A graphical calculus for quantum observables
by
Bob Coecke, Ross Duncan
Computing Laboratory, University of Oxford

Not too long ago the distinction between pure ^^] and pure ^^] data has been characterised by the availability of copying [WZ82] and deleting [PB00] operations. This manner of distinguishing classical and quantum data is very different from the usual ones (commutativity vs. non-commutativity; distributivity vs. non-distributivity; etc.). Since both copying and deleting are actual processes, axiomatising these requires a framework for describing processes rather than properties. The notion of classical object [CP07] axiomatises copying and deleting operations (and their adjoints) as a special -Frobenius algebra over the state space in question, within the categorical quantum axiomatics initiated in [AC04]. Any such classical structure corresponds to a basis of the state space of the quantum system, or from an operational point of view, an observable. Based on the notion of classical object one can construct cloning and deleting operations corresponding to any given observable. However, consideration of the classical structures alone misses a key ingredient of quantum mechanics: the interaction of distinct, non-commuting observables. In this work we address that question by means of a bialgebraic formulation describing the interaction of a pair maximally incompatible observables; or equivalently, a pair of mutually unbiased bases [Sch60]. In this fashion we provide a genuinely quantum structure which provides a categorical counterpart to the nondistributive Birkhoff-von Neumann lattice structure [BvN36] for non-commuting observables.

This bialgebraic structure comes with a graphical calculus derived from those found in [Pen71, Coe05, Sel05, CP06, Dun07] which provides sound representation of the underlying algebra, while also retaining a quantitative element: one can truly reason about probabilities in this setting. Indeed it is possible to prove the correctness of various protocols, for example, teleportation and key exchange, without any additional extensional assumptions. In the same way one can derive elementary properties of quantum gates and prove equivalences between quantum circuits and measurement-based quantum computations.

Strictly speaking, the structure introduced is not a true bialgebra, but a scaled variant thereof, where each equation introduces a quantitative element corresponding to the degree of incompatibility between the non-commuting classical structures. This scalar is the square root of the dimension of the space upon which the classical operations act. This scalar element aside, the resulting calculus strongly resembles existing structures studied in the context of quantum groups, knot theory, and quantum field theories.

References:

[AC04] S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science: LICS 2004, pages 415^S425. IEEE Computer Society, 2004.

[BvN36] G. Birkhoff and J. von Neumann. The logic of quantum mechanics. Annals of Mathematics, 37(4):823^S843, October 1936.

[Coe05] B. Coecke. Kindergarten quantum mechanics. Lecture Notes, 2005.

[CP06] B. Coecke and E. O. Paquette. POVMs and Naimark^Ys theorem without sums. In Proceedings of the 4th International Workshop on Quantum Programming Languages, 2006.

[CP07] B. Coecke and D. Pavlovic. Quantum measurements without sums. In G. Chen, L. H. Kauffman, and Jr Lomonaco, S.J., editors, The Mathematics of Quantum Computation and Technology, CRC Applied Mathematics Nonlinear Science. Taylor and Francis, 2007.

[Dun07] R Duncan. Types for Quantum Computation. PhD thesis, Oxford University, 2007.

[PB00] A.K. Pati and S. L. Braunstein. Impossibility of deleting an unknown quantum state. Nature, 404:164^S165, 2000.

[Pen71] R. Penrose. Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications, pages 221^S244. Academic Press, 1971.

[Sch60] J. Schwinger. Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A., 46:570^S579, 1960.

[Sel05] P. Selinger. Dagger compact closed categories and completely positive maps. In Proceedings of the 3rd International Workshop on Quantum Programming Languages, 2005.

[WZ82] W. Wootters and W. Zurek. A single quantum cannot be cloned. Nature, 299:802^S803, 1982.

Date received: June 3, 2007