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arXiv:1203.4056 · Quantum Information Processing

Local controllability of quantum systems

Zbigniew Puchała

Published Mar 22, 2012• 6 citations
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Abstract

We give a criterion that is sufficient for controllability of multipartite quantum systems. We generalize the graph infection criterion to the quantum systems that cannot be described with the use of a graph theory. We introduce the notation of hypergraphs and reformulate the infection property in this setting. The introduced criterion has a topological nature and therefore it is not connected to any particular experimental realization of quantum information processing.

DOI
10.1007/s11128-012-0391-x

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We give a criterion that is sufficient for controllability of multipartite quantum systems. We generalize the graph infection criterion to the quantum systems that cannot be described with the use of a graph theory. We introduce the notation of hypergraphs and reformulate the infection property in this setting. The introduced criterion has a topological nature and therefore it is not connected to any particular experimental realization of quantum information processing.
Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary (LU) and stochastic local operations with classical communication (SLOCC), the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation (MBQC), and to their non-stabilizer character - quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.
Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary and stochastic local operations with classical communication, the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation, and to their non-stabilizer character, quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.
Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a different family of quantum states, called the hypergraph states. Moreover, we also try to do the following: (i) show that real equally weighted states, which occur in Grover and Deutsch-Jozsa algorithms, are equivalent to hypergraph states; (ii) describe the relations among hypergraph states, graph states, and stabilizer states; (iii) provide some transformation rules, stated in purely hypergraph theoretical terms, which completely characterize the evolution of hypergraph states under some local operations, including operators in the Pauli group and some special local Pauli measurements; and (iv) investigate some properties of multipartite entanglement of hypergraph states by hypergraph theory.
Hypergraph states are multiqubit states whose combinatorial description and entanglement properties generalize the well-studied class of graph states. Graph states are important in applications such as measurement-based quantum computation and quantum error correction. The study of hypergraph states, with their richer multipartite entanglement and other nonlocal properties, has a promising outlook for new insight into multipartite entanglement. We present results analyzing local unitary symmetries of hypergraph states, including both continuous and discrete families of symmetries. In particular, we show how entanglement types can be detected and distinguished by certain configurations in the hypergraphs from which hypergraph states are constructed.
In this paper, we consider discrete time quantum walks on graphs with coin focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. We then prove general results and criteria relating controllability to the algebraic and topological properties of the walk. As a consequence of these results, we prove that if the degree of the underlying graph is larger than N2\frac{N}{2}2N​, where NNN is the number of nodes, the quantum walk is always completely controllable, i.e., it is possible to having it to evolve according to an arbitrary unitary evolution. Another result is that controllability for decentralized models only depends on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.