31 research outputs found
Dynamical Correspondence in a Generalized Quantum Theory
In order to figure out why quantum physics needs the complex Hilbert space,
many attempts have been made to distinguish the C*-algebras and von Neumann
algebras in more general classes of abstractly defined Jordan algebras (JB- and
JBW-algebras). One particularly important distinguishing property was
identified by Alfsen and Shultz and is the existence of a dynamical
correspondence. It reproduces the dual role of the selfadjoint operators as
observables and generators of dynamical groups in quantum mechanics. In the
paper, this concept is extended to another class of nonassociative algebras,
arising from recent studies of the quantum logics with a conditional
probability calculus and particularly of those that rule out third-order
interference. The conditional probability calculus is a mathematical model of
the Lueders-von Neumann quantum measurement process, and third-order
interference is a property of the conditional probabilities which was
discovered by R. Sorkin in 1994 and which is ruled out by quantum mechanics. It
is shown then that the postulates that a dynamical correspondence exists and
that the square of any algebra element is positive still characterize, in the
class considered, those algebras that emerge from the selfadjoint parts of
C*-algebras equipped with the Jordan product. Within this class, the two
postulates thus result in ordinary quantum mechanics using the complex Hilbert
space or, vice versa, a genuine generalization of quantum theory must omit at
least one of them.Comment: 11 pages in Foundations of Physics 201
A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras
Quantum theory's Hilbert space apparatus in its finite-dimensional version is
nearly reconstructed from four simple and quantum-mechanically motivated
postulates for a quantum logic. The reconstruction process is not complete,
since it excludes the two-dimensional Hilbert space and still includes the
exceptional Jordan algebras, which are not part of the Hilbert space apparatus.
Options for physically meaningful potential generalizations of the apparatus
are discussed.Comment: 19 page
Quantum teleportation and Grover's algorithm without the wavefunction
In the same way as the quantum no-cloning theorem and quantum key
distribution in two preceding papers, entanglement-assisted quantum
teleportation and Grover's search algorithm are generalized by transferring
them to an abstract setting, including usual quantum mechanics as a special
case. This again shows that a much more general and abstract access to these
quantum mechanical features is possible than commonly thought. A non-classical
extension of conditional probability and, particularly, a very special type of
state-independent conditional probability are used instead of Hilbert spaces
and wavefunctions.Comment: 21 pages, including annex, important typo in annex corrected in v2,
Found Phys (2017
Quantum key distribution without the wavefunction
A well-known feature of quantum mechanics is the secure exchange of secret
bit strings which can then be used as keys to encrypt messages transmitted over
any classical communication channel. It is demonstrated that this quantum key
distribution allows a much more general and abstract access than commonly
thought. The results include some generalizations for the Hilbert space version
of quantum key distribution,but base upon a general non-classical extension of
conditional probability. A special state-independent conditional probability is
identified as origin of the superior security of quantum key distribution and
may have more profound implications for the foundations and interpretation of
quantum mechanics,quantum information theory, and the philosophical question
what actually constitutes physical reality.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1502.0215
Non-classical conditional probability and the quantum no-cloning theorem
The quantum mechanical no-cloning theorem for pure states is generalized and
transfered to the quantum logics with a conditional probability calculus in a
rather abstract, though simple and basic fashion without relying on a tensor
product construction or finite dimension as required in other generalizations.Comment: 6 page
Local tomography and the role of the complex numbers in quantum mechanics
Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but not as strong as the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors
Quantum probability's algebraic origin
Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further novel cases.
A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions