Strongly compact closed semantics (Q2852356)

The author, together with \textit{S. Abramsky}, has axiomatized the notion of quantum information flow in category-theoretic terms in [``A categorical semantics of quantum protocols'', in: Proceedings of the 19th annual IEEE symposium on logic in computer science, LiCS'04. Los Alamitos: IEEE Press. 415--425 (2004), \url{arXiv:quant-ph/0402130}] and [\textit{S. Abramsky} and \textit{B. Coecke}, Theory Appl. Categ. 14, 111--124 (2005; Zbl 1065.18005)].NEWLINEThis justifies, via the work of [\textit{G. M. Kelly} and \textit{M. L. Laplaza}, J. Pure Appl. Algebra 19, 193--213 (1980; Zbl 0447.18005)] and [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 88, No. 1, 55--112 (1991; Zbl 0738.18005)], the graphical calculus in [``A categorical semantics of quantum protocols'', in:NEWLINEProceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Science Press, (2004)], [``Quantum information flow, concretely,NEWLINEabstractly'', in: Proceedings of the 2nd Workshop on Quantum Programming Languages, 57--74 (2004)] and [\textit{S. Abramsky} and \textit{B. Coecke}, Theory Appl. Categ. 14, 111--124 (2005; Zbl 1065.18005)], which is no other than aNEWLINEtranscription of strong compact closure or what Selinger calls dagger compact closure. It is well known that strong compact closure implies compact closureNEWLINE[\textit{G. M. Kelly}, Lect. Notes Math. 281, 66--105 (1972; Zbl 0243.18015)] while compact closure implies *-autonomy [\textit{M. Barr}, *-autonomous categories. With an appendix by Po-Hsiang Chu. Lecture Notes in Mathematics. 752. Berlin-Heidelberg-New York: Springer-Verlag. (1979; Zbl 0415.18008)], which is the semantics for the multiplicative fragment of linear logic [\textit{R. A. G. Seely}, in: Categories in computer science and logic, Proc. AMS--IMS--SIAM Jt. Summer Res. Conf., Boulder/Colo. 1987, Contemp. Math. 92, 371--382 (1989; Zbl 0674.03007)]. A very substantial contribution to the author's program was [\textit{P. Selinger}, in: Proceedings of the 3rd international workshop on quantum programming languages (QPL 2005), DePaul University, Chicago, IL, USA, June 30 -- July 1, 2005. Selinger, Peter (ed.), Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 170, 139--163 (2007; Zbl 1277.18008)], which has discovered the construction turning any stronglyNEWLINEcompact closed category of pure states and pure operations into that of mixed states and completely positive maps. The author has discovered theNEWLINEpreparation-state agreement axiom in [\textit{B. Coecke}, in: Proceedings of the 3rd international workshop on quantum programming languages (QPL 2005), DePaul University, Chicago, IL, USA, June 30 -- July 1, 2005. Selinger, Peter (ed.), Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 170, 49--72 (2007; Zbl 1277.18006)], which was combined by the author with the structural content of Selinger's construction in [\textit{B. Coecke}, in: Proceedings of the 4th international workshop on quantum programming languages (QPL 2006), Oxford, UK, 17--19 July 2006. Selinger, Peter (ed.), Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 210, 3--13 (2008; Zbl 1279.81010)]. This paper is a survey of this ongoing work of the author and Samson Abramsky, as it was presented at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005). For a more extensive exposition, one is referred to [\textit{S. Abramsky} and \textit{B. Coecke}, in: Handbook of quantum logic and quantum structures. Quantum logic. With a foreword by Anatolij Dvurečenskij. Engesser, Kurt (ed.) et al., Amsterdam: Elsevier/North--Holland 261--323 (2009; Zbl 1273.81014)].NEWLINENEWLINEFor the entire collection see [Zbl 1272.68017].