Following the B. Hiley belief that unresolved problems of conventional quantum mechanics could be the result of a wrong mathematical structure, an alternative basic structure is suggested. Critical part of the structure is modification of the sense of commonly used terms state, observable, measurement giving them a clear unambiguous definition. This concrete definition, along with using of variable complex plane, is quite natural in geometric algebra terms. It helps to establish a feasible language for the area of quantum computing. The suggested approach is used then in the fiber optics quantum information processing scenario.
Following the B. Hiley belief that unresolved problems of conventional quantum mechanics could be the result of a wrong mathematical structure, an alternative basic structure is suggested. Critical part of the structure is modification of the sense of commonly used terms state, observable, measurement giving them a clear unambiguous definition. This concrete definition, along with using of variable complex plane, is quite natural in geometric algebra terms. It helps to establish a feasible language for the area of quantum computing. The suggested approach is used then in the fiber optics quantum information processing scenario.
Recently suggested scheme [1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.
When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system.
The Geometric Algebra formalism opens the door to developing a theory upgrading conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions; unambiguous definition of states, observables, measurements bring into reality clear explanations of conventional weird quantum mechanical features, particularly the results of double split experiments where particles create diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of quantum mechanics.
Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra.
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.