Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in ...terms of classical mechanics. We survey this programme in terms of algebraic quantum theory.
Purity through Factorisation Cunningham, Oscar; Heunen, Chris
Electronic proceedings in theoretical computer science,
02/2018, Letnik:
266, Številka:
Proc. QPL 2017
Journal Article
Odprti dostop
We give a construction that identifies the collection of pure processes (i.e. those which are deterministic, or without randomness) within a theory containing both pure and mixed processes. Working ...in the framework of symmetric monoidal categories, we define a pure subcategory. This definition arises elegantly from the categorical notion of a weak factorisation system. Our construction gives the expected result in several examples, both quantum and classical.
The standard formalism of quantum theory treats space and time in fundamentally different ways. In particular, a composite system at a given time is represented by a joint state, but the formalism ...does not prescribe a joint state for a composite of systems at different times. If there were a way of defining such a joint state, this would potentially permit a more even-handed treatment of space and time, and would strengthen the existing analogy between quantum states and classical probability distributions. Under the assumption that the joint state over time is an operator on the tensor product of single-time Hilbert spaces, we analyse various proposals for such a joint state, including one due to Leifer and Spekkens, one due to Fitzsimons, Jones and Vedral, and another based on discrete Wigner functions. Finding various problems with each, we identify five criteria for a quantum joint state over time to satisfy if it is to play a role similar to the standard joint state for a composite system: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time steps is associative. We show that no construction satisfies all these requirements. If Hermiticity is dropped, then there is an essentially unique construction that satisfies the remaining four criteria.
Axioms for the category of Hilbert spaces Heunen, Chris; Kornell, Andre
Proceedings of the National Academy of Sciences - PNAS,
03/2022, Letnik:
119, Številka:
9
Journal Article
Recenzirano
Odprti dostop
We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical ...structure. This addresses a question about the mathematical foundations of quantum theory raised in reconstruction programs such as those of von Neumann, Mackey, Jauch, Piron, Abramsky, and Coecke.
We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are ...often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics.
Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions.
We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the ...correspondence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal'cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quantum features. These models are closer to Hilbert spaces than relations over sets in several respects: Heisenberg uncertainty, impossibility of broadcasting, and behavedness of rank one morphisms.
Axiomatizing complete positivity Cunningham, Oscar; Heunen, Chris
Electronic proceedings in theoretical computer science,
11/2015, Letnik:
195, Številka:
Proc. QPL 2015
Journal Article
Odprti dostop
There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, ...whereas the other also allows classical systems on an equal footing. The former has been axiomatized using environment structures. We extend this axiomatization to the latter by introducing decoherence structures.
The CBH theorem characterises quantum theory within a C*-algebraic framework. Namely, mathematical properties of C*-algebras modelling quantum systems are equivalent to constraints that are ...information-theoretic in nature: (1) noncommutativity of subalgebras is equivalent to impossibility of signalling; (2) noncommutativity of the whole algebra is equivalent to impossibility of broadcasting; (3) the existence of entangled states is implied by the impossibility of secure bit commitment (with the converse conjectured). However, the C*-algebraic framework has drawn criticism as it already contains much of the mathematical structure of quantum theory such as complex linearity. We address this issue by a generalising C*-algebras categorically. In this framework, equivalence (1) holds, equivalence (2) becomes a strict implication, and implication (3) fails in general. Thus we identify exactly what work is being done by the complex-linear structure of C*-algebras. In doing so, we uncover a richer hierarchy of notions of ‘classicality’ and ‘quantumness’ of information than visible in the concrete case.
Compact categories have lately seen renewed interest via applications to
quantum physics. Being essentially finite-dimensional, they cannot accomodate
(co)limit-based constructions. For example, they ...cannot capture protocols such
as quantum key distribution, that rely on the law of large numbers. To overcome
this limitation, we introduce the notion of a compactly accessible category,
relying on the extra structure of a factorisation system. This notion allows
for infinite dimension while retaining key properties of compact categories:
the main technical result is that the choice-of-duals functor on the compact
part extends canonically to the whole compactly accessible category. As an
example, we model a quantum key distribution protocol and prove its correctness
categorically.
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