Maximum stabilizer dimension for nonproduct states

Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for n-qubit pure states, and find its maximum dimension to be n-1 for nonproduct states of three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: we show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the n-qubit GHZ state.Comment: 6 page

Lightweight Multilingual Software Analysis

Developer preferences, language capabilities and the persistence of older languages contribute to the trend that large software codebases are often multilingual, that is, written in more than one computer language. While developers can leverage monolingual software development tools to build software components, companies are faced with the problem of managing the resultant large, multilingual codebases to address issues with security, efficiency, and quality metrics. The key challenge is to address the opaque nature of the language interoperability interface: one language calling procedures in a second (which may call a third, or even back to the first), resulting in a potentially tangled, inefficient and insecure codebase. An architecture is proposed for lightweight static analysis of large multilingual codebases: the MLSA architecture. Its modular and table-oriented structure addresses the open-ended nature of multiple languages and language interoperability APIs. We focus here as an application on the construction of call-graphs that capture both inter-language and intra-language calls. The algorithms for extracting multilingual call-graphs from codebases are presented, and several examples of multilingual software engineering analysis are discussed. The state of the implementation and testing of MLSA is presented, and the implications for future work are discussed.Comment: 15 page

Lightweight Call-Graph Construction for Multilingual Software Analysis

Analysis of multilingual codebases is a topic of increasing importance. In prior work, we have proposed the MLSA (MultiLingual Software Analysis) architecture, an approach to the lightweight analysis of multilingual codebases, and have shown how it can be used to address the challenge of constructing a single call graph from multilingual software with mutual calls. This paper addresses the challenge of constructing monolingual call graphs in a lightweight manner (consistent with the objective of MLSA) which nonetheless yields sufficient information for resolving language interoperability calls. A novel approach is proposed which leverages information from a compiler-generated AST to provide the quality of call graph necessary, while the program itself is written using an Island Grammar that parses the AST providing the lightweight aspect necessary. Performance results are presented for a C/C++ implementation of the approach, PAIGE (Parsing AST using Island Grammar Call Graph Emitter) showing that despite its lightweight nature, it outperforms Doxgen, is robust to changes in the (Clang) AST, and is not restricted to C/C++.Comment: 10 page

Werner state structure and entanglement classification

We present applications of the representation theory of Lie groups to the analysis of structure and local unitary classification of Werner states, sometimes called the {\em decoherence-free} states, which are states of $n$ quantum bits left unchanged by local transformations that are the same on each particle. We introduce a multiqubit generalization of the singlet state, and a construction that assembles these into Werner states.Comment: 9 pages, 2 figures, minor changes and corrections for version

Pleasure, profit and pain: Alcohol in New Zealand and the contemporary culture of intoxication

This book details the rich, complex and often contested role of alcohol in New Zealand society. It explores the three fundamental alcohol rights that continue to fight for dominance of the national drinking culture: the rights of individual drinkers to enjoy the pleasures of alcohol, the rights of society to protect itself from the harms of alcohol, and the rights of the alcohol industry to profit from the sale of a legal commodity. Historically, most of our intoxicated drinkers were adult males and drinking was typically separated from family, food and entertainment. With the sweeping social changes of the 1960s and 1970s, women and later young people, increasingly engaged with alcohol. A growing proportion of these groups have since joined men in a culture of intoxication, or binge drinking culture as it is often termed. New Zealand is not alone however, in having a culture of intoxication, with similar alcohol consumption patterns evident in many other developed nations. This book identifies the local and the global influences that have affected New Zealand society (and much of the rest of the world) since the late 1900s and details how these influences have sustained the contemporary culture of intoxication. Finally, this book will propose that to implement effective change to our national drinking culture, the rights of the alcohol industry and of individual drinkers will need to be pulled back from the liberal excesses that the 1980s and 1990s provided. A re-balancing is required in order to strengthen and sustain society’s right to protect itself from alcohol-related harm

Classification of nonproduct states with maximum stabilizer dimension

Nonproduct n-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized n-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for n greater than or equal to 3 but not equal to 4. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.Comment: 6 pages, version 3 has a typographical correction in the displayed equation just after numbered equation (2), and other minor correction

Minimum orbit dimension for local unitary action on n-qubit pure states

The group of local unitary transformations partitions the space of n-qubit quantum states into orbits, each of which is a differentiable manifold of some dimension. We prove that all orbits of the n-qubit quantum state space have dimension greater than or equal to 3n/2 for n even and greater than or equal to (3n + 1)/2 for n odd. This lower bound on orbit dimension is sharp, since n-qubit states composed of products of singlets achieve these lowest orbit dimensions.Comment: 19 page

Classification of n-qubit states with minimum orbit dimension

The group of local unitary transformations acts on the space of n-qubit pure states, decomposing it into orbits. In a previous paper we proved that a product of singlet states (together with an unentangled qubit for a system with an odd number of qubits) achieves the smallest possible orbit dimension, equal to 3n/2 for n even and (3n + 1)/2 for n odd, where n is the number of qubits. In this paper we show that any state with minimum orbit dimension must be of this form, and furthermore, such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets.Comment: 15 pages, latex, revision 2, conclusion added, some proofs shortene

Multiparty quantum states stabilized by the diagonal subgroup of the local unitary group

We classify, up to local unitary equivalence, the set of $n$-qubit states that is stabilized by the diagonal subgroup of the local unitary group. We exhibit a basis for this set, parameterized by diagrams of nonintersecting chords connecting pairs of points on a circle, and give a criterion for when the stabilizer is precisely the diagonal subgroup and not larger. This investigation is part of a larger program to partially classify entanglement type (local unitary equivalence class) via analysis of stabilizer structure.Comment: 4 pages, 3 figures. Version 2 has numerous small changes and correction