33,673 research outputs found
Energy Quantisation and Time Parameterisation
We show that if space is compact, then trajectories cannot be defined in the
framework of quantum Hamilton--Jacobi equation. The starting point is the
simple observation that when the energy is quantized it is not possible to make
variations with respect to the energy, and the time parameterisation
t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group
velocity, is ill defined. It should be stressed that this follows directly form
the quantum HJ equation without any axiomatic assumption concerning the
standard formulation of quantum mechanics. This provides a stringent connection
between the quantum HJ equation and the Copenhagen interpretation. Together
with tunneling and the energy quantization theorem for confining potentials,
formulated in the framework of quantum HJ equation, it leads to the main
features of the axioms of quantum mechanics from a unique geometrical
principle. Similarly to the case of the classical HJ equation, this fixes its
quantum analog by requiring that there exist point transformations, rather than
canonical ones, leading to the trivial hamiltonian. This is equivalent to a
basic cocycle condition on the states. Such a cocycle condition can be
implemented on compact spaces, so that continuous energy spectra are allowed
only as a limiting case. Remarkably, a compact space would also imply that the
Dirac and von Neumann formulations of quantum mechanics essentially coincide.
We suggest that there is a definition of time parameterisation leading to
trajectories in the context of the quantum HJ equation having the probabilistic
interpretation of the Copenhagen School.Comment: 11 pages. The main addition concerns a discussion on the variational
principle in the case of discrete energy spectra (Jacobi's Theorem).
References adde
Probabilistic Parsing Strategies
We present new results on the relation between purely symbolic context-free
parsing strategies and their probabilistic counter-parts. Such parsing
strategies are seen as constructions of push-down devices from grammars. We
show that preservation of probability distribution is possible under two
conditions, viz. the correct-prefix property and the property of strong
predictiveness. These results generalize existing results in the literature
that were obtained by considering parsing strategies in isolation. From our
general results we also derive negative results on so-called generalized LR
parsing.Comment: 36 pages, 1 figur
Computation in generalised probabilistic theories
From the existence of an efficient quantum algorithm for factoring, it is
likely that quantum computation is intrinsically more powerful than classical
computation. At present, the best upper bound known for the power of quantum
computation is that BQP is in AWPP. This work investigates limits on
computational power that are imposed by physical principles. To this end, we
define a circuit-based model of computation in a class of operationally-defined
theories more general than quantum theory, and ask: what is the minimal set of
physical assumptions under which the above inclusion still holds? We show that
given only an assumption of tomographic locality (roughly, that multipartite
states can be characterised by local measurements), efficient computations are
contained in AWPP. This inclusion still holds even without assuming a basic
notion of causality (where the notion is, roughly, that probabilities for
outcomes cannot depend on future measurement choices). Following Aaronson, we
extend the computational model by allowing post-selection on measurement
outcomes. Aaronson showed that the corresponding quantum complexity class is
equal to PP. Given only the assumption of tomographic locality, the inclusion
in PP still holds for post-selected computation in general theories. Thus in a
world with post-selection, quantum theory is optimal for computation in the
space of all general theories. We then consider if relativised complexity
results can be obtained for general theories. It is not clear how to define a
sensible notion of an oracle in the general framework that reduces to the
standard notion in the quantum case. Nevertheless, it is possible to define
computation relative to a `classical oracle'. Then, we show there exists a
classical oracle relative to which efficient computation in any theory
satisfying the causality assumption and tomographic locality does not include
NP.Comment: 14+9 pages. Comments welcom
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