Abstract
We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.
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Notes
By an abstract C*-algebra, we mean an object in a monoidal category satisfying certain requirements. By a concrete one, we mean an object satisfying those requirements in the category of (finite-dimensional) Hilbert spaces. This is not to be confused with terminology from functional analysis. There, a concrete C*-algebra is a *-subalgebra of the algebra \(\mathcal {B}(H)\) of bounded operators on a Hilbert space \(H\) that is uniformly closed, whereas an abstract C*-algebra is any Banach algebra with an involution satisfying \(\Vert a^*a\Vert =\Vert a\Vert ^2\); these notions are equivalent by the Gelfand–Naimark–Segal construction; see e.g. [16, Theorem I.9.12].
There is a closely related notion called specialness. A dagger Frobenius algebra is normal if and only if it is special and symmetric. In \(\mathbf {FHilb} \), normal and special coincide for dagger Frobenius algebras.
Commutativity might be too strong a notion of “completely classical” system in the abstract. A weaker notion of broadcastability, that coincides with commutativity in \(\mathbf {FHilb} \), seems more reasonable. Subsequent work will investigate such more operational notions of classicality.
Notice also that \(\mathbb {R}_{\ge 0}\) is not a quantale under its usual ordering.
References
Abramsky, S., Coecke, B.: Categorical Quantum Mechanics. Elsevier, Amsterdam (2008)
Abramsky, S., Heunen, C.: H*-algebras and nonunital frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Clifford Lect. 71, 1–24 (2012)
Alicki, R.: Comment on ‘reduced dynamics need not be completely positive’. Phys. Rev. Lett. 75, 3020 (1995)
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Boixo, S., Heunen, C.: Entangled and sequential quantum protocols with dephasing. Phys. Rev. Lett. 108, 120–402 (2012)
Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)
Coecke, B.: Axiomatic description of mixed states from Selinger’s CPM-construction. Electron. Notes Theor. Comput. Sci. 210, 3–13 (2008)
Coecke, B. (ed.): New Structures for Physics. No. 813 in Lecture Notes in Physics. Springer, Berlin (2009).
Coecke, B., Duncan, R.: Interacting quantum observables: categorical algebra and diagrammatics. New J. Phys. 13, 043,016 (2011).
Coecke, B., Heunen, C.: Pictures of complete positivity in arbitrary dimension. Electron. Proc. Theor. Comput. Sci. 95, 27–35 (2012)
Coecke, B., Heunen, C., Kissinger, A.: Compositional quantum logic. In: Coecke, B., Ong, L., Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundationsno. 7860 in Lectures Notes in Computer Science, pp. 21–36. Springer, New York (2013).
Coecke, B., Paquette, É.O., Pavlović, D.: Classical and quantum structuralism. In: Gay, S., Mackey, I. (eds.) Semantic Techniques in Quantum Computation, pp. 29–69. Cambridge University Press, Cambridge (2010)
Coecke, B., Pavlović, D.: Quantum measurements without sums. Mathematics of Quantum Computing and Technology. Taylor and Francis, New York (2007)
Coecke, B., Pavlović, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. 23(3), 555–567 (2012)
Coecke, B., Perdrix, S.: Environment and classical channels in categorical quantum mechanics. Computer Science Logic, pp. 230–244. Springer, New York (2010).
Davidson, K.R.: C*-algebras by Example. American Mathematical Society, Providence (1991)
Duncan, R.: Types for Quantum Computing. Ph.D. thesis, Oxford University, Oxford (2006).
Heunen, C., Contreras, I., Cattaneo, A.S.: Relative frobenius algebras are groupoids. J. Pure Appl. Algebra 217, 114–124 (2013)
Heunen, C., Kissinger, A., Selinger, P.: Completely positive projections and biproducts. In: Proceedings of Quantum Physics and Logic X (2013) (to appear) arxiv:1308.4557
Heunen, C., Vicary, J.: Introduction to Categorical Quantum Mechanics. Oxford University Press (to appear)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Keyl, M.: Fundamentals of quantum information theory. Phys. Rep. 369, 431–548 (2002)
Keyl, M., Werner, R.F.: Channels and maps. In: Bruß, D., Leuchs, G. (eds.) Lectures on Quantum Information, pp. 73–86. Wiley, Hoboken (2007)
Li, B.: Real Operator Algebras. World Scientific, Singapore (2003)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1971).
Panangaden, P., Paquette, É.O.: New structures for physics. In: Coecke, B. (ed.) New Structures for Physics. No. 813 in Lecture Notes in Physics, pp. 939–979. Springer, New York (2009).
Paulsen, V.: Completely Bounded Maps and Operators Algebras. Cambridge University Press, Cambridge (2002)
Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 74, 1060–1062 (1994)
Redei, M.: Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud. Hist. Philos. Sci. Part B 27, 493–510 (1996)
Rosenthal, K.I.: Quantales and their Applicatoins. Pitman Research Notes in Mathematics. Longman Scientific & Technical, Harlow (1990)
Ruan, Z.J.: On real operator spaces. Acta Math. Sinica 19(3), 485–496 (2003)
Selinger, P.: Dagger compact closed categories and completely positive maps. Quantum Programming Languages, Electronic Notices in Theoretical Computer Science, pp. 139–163. Elsevier, Amsterdam (2007).
Selinger, P.: Idempotents in dagger categories. Quantum Programming Languages, Electronic Notes in Theoretical Computer Science, pp. 107–122. Elsevier, Amsterdam (2008).
Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics. No. 813 in Lecture Notes in Physics, pp. 289–356. Springer, New York (2009).
Shaji, A., Sudarshan, E.C.G.: Who’s afraid of not completely positive maps? Phys. Lett. A 341(1–4), 48–54 (2005)
Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)
Størmer, E.: Positive Linear Maps of Operator Algebras. Springer, New York (2013).
Vicary, J.: Categorical formulation of finite-dimensional quantum algebras. Commun. Math. Phys. 304(3), 765–796 (2011)
Zakrzewski, S.: Quantum and classical pseudogroups I. Commun. Math. Phys. 134, 347–370 (1990)
Życzkowski, K., Bengtsson, I.: On duality between quantum states and quantum maps. Open Syst. Inf. Dyn. 11, 3–42 (2004)
Acknowledgments
This research was supported by the Engineering and Physical Sciences Research Council Fellowship EP/L002388/1, and the John Templeton Foundation.
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Coecke, B., Heunen, C. & Kissinger, A. Categories of quantum and classical channels. Quantum Inf Process 15, 5179–5209 (2016). https://doi.org/10.1007/s11128-014-0837-4
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DOI: https://doi.org/10.1007/s11128-014-0837-4