We consider a typical realization of a qubit as a single particle in two-path interferometric circuits built from phase shifters, beam splitters and detectors. This framework is often taken as a standard example illustrating various paradoxes and qua
ntum effects, including non-locality. In this paper we show that it is possible to simulate the behaviour of such circuits in a classical manner using stochastic gates and two kinds of particles, real ones and ghosts, which interact only locally. The model has built-in limited information gain and state disturbance in measurements which are blind to ghosts. We demonstrate that predictions of the model are operationally indistinguishable from the quantum case of a qubit, and allegedly non-local effects arise only on the epistemic level of description by the agent whose knowledge is incomplete due to the restricted means of investigating the system.
We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of
modern enumerative combinatorics and their applications to computational problems.
Contextuality lays at the heart of quantum mechanics. In the prevailing opinion it is considered as a signature of quantumness that classical theories lack. However, this assertion is only partially justified. Although contextuality is certainly true
of quantum mechanics, it cannot be taken by itself as discriminating against classical theories. Here we consider a representative example of contextual behaviour, the so-called Mermin-Peres square, and present a discrete toy model of a bipartite system which reproduces the pattern of quantum predictions that leads to contradiction with the assumption of non-contextuality. This illustrates that quantum-like contextual effects have their analogues within classical models with epistemic constraints such as limited information gain and measurement disturbance.
We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpret
ation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.
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 mo
del 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.
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial structure
s and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or diagrams, that are composed of elementary gates. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to q-analogues, multivariate frameworks, and urn models are also briefly discussed.
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication laws, the
reby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H),
the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.
The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decompositi
on rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.
We consider properties of the operators D(r,M)=a^r(a^dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^dag are boson annihilation and creation operators respectively, satisfying [a,a^dag]=1. We o
btain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.
