantum separability problem from a computational perspective and identify relevant open problems. Features and new results include: • the first specification of an exhaustive-search algorithm for the (approximate) quantum separability based on the recent algorithm of Hulpke and Bruß, as well as an upper bound on Doherty et al.’s semidefinite programming algorithm using König’s and Renner’s recent finite quantum de Finetti theorem; • a discussion of why the natural formulation of the quantum separability problem is not in NP, as well as a first NP-contained formulation of the quantum separability problem, thus obtaining a proof that (a meaningful formulation of) the quantum separability problem is not in co-NP unless NP=co-NP;