The logic of quantum mechanics - take II

Abstract

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. One year later, together with Birkhoff, they published "The logic of quantum mechanics". However, it is fair to say that this program was never successful nor does it have anything to do with logic. So what is logic? We will conceive logic in two manners: (1) Something which captures the mathematical content of language (cf `and', `or', `no', `if ... then' are captured by Boolean algebra); (2) something that can be encoded in a `machine' and enables it to reason. Recently we have proposed a new kind of `logic of quantum mechanics' [4]. It follows Schrodinger in that the behavior of compound quantum systems, described by the tensor product [2, again 75 years ago], that captures the quantum behaviors. Over the past couple of years we have played the following game: how much quantum phenomena can be derived from `composition + 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. 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. Most importantly, it paved the way to automate quantum reasoning [5,6], and also enables to model meaning for natural languages [7,8]. That is, we are truly taking `quantum logic' now! If time permits, we also discuss how this logical view has helped to solve concrete problems in quantum information. [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] G Birkhoff and J von Neumann (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823843. [3] E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451. [4] B Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83. arXiv:0908.1787 [5] L Dixon, R Duncan, A Kissinger and A Merry. http://dream.inf.ed.ac.uk/projects/quantomatic/ [6] L Dixon and R Duncan (2009) Graphical reasoning in compact closed categories for quantum computation. Annals of Mathematics and Articial Intelligence 56, 2342. [7] B Coecke, M Sadrzadeh & S Clark (2010) Linguistic Analysis 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394 [8] New scientist (11 Dec 2011) Quantum links let computers read.