The computational power of the W and GHZ states

D’Hondt, Ellie; Panangaden, Prakash

doi:10.26421/qic6.2-3

Cited by 76 publications

(65 citation statements)

References 14 publications

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“…11|) is a separable state. On the other hand, W states still retain a considerable amount of entanglement after the removal of an arbitrary particle, and it has been shown that the n-qubit W state is the optimal state for leader election [24]. This shows that the entanglement of GHZ and W states is of a different nature, and such qualitative aspects of entanglement and their characterisation will be investigated in the next section.…”

Section: Bipartite and Multipartite Systemsmentioning

confidence: 95%

“…Returning to the concept of symmetry in physics, we recall that permutation-symmetric quantum states appear naturally in some systems [22,23], that it is possible to prepare them experimentally [3][4][5][6], and that they have found some applications [24][25][26][27]. Many canonical states that appear in quantum information science are symmetric, e.g.…”

Section: Motivationmentioning

confidence: 99%

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Classification of Entanglement in Symmetric States

Aulbach

2012

Int. J. Quantum Inform.

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Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.

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Section: Bipartite and Multipartite Systemsmentioning

confidence: 95%

Section: Motivationmentioning

confidence: 99%

Classification of Entanglement in Symmetric States

Aulbach

2012

Int. J. Quantum Inform.

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“…that will disconnect the other two with certainty [BR01]. This enables the state | W to support protocols for which | GHZ is inadequate, see for example [DP06].…”

Section: The Entanglement Classification Problemmentioning

confidence: 99%

The algebra of entanglement and the geometry of composition

Hadzihasanovic

2017

Preprint

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String diagrams turn algebraic equations into topological moves. These moves have often-recurring shapes, involving the sliding of one diagram past another. In the past, this fact has mostly been considered in terms of its computational convenience; this thesis investigates its deeper reasons.In the first part of the thesis, we individuate, at its root, the dual nature of polygraphs -freely generated higher categories -as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces": CW complexes whose cells have their boundary subdivided into an input and an output region. Operations of polygraphs modelled on operations of topological spaces, including an asymmetric tensor product refining the topological product of spaces, can then be used as the foundation of a compositional universal algebra, where sliding moves of string diagrams for an algebraic theory arise from tensor products of sub-theories. Such compositions come automatically with higher-dimensional coherence cells. We provide several examples of compositional reconstructions of higher algebraic theories, including theories of braids, homomorphisms of algebras, and distributive laws of monads, from operations on polygraphs.In this regard, the standard formalism of polygraphs, based on strict ω-categories, suffers from some technical problems and inadequacies, including the difficulty of computing tensor products. As a solution, we propose a notion of regular polygraph, barring cell boundaries that are degenerate, that is, not homeomorphic to a disk of the appropriate dimension. We develop the theory of globular posets, based on ideas of poset topology, in order to specify a category of shapes for cells of regular polygraphs. We prove that these shapes satisfy our non-degeneracy requirement, and show how to calculate their tensor products. Then, we introduce a notion of weak unit for regular polygraphs, allowing us to recover weakly degenerate boundaries in low dimensions. We prove that the existence of weak units is equivalent to the existence of cells satisfying certain divisibility properties -an elementary notion of equivalence cell -which prompts new questions on the relation between units and equivalences in higher categories.In the second part of the thesis, we turn to specific applications of diagrammatic algebra to quantum theory. First, we re-evaluate certain aspects of quantum theory from the point of view of categorical universal algebra, which leads us to define a 2-dimensional refinement of the category of Hilbert spaces. Then, we focus on the problem of axiomatising fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits.The ZW calculus has specific advantages over its predecessors, the ZX calculi, including the existence of a computationally meaningful normal form, and of a fragment whose diagrams can be interpreted physically as setups of fermionic oscillators. Moreover, the choice of its generators reflects an operational clas...

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“…While quantum communication often relies on entanglement distribution between two end-nodes, there are a several possible applications that go beyond the two-party paradigm and require multipartite entanglement, which has no classical analogue. Important examples of these applications are quantum sensor networks [7][8][9][10], multi-party quantum communication [11][12][13] and computation [14,15].…”

Section: Introductionmentioning

confidence: 99%