53 research outputs found
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Urn models & operator ordering procedures
Ordering of operators is purely combinatorial task involving a number
of commutators shuffling components of operator expression to desired
form. Here we show how it can be illustrated by simple urn models in
which normal ordering procedure is equivalent to enumeration of urn
histories
On Urn Models, Non-commutativity and Operator Normal Forms
Non-commutativity is ubiquitous in mathematical modeling of reality and in
many cases same algebraic structures are implemented in different situations.
Here we consider the canonical commutation relation of quantum theory and
discuss a simple urn model of the latter. It is shown that enumeration of urn
histories provides a faithful realization of the Heisenberg-Weyl algebra.
Drawing on this analogy we demonstrate how the operator normal forms facilitate
counting of histories via generating functions, which in turn yields an
intuitive combinatorial picture of the ordering procedure itself.Comment: 7 pages, 2 figure
On safe post-selection for Bell tests with ideal detectors: Causal diagram approach
Reasoning about Bell nonlocality from the correlations observed in
post-selected data is always a matter of concern. This is because conditioning
on the outcomes is a source of non-causal correlations, known as a selection
bias, rising doubts whether the conclusion concerns the actual causal process
or maybe it is just an effect of processing the data. Yet, even in the
idealised case without detection inefficiencies, post-selection is an integral
part of experimental designs, not least because it is a part of the
entanglement generation process itself. In this paper we discuss a broad class
of scenarios with post-selection on multiple spatially distributed outcomes. A
simple criterion is worked out, called the all-but-one principle, showing when
the conclusions about nonlocality from breaking Bell inequalities with
post-selected data remain in force. Generality of this result, attained by
adopting the high-level diagrammatic tools of causal inference, provides safe
grounds for systematic reasoning based on the standard form of multipartite
Bell inequalities in a wide array of entanglement generation schemes, without
worrying about the dangers of selection bias. In particular, it can be applied
to post-selection defined by single-particle events in each detection chanel
when the number of particles in the system is conserved.Comment: 16 pages, 9 figure
One-parameter groups and combinatorial physics
In this communication, we consider the normal ordering of sums of elements of
the form (a*^r a a*^s), where a* and a are boson creation and annihilation
operators. We discuss the integration of the associated one-parameter groups
and their combinatorial by-products. In particular, we show how these groups
can be realized as groups of substitutions with prefunctions.Comment: 15 pages, 23 references. Presented at the Third International
Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3),
Porto-Novo (Benin), November 200
Normal Order: Combinatorial Graphs
A conventional context for supersymmetric problems arises when we consider
systems containing both boson and fermion operators. In this note we consider
the normal ordering problem for a string of such operators. In the general
case, upon which we touch briefly, this problem leads to combinatorial numbers,
the so-called Rook numbers. Since we assume that the two species, bosons and
fermions, commute, we subsequently restrict ourselves to consideration of a
single species, single-mode boson monomials. This problem leads to elegant
generalisations of well-known combinatorial numbers, specifically Bell and
Stirling numbers. We explicitly give the generating functions for some classes
of these numbers. In this note we concentrate on the combinatorial graph
approach, showing how some important classical results of graph theory lead to
transparent representations of the combinatorial numbers associated with the
boson normal ordering problem.Comment: 7 pages, 15 references, 2 figures. Presented at "Progress in
Supersymmetric Quantum Mechanics" (PSQM'03), Valladolid, Spain, July 200
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