On the Hardness of the Quantum Separability Problem and the Global Power of Locally Invariant Unitary Operations

quantum information;quantum computing;quantum separability problem;cyclic unitary operations;Computer Science

Gharibian, Sevag

University of Waterloo

关键词: quantum information;    quantum computing;    quantum separability problem;    cyclic unitary operations;    Computer Science;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/3991/1/SevagGharibianMMathThesis.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

Given a bipartite density matrix ρ of a quantum state, the Quantum Separabilityproblem (QUSEP) asks — is ρ entangled, or separable? In this thesis, we firststrengthen Gurvits’ 2003 NP-hardness result for QUSEP by showing that the WeakMembership problem over the set of separable bipartite quantum states is stronglyNP-hard, meaning it is NP-hard even when the error margin is as large as inversepolynomial in the dimension, i.e. is ;;moderately large”. Previously, this NP-hardness was known only to hold in the case of inverse exponential error. We observethe immediate implication of NP-hardness of the Weak Membership problem over the set of entanglement-breaking maps, as well as lower bounds on the maximum (Euclidean) distance possible between a bound entangled state and the separable set of quantum states (assuming P ≠ NP).We next investigate the entanglement-detecting capabilities of locally invariantunitary operations, as proposed by Fu in 2006. Denoting the subsystems of ρ asA and B, such that ρ_B = Tr_A(ρ), a locally invariant unitary operation U^B is onewith the property U^B ρ_B (U^B)^† = ρ_B. We investigate the maximum shift (in Euclideandistance) inducible in ρ by applying I⊗U^B, over all locally invariant choices of U^B.We derive closed formulae for this quantity for three cases of interest: (pseudo)purequantum states of arbitrary dimension, Werner states of arbitrary dimension, andtwo-qubit states. Surprisingly, similar to recent anomalies detected for non-localitymeasures, the first of these formulae demonstrates the existence of non-maximallyentangled states attaining shifts as large as maximally entangled ones. Using the latter of these formulae, we demonstrate for certain classes of two-qubit states an equivalence between the Fu criterion and the CHSH inequality. Among other results, we investigate the ability of locally invariant unitary operations to detect bound entanglement.