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

Network Synthesis of Linear Dynamical Quantum Stochastic Systems

The purpose of this paper is to develop a synthesis theory for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of coherent/fully quantum linear stochastic controllers for quantum control, amongst other potential applications. We show how general linear dynamical quantum stochastic systems can be constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, discuss how such a network of oscillators can be approximately synthesized or implemented in a systematic way from some linear and non-linear quantum optical elements. An example is also provided to illustrate the theory.Comment: Revised and corrected version, published in SIAM Journal on Control and Optimization, 200

A new bound of the ℒ2[0, T]-induced norm and applications to model reduction

We present a simple bound on the finite horizon ℒ2/[0, T]-induced norm of a linear time-invariant (LTI), not necessarily stable system which can be efficiently computed by calculating the ℋ∞ norm of a shifted version of the original operator. As an application, we show how to use this bound to perform model reduction of unstable systems over a finite horizon. The technique is illustrated with a non-trivial physical example relevant to the appearance of time-irreversible phenomena in statistical physics

Influence of convective transport on tropospheric ozone and its precursors in a chemistry-climate model

The impact of convection on tropospheric O₃ and its precursors has been examined in a coupled chemistry-climate model. There are two ways that convection affects O₃. First, convection affects O₃ by vertical mixing of O₃ itself. Convection lifts lower tropospheric air to regions where the O₃ lifetime is longer, whilst mass-balance subsidence mixes O₃-rich upper tropospheric (UT) air downwards to regions where the O₃ lifetime is shorter. This tends to decrease UT O₃ and the overall tropospheric column of O₃. Secondly, convection affects O₃ by vertical mixing of O₃ precursors. This affects O₃ chemical production and destruction. Convection transports isoprene and its degradation products to the UT where they interact with lightning NO_x to produce PAN, at the expense of NO_x. In our model, we find that convection reduces UT NO_x through this mechanism; convective down-mixing also flattens our imposed profile of lightning emissions, further reducing UT NO_x. Over tropical land, which has large lightning NO_x emissions in the UT, we find convective lofting of NO_x from surface sources appears relatively unimportant. Despite UT NO_x decreases, UT O₃ production increases as a result of UT HO_x increases driven by isoprene oxidation chemistry. However, UT O₃ tends to decrease, as the effect of convective overturning of O₃ itself dominates over changes in O₃ chemistry. Convective transport also reduces UT O₃ in the mid-latitudes resulting in a 13% decrease in the global tropospheric O₃ burden. These results contrast with an earlier study that uses a model of similar chemical complexity. Differences in convection schemes as well as chemistry schemes – in particular isoprene-driven changes are the most likely causes of such discrepancies. Further modelling studies are needed to constrain this uncertainty range

Optimal tracking for pairs of qubit states

In classical control theory, tracking refers to the ability to perform measurements and feedback on a classical system in order to enforce some desired dynamics. In this paper we investigate a simple version of quantum tracking, namely, we look at how to optimally transform the state of a single qubit into a given target state, when the system can be prepared in two different ways, and the target state depends on the choice of preparation. We propose a tracking strategy that is proved to be optimal for any input and target states. Applications in the context of state discrimination, state purification, state stabilization and state-dependent quantum cloning are presented, where existing optimality results are recovered and extended.Comment: 15 pages, 8 figures. Extensive revision of text, optimality results extended, other physical applications include

Two-qutrit Entanglement Witnesses and Gell-Mann Matrices

The Gell-Mann $\lambda$ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the $\lambda$-diagonal EWs presented in this paper are all decomposable but there exist non-decomposable ones among $\lambda$-non-diagonal EWs.Comment: 25 page

Applying matrix product operators to model systems with long-range interactions

An algorithm is presented which computes a translationally invariant matrix product state approximation of the ground state of an infinite 1D system; it does this by embedding sites into an approximation of the infinite ``environment'' of the chain, allowing the sites to relax, and then merging them with the environment in order to refine the approximation. By making use of matrix product operators, our approach is able to directly model any long-range interaction that can be systematically approximated by a series of decaying exponentials. We apply our techniques to compute the ground state of the Haldane-Shastry model and present results.Comment: 7 pages, 3 figures; manuscript has been expanded and restructured in order to improve presentation of the algorith