|
Organizers |
The logic of quantum mechanics - take 2
by
Bob Coecke
Oxford, UK
It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: ``I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.'' (sic) [1] His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. So what are these key quantum behaviors then? [2,3]
For Schrodinger this is the behavior of compound quantum systems, described by the tensor product [4, again 75 years ago]. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0's and 1's in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct.
Over the past couple of years we have played the following game: how much quantum phenomena can be derived from `compoundness + epsilon'. It turned out that epsilon can be taken to be `very little', surely not involving anything like continuum, fields, vector spaces, but merely a `two-dimensional space' of temporal composition (cf `and then') and compoundness (cf `while'), together with some very natural purely operational assertion, including one which in a constructive manner asserts entanglement; among many other things, trace structure (cf von Neumann above) then follow [5, survey]. In a very short time, this radically different approach has produced a universal graphical language for quantum theory which helped to resolve some open problems [6,7,8], and give a particularly elegant account on quantum classical interaction, on the basis of complementarity [9]. It also paved the way to automate quantum reasoning [10] and has even helped to solve problems outside physics, most notably in modeling meaning for natural languages [11].
This `categorical quantum mechanics' research program started with [12].
[1] M Redei (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Phil Mod Phys 27, 493-510.
[2] For von Neumann, initially these were the propositions that one could measure with certainty, an idea that he later abandoned in favor of the trace structure, which generates probability [1].
[3] Still, today for most physicists `quantum' is synonym for `Hilbert space', which of course is not unrelated to the dominant ``shut up and calculate''-conception of quantum theory.
[4] E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451.
[5] B Coecke (2010) Quantum picturalism. Cont Phys 51, 59-83. arXiv:0908.1787
[6] B Coecke, B Edwards and RW Spekkens (2010) Phase groups and the origin of non-locality for qubits. ENTCS, to appear. arXiv:1003.5005
[7] R Duncan and S Perdrix (2010) Rewriting measurement-based quantum computations with generalised flow. ICALP'10.
[8] B Coecke and A Kissinger (2010) The compositional structure of multipartite quantum entanglement. ICALP'10. arXiv:1002.2540
[9] B Coecke and S Perdrix (2010) Environment and classical channels in categorical quantum mechanics. CSL'10. arXiv:1004.1598
[10] L Dixon, R Duncan & A Kissinger. dream.inf.ed.ac.uk/projects/quantomatic/
[11] B Coecke, S Clark & M Sadrzadeh (2010) Ling Anal 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394
[12] S Abramsky & B Coecke (2004) A categorical semantics of quantum protocols. LiCS '04. arXiv:0808.1023
Date received: November 8, 2010
Copyright © 2010 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbbr-12.