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arXiv:quant-ph/0211020 · Physical Review A

Entangled graphs: Bipartite entanglement in multiqubit systems

Martin Plesch, Vladimír Bužek

Published Jan 29, 2003• 55 citations
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Abstract

Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits, we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex), while a bipartite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multiqubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superposition of vectors from a subspace of the ${2}^{N}$-dimensional Hilbert space, whose dimension grows linearly with the number of entangled pairs.

DOI
10.1103/physreva.67.012322

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Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits, we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex), while a bipartite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multiqubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superposition of vectors from a subspace of the 2N{2}^{N}2N-dimensional Hilbert space, whose dimension grows linearly with the number of entangled pairs.
Bipartite correlations in multiqubit systems cannot be shared freely. The presence of entanglement or classical correlation on certain pairs of qubits may imply correlations on other pairs. We present a method of characterization of bipartite correlations in multiqubit systems using a concept of entangled graphs which has been introduced in our earlier work [M. Plesch and V. Bu\ifmmode \check{z}\else \v{z}\fi{}ek, Phys. Rev. A 67, 012322 (2003)]. In entangled graphs, each qubit is represented by a vertex while the entanglement and classical correlations are represented by two types of edges. We prove by construction that any entangled graph with classical correlations can be represented by a mixed state of N qubits. However, not all entangled graphs with classical correlations can be represented by a pure state.
We study how bi-partite quantum entanglement (measured in terms of a concurrence) can be shared in multi-qubit systems. We introduce a concept of the entangled graph such that each qubit of a multi-partite system is associated with a vertex while a bi-partite entanglement between two specific qubits is represented by an edge. We prove that any entangled graph can be associated with a pure state of a multi-qubit system. We also derive bounds on the concurrence for some weighted entangled graphs (the weight corresponds to the value of concurrence associated with the given edge).
Bipartite correlations in multiqubit systems cannot be shared freely. The presence of entanglement or classical correlation on certain pairs of qubits may imply correlations on other pairs. We present a method of characterization of bipartite correlations in multiqubit systems using a concept of entangled graphs which has been introduced in our earlier work @M. Plesch and V. Bužek, Phys. Rev. A 67, 012322 ~2003!#. In entangled graphs, each qubit is represented by a vertex while the entanglement and classical correlations are represented by two types of edges. We prove by construction that any entangled graph with classical correlations can be represented by a mixed state of N qubits. However, not all entangled graphs with classical correlations can be represented by a pure state.
Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition we present a logic network with O(N2) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.
We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.
Graph states are intuitive representations of quantum states that are useful in quantum computing and quantum information. In this paper, we investigate the role of entanglement and local unitary equivalence in quantum computing through graph states in high-dimensional Hilbert spaces. We introduce bipartitions on the vertices sets and demonstrate separability and local unitary equivalence based on different choices of partitions. By performing the two-qubit controlled phase shift gates on various pairs of multiqubit system, one can generate a genuinely entangled graph state based on simple connected graphs. We extend 2-dimensional Hilbert spaces to higher and mixed dimensions and examine pure quantum states that cannot be separated by any bipartition. In addition, we generalize the analysis on classification of local unitary equivalence beyond the controlled-Z gates to controlled phase gates. By examining graphs constructed from bipartition graphs, we have shown that graphs with identical bipartition subgraph structures are local unitary equivalent.