A complete family of separability criteria

We introduce a new family of separability criteria that are based on the existence of extensions of a bipartite quantum state $\rho$ to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the non-existence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program (SDP). Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known Positive Partial Transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that in turn allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP-hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.Comment: 21 pages. Expanded introduction. References added, typos corrected. Accepted for publication in Physical Review

Constructing Driver Hamiltonians for Optimization Problems with Linear Constraints

Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealers has centered around using more advanced and novel Hamiltonian representations to solve optimization problems. One of these advances has centered around the development of driver Hamiltonians that commute with the constraints of an optimization problem - allowing for another avenue to satisfying those constraints instead of imposing penalty terms for each of them. In particular, the approach is able to use sparser connectivity to embed several practical problems on quantum devices than other common practices. However, designing the driver Hamiltonians that successfully commute with several constraints has largely been based on strong intuition for specific problems and with no simple general algorithm to generate them for arbitrary constraints. In this work, we develop a simple and intuitive algebraic framework for reasoning about the commutation of Hamiltonians with linear constraints - one that allows us to classify the complexity of finding a driver Hamiltonian for an arbitrary set of constraints as NP-Complete. Because unitary operators are exponentials of Hermitian operators, these results can also be applied to the construction of mixers in the Quantum Alternating Operator Ansatz (QAOA) framework.Comment: 20 pages, 2 figure

Detecting separable states via semidefinite programs

We introduce a new technique to detect separable states using semidefinite programs. This approach provides a sufficient condition for separability of a state that is based on the existence of a certain local linear map applied to a known separable state. When a state is shown to be separable, a proof of this fact is provided in the form of an explicit convex decomposition of the state in terms of product states. All states in the interior of the set of separable states can be detected in this way, except maybe for a set of measure zero. Even though this technique is more suited for a numerical approach, a new analytical criterion for separability can also be derived.Comment: 8 pages, accepted for publication in Physical Review

Alternate Scheme for Optical Cluster-State Generation without Number-Resolving Photon Detectors

We design a controlled-phase gate for linear optical quantum computing by using photodetectors that cannot resolve photon number. An intrinsic error-correction circuit corrects errors introduced by the detectors. Our controlled-phase gate has a 1/4 success probability. Recent development in cluster-state quantum computing has shown that a two-qubit gate with non-zero success probability can build an arbitrarily large cluster state with only polynomial overhead. Hence, it is possible to generate optical cluster states without number-resolving detectors and with polynomial overhead.Comment: 10 pages, 4 figures, 4 tables; made significant revisions and changed forma