Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces.Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product.The corresponding axioms define classical spaces as objects that allow copying and deleting data.Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data.The sums turn out to be an implicit implementation of this capability.Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus.In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras. QUANTUM MEASUREMENTS WITHOUT SUMS1 But while computing devices do manipulate strings of 0s and 1s, and high-level modern programming is a matter of providing a convenient interface with that process, the language for quantum information and computation we seek is not a convenient superstructure, but the meaningful infrastructure. 2A closely related knot-theoretical scheme has been put forward by Kauffman in [23].7 See [17] for a discussion why we call this 'formal decohering'.
Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces.Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product.The corresponding axioms define classical spaces as objects that allow copying and deleting data.Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data.The sums turn out to be an implicit implementation of this capability.Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus.In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras. QUANTUM MEASUREMENTS WITHOUT SUMS1 But while computing devices do manipulate strings of 0s and 1s, and high-level modern programming is a matter of providing a convenient interface with that process, the language for quantum information and computation we seek is not a convenient superstructure, but the meaningful infrastructure. 2A closely related knot-theoretical scheme has been put forward by Kauffman in [23].7 See [17] for a discussion why we call this 'formal decohering'.
Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are dagger-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.
Abstract Sums play a prominent role in the formalisms of quantum mechanics, whether for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capability. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
In recent work, several researchers including the authors have developed a categorical formalization of quantum mechanics in terms of symmetric monoidal dagger categories. In this framework, classical data turned out to be represented by an algebraic structure, that of special commutative dagger Frobenius algebras. This structure captures the distinct capabilities that apply to classical data – that they can be copied and deleted. In the present paper, we provide categorical semantics and diagrammatic representations of deterministic, nondeterministic, and probabilistic operations over classical data represented in this way. Moreover, a combination of some fundamental categorical constructions (the Kleisli construction of the category of free algebras and the Grothendieck construction of the total category of an indexed category) with the specific categorical presentations of pure and mixed quantum states provides a resource-sensitive categorical account of classical control of quantum data and of classical data resulting from quantum measurements, as well as of the classical data processing that may happen in between measurements and controls. Along the way we also discover some apparently novel quantum typing structures. One of the salient features of categorical quantum mechanics is still its graphic calculus, which allows succinct presentations of diverse quantum protocols. The elements of an abstract stochastic calculus are beginning to emerge from it, pointing toward convenient refinements of resource-sensitive logics that are hoped to capture the probabilistic content and limited observability of quantum data.
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.
Abstract Measurement lies at the core of the quantum formalism. This may seem a statement of the intuitively obvious for any theoretical structure with a pretension to predict the outcomes of observations made or experiments performed on objects in the real world. But classical mechanics is a theorys of the physical properties of objects and their behaviour when exposed to different kinds of forces, and there is no particularly significant role for measurement other than a passive role involving the recording of those properties or behaviour. In contrast, measurement in quantum theory takes on a far deeper, almost ominous, significance