These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).2 It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).2 It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds'. This picture calculus can be seen as a very substantial extension of Dirac's notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph/0402014. For a survey on the `what', the `why' and the `hows' I refer to a previous set of lecture notes quant-ph/0506132. In a last section we provide some pointers to the body of technical literature on the subject.
Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843]. Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of so‐called Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why’ and the ‘hows’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.
This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in low-level formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly important in this treatment, as is appropriate, since entanglement is a fundamental driver of quantum computation. The benefits two-fold: as well as producing foundational tools for the study of quantum computation this work also connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting. We describe, following Abramsky and Coecke, how quantum mechanics may be carried out without reference to Hilbert space, in a strongly compact closed category. In particular we show how to freely construct a categorical model of abstract quantum mechanics from an arbitrary category. We introduceMultiplicative Categorical Quantum Logic (mCQL), a sequent calculus whose proof rules capture the structure of compact closed categories. This sequent calculus is interpreted in a freely generated compact closed category, and its semantics is sound with respect to cut elimination. We define an equivalent graphical syntax, similar to linear logic’s proof-nets, and prove that these proof-nets provide a full and faithful representation of any freely generated compact closed category. Further analysis of the structure of quantum states which correspond to mCQL proofs using multiplicative linear logic shows that the linear type system describes the quantum entanglement found in such states. We show that the entanglement present in these states is always of a particularly simple form: collections of entangled pairs. In order to tackle arbitrary entanglement, we generalise the work of Kelly and Laplaza to give a representation theorem for the free compact closed category by a polycategory. Such categories are shown to be equivalent to a generalised system of proof-nets whose axioms may have more than one premise
This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet emerged. The picture is complicated by a diversity of examples that lacks a unifying framework for proving theorems and discovering new protocols. We use category theory to construct a high-level language that abstracts the quantum mechanical properties of braided systems. We exploit this framework to propose an axiomatic description of braided quantum information intended for topological quantum computation. In this installment we first generalize the primordial Abramsky-Coecke "quantum information flow" paradigm from compact closed categories to right-rigid strict monoidal categories. We then study dagger structures for rigid and/or braided categories and formulate a graphical dagger calculus. We then propose two generalizations of strongly compact closed categories. Finally we study partial traces in the context of dagger categories.
Dagger compact closed categories were recently introduced by Abramsky and Coecke, under the name “strongly compact closed categories”, as an axiomatic framework for quantum mechanics. We present a graphical language for dagger compact closed categories, and sketch a proof of its completeness for equational reasoning. We give a general construction, the CPM construction, which associates to each dagger compact closed category its “category of completely positive maps”, and we show that the resulting category is again dagger compact closed. We apply these ideas to Abramsky and Coecke's interpretation of quantum protocols, and to D'Hondt and Panangaden's predicate transformer semantics.