Abstract
Multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols. However obtaining a generic, structural understanding of entanglement in N-qubit systems is a long-standing open problem in quantum computer science. Here we show that multipartite quantum entanglement admits a compositional structure, and hence is subject to modern computer science methods.
Recall that two N-qubit states are SLOCC-equivalent if they can be inter-converted by stochastic local (quantum) operations and classical communication. There are only two SLOCC-equivalence classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, and we show that these exactly correspond with two kinds of internal commutative Frobenius algebras on ℂ2 in the symmetric monoidal category of Hilbert spaces and linear maps, namely ‘special’ ones and ‘anti-special’ ones. Within the graphical language of symmetric monoidal categories, the distinction between ‘special’ and ‘anti-special’ is purely topological, in terms of ‘connected’ vs. ‘disconnected’.
These GHZ and W Frobenius algebras form the primitives of a graphical calculus which is expressive enough to generate and reason about representatives of arbitrary N-qubit states. This calculus refines the graphical calculus of complementary observables in [5, ICALP’08], which has already shown itself to have many applications and admit automation. Our result also induces a generalised graph state paradigm for measurement-based quantum computing.
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References
Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: LiCS (2004), Revision: arXiv:quant-ph/0808.1023
Anders, J., Browne, D.E.: Computational power of correlations. Physical Review Letters 102 (2009), 050502. arXiv:0805.1002
Browne, D.E., Kashefi, E., Mhalla, M., Perdrix, S.: Generalized flow and determinism in measurement-based quantum computation. New Journal Physics 9, 250 (2007) arXiv:quant-ph/0702212
Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49, 11–32 (1987)
Coecke, B., Duncan, R.W.: Interacting quantum observables. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 298–310. Springer, Heidelberg (2008); Extended version: arXiv:0906.4725
Coecke, B., Edwards, B., Spekkens, R.W.: Phase groups and the origin of non-locality for qubits. In: QPL 2009. ENTCS (to appear, 2010), arXiv:1003.5005
Coecke, B., Paquette, E.O., Pavlovic, D.: Classical and quantum structuralism. In: Mackie, I., Gay, S. (eds.) Semantic Techniques for Quantum Computation, pp. 29–69. Cambridge University Press, Cambridge (2009), arXiv:0904.1997
Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology, pp. 567–604. Taylor and Francis, Abington (2007), arXiv:quant-ph/0608035
Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases (2008), arXiv:0810.0812
Coecke, B., Perdrix, S.: Environment and classical channels in categorical quantum mechanics (2010) arXiv:1004.1598
Coecke, B., Wang, B.-S., Wang, Q.-L., Wang, Y.-J., Zhang, Q.-Y.: Graphical calculus for quantum key distribution. In: QPL 2009. ENTCS (to appear, 2010)
D’Hondt, E., Panangaden, P.: The computational power of the W and GHZ states. Quantum Information and Computation 6, 173–183 (2006)
Dixon, L., Duncan, R., Kissinger, A.: quantomatic, http://dream.inf.ed.ac.uk/projects/quantomatic/
Duncan, R., Perdrix, S.: Graph states and the necessity of Euler decomposition. In: CiE 2009. LNCS, vol. 5635. Springer, Heidelberg (2009) arXiv:0902.0500
Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with generalised flow. Accepted for ICALP 2010 (2010)
Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62 (2000), 62314.arXiv:quant-ph/0005115
Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. American Journal of Physics 58, 1131–1143 (1990)
Joyal, A., Street, R.: The Geometry of tensor calculus I. Advances in Mathematics 88, 55–112 (1991)
Hein, M., Dür, W., Eisert, J., Raussendorf, R., Van den Nest, M., Briegel, H.-J.: Entanglement in graph states and its applications (2006) arXiv:quant-ph/0602096v1
Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)
Kock, J.: Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge University Press, Cambridge (2003)
Lamata, L., Leon, J., Salgado, D., Solano, E.: Inductive entanglement classification of four qubits under SLOCC. Physical Review A 75, 22318 (2007) arXiv:quant-ph/0610233
Lamata, L., Leon, J., Salgado, D., Solano, E.: Inductive classification of multipartite entanglement under SLOCC. Physical Review A 74, (2006), 52336.arXiv:quant-ph/0603243
Markham, D., Sanders, B.C.: Graph states for quantum secret sharing. Physical Review A 78 (2008), 42309.arXiv:0808.1532
Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Combinatorial Mathematics and its Applications, pp. 221–244. Academic Press, London (1971)
Selinger, P.: Finite dimensional Hilbert spaces are complete for dagger compact closed categories. In: QPL 2008. ENTCS (to appear, 2010)
Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, pp. 275–337. Springer, Heidelberg (2009) arXiv:0908.3347
Verstraete, F., Dehaene, J., De Moor, B., Verschelde, H.: Four qubits can be entangled in nine different ways. Physical Review A 65 (2002) 52112.arXiv:quant-ph/0109033
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Coecke, B., Kissinger, A. (2010). The Compositional Structure of Multipartite Quantum Entanglement. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6199. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14162-1_25
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