241,008 research outputs found
Interior and Exterior Differential Systems for Lie Algebroids
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that
any vector subbundle of a Lie algebroid is called interior differential system
(IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending
the classical notion of exterior differential system (EDS) to Lie algebroids, a
theorem of Cartan type is obtained.Comment: 10 pages. Submitted and accepted to Advances in Pure Mathematics;
May, 201
Markovian master equations for quantum-classical hybrid systems
The problem of constructing a consistent quantum-classical hybrid dynamics is
afforded in the case of a quantum component in a separable Hilbert space and a
continuous, finite-dimensional classical component. In the Markovian case, the
problem is formalized by the notion of hybrid dynamical semigroup. A classical
component can be observed without perturbing the system and information on the
quantum component can be extracted, thanks to the quantum-classical
interaction. This point is formalized by showing how to introduce positive
operator valued measures and operations compatible with the hybrid dynamical
semigroup; in this way the notion of hybrid dynamics is connected to quantum
measurements in continuous time. Then, the case of the most general quasi-free
generator is presented and the various quantum-classical interaction terms are
discussed. To bee quasi-free means to send, in the Heisenberg description,
hybrid Weyl operators into multiples of Weyl operators; the results on the
structure of quasi-free semigroups were proved in the article arXiv:2307.02611.
Even in the pure quantum case, a quasi-free semigroup is not restricted to have
only a Gaussian structure, but also jump-type terms are allowed. An important
result is that, to have interactions producing a flow of information from the
quantum component to the classical one, suitable dissipative terms must be
present in the generator. Finally, some possibilities are discussed to go
beyond the quasi-free case.Comment: 12 pages. Section 2.1 has been reformulated. Some typos have been
correcte
Recommended from our members
Efficient classical simulation of a variant of cluster state quantum computation
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityQuantum computers are known for their ability to solve some computational problems faster than classical computers. There is a race to build quantum computers because it is believed they might be better than classical; but it remains unknown
whether quantum computers are in fact better than conventional computers. To understand this problem, we develop a new method of classically simulating certain types of quantum system that are previously unknown to be efficiently simulatable
on classical computers.
We adjust a part of cluster state quantum computation to study the computational power and we demonstrate that there is a finite region of pure states j i around the Z-eigenstates for which the setup can be efficiently simulated classically, given
that the measurements are limited to Z and X - Y plane measurements. This classical simulation works by considering alternative local state spaces that we called "cylinders" and different notion of entanglement to normal quantum entanglement.
Then, we work out similar regions for states created using other diagonal gates instead of the CZ. These diagonal gates are represented by V (θ) = |0><1|. It turns out that almost all inputs are classically
simulatable when θ is small.
In addition, we nd that classical simulation also works by considering new type of non-quantum state spaces other than cylinders and maintaining non-entangled representation by growing the size of these state spaces. We search over some state spaces to try optimize our classical simulation and it turns out that, among the state spaces that we searched through, the cylinder is the most optimal state space.
And finally, we will look at a coarse graining version of construction which increases the efficiently simulatable region
A Categorical Framework for Quantum Theory
Underlying any theory of physics is a layer of conceptual frames. They
connect the mathematical structures used in theoretical models with physical
phenomena, but they also constitute our fundamental assumptions about reality.
Many of the discrepancies between quantum physics and classical physics
(including Maxwell's electrodynamics and relativity) can be traced back to
these categorical foundations. We argue that classical physics corresponds to
the factual aspects of reality and requires a categorical framework which
consists of four interdependent components: boolean logic, the
linear-sequential notion of time, the principle of sufficient reason, and the
dichotomy between observer and observed. None of these can be dropped without
affecting the others. However, in quantum theory the reduction postulate also
addresses the "status nascendi" of facts, i.e., their coming into being.
Therefore, quantum phyics requires a different conceptual framework which will
be elaborated in this article. It is shown that many of its components are
already present in the standard formalisms of quantum physics, but in most
cases they are highlighted not so much from a conceptual perspective but more
from their mathematical structures. The categorical frame underlying quantum
physics includes a profoundly different notion of time which encompasses a
crucial role for the present.Comment: 35 pages, 1 figur
Environment and classical channels in categorical quantum mechanics
We present a both simple and comprehensive graphical calculus for quantum
computing. In particular, we axiomatize the notion of an environment, which
together with the earlier introduced axiomatic notion of classical structure
enables us to define classical channels, quantum measurements and classical
control. If we moreover adjoin the earlier introduced axiomatic notion of
complementarity, we obtain sufficient structural power for constructive
representation and correctness derivation of typical quantum informatic
protocols.Comment: 26 pages, many pics; this third version has substantially more
explanations than previous ones; Journal reference is of short 14 page
version; Proceedings of the 19th EACSL Annual Conference on Computer Science
Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag (2010
Quantum Spacetime Pictures and Dynamics from a Relativity Perspective
Based on an identified quantum relativity symmetry the contraction of which
gives the Newtonian approximation of Galilean relativity, a quantum model of
the physical space can be formulated with the Newtonian space seen in a way as
the classical approximation. Matching picture for the observable algebra as the
corresponding representation of the group C* -algebra, describes the full
dynamical pictures equally successfully. Extension of the scheme to a Lorentz
covariant setting and beyond will also be addressed.The formulation of quantum
mechanics allows the theory to be seen in a new picture in line with the notion
of a noncommutative spacetime.Comment: For submission to Proc. of 10th Meeting of Balkan Physics Union,
Sofia, Bulgari
Do We Understand Quantum Mechanics - Finally?
After some historical remarks concerning Schroedinger's discovery of wave
mechanics, we present a unified formalism for the mathematical description of
classical and quantum-mechanical systems, utilizing elements of the theory of
operator algebras. We then review some basic aspects of quantum mechanics and,
in particular, of its interpretation. We attempt to clarify what Quantum
Mechanics tells us about Nature when appropriate experiments are made. We
discuss the importance of the mechanisms of "dephasing" and "decoherence" in
associating "facts" with possible events and rendering complementary possible
events mutually exclusive.Comment: 42 pages, contribution to the Proceedings of a conference in memory
of Erwin Schroedinger, Vienna, January 201
- …