arXiv subjects

A. Vourdas

Publications and source records attributed to A. Vourdas.

At least 37 records · Page 2Linked to original sources

arXiv2018

Selective correlations in finite quantum systems and the Desargues property

The Desargues property is well known in the context of projective geometry. An analogous property is presented in the context of both classical and Quantum Physics. In a classical context, the Desargues property implies that two logical circuits with the same input, show in their outputs selective correlations. In general their outputs are uncorrelated, but if the output of one has a particular value, then the output of the other has another particular value. In a quantum context, the Desargues property implies that two experiments each of which involves two successive projective measurements, have selective correlations. For a particular set of projectors, if in one experiment the second measurement does not change the output of the first measurement, then the same is true in the other experiment.

math-ph↗

arXiv2018

Independence and totalness of subspaces in phase space methods

The concepts of independence and totalness of subspaces are introduced in the context of quasi-probability distributions in phase space, for quantum systems with finite-dimensional Hilbert space. It is shown that due to the non-distributivity of the lattice of subspaces, there are various levels of independence, from pairwise independence up to (full) independence. Pairwise totalness, totalness and other intermediate concepts are also introduced, which roughly express that the subspaces overlap strongly among themselves, and they cover the full Hilbert space. A duality between independence and totalness, that involves orthocomplementation (logical NOT operation), is discussed. Another approach to independence is also studied, using Rota's formalism on independent partitions of the Hilbert space. This is used to define informational independence, which is proved to be equivalent to independence. As an application, the pentagram (used in discussions on contextuality) is analyzed using these concepts.

math-ph↗

arXiv2017

Renormalization of total sets of states into generalized bases with a resolution of the identity

A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a `generalized basis', which is practically useful. The dresssing mechanism is inspired by Shapley's methodology in cooperative game theory, and it uses Möbius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases, are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.

math-ph↗

arXiv2017

The groupoid of bifractional transformations

Bifractional transformations which lead to quantities that interpolate between other known quantities, are considered. They do not form a group, and groupoids are used to described their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context, is also studied.

quant-ph↗

arXiv2017

Dressed coherent states in finite quantum systems: a cooperative game theory approach

A quantum system with variables in Z(d) is considered. Coherent density matrices and coherent projectors of rank n are introduced, and their properties (e.g., the resolution of the identity) are dis- cussed. Cooperative game theory and in particular the Shapley methodology, is used to renormalize coherent states, into a particular type of coherent density matrices (dressed coherent states). The Q-function of a Hermitian operator, is then renormalized into a physical analogue of the Shapley values. Both the Q-function and the Shapley values, are used to study the relocation of a Hamilto- nian in phase space as the coupling constant varies, and its effect on the ground state of the system. The formalism is also generalized for any total set of states, for which we have no resolution of the identity. The dressing formalism leads to density matrices that resolve the identity, and makes them practically useful.

math-ph↗

arXiv2016

Coherent spaces, Boolean rings and quantum gates

Coherent spaces spanned by a finite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean ring (Stone\rq{}s formalism). Applications of this Boolean ring into classical CNOT gates with $n$-ary variables, and also quantum CNOT gates with coherent states, are discussed.

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arXiv2016

An analytic function approach to weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is $1/\sqrt{k}$ (where $k|d$) or $0$. The second is maximal lines through the origin in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where $d=p_1\times p_2$, where $p_1,p_2$ are odd prime numbers different from each other, is considered.

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arXiv2016

Comonotonicity and Choquet integrals of Hermitian operators and their applications

In a quantum system with d-dimensional Hilbert space, the Q-function of a Hermitian positive semidefinite operator ?, is defined in terms of the d2 coherent states in this system. The Choquet integral CQ of the Q-function, is introduced using a ranking of the values of the Q-function, and Mobius transforms which remove the overlaps between coherent states. It is a figure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q-function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.

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arXiv2016

Paths of zeros of analytic functions describing finite quantum systems

Quantum systems with positions and momenta in Z(d), are described by the d zeros of analytic functions on a torus. The d paths of these zeros on the torus, describe the time evolution of the system. A semi-analytic method for the calculation of these paths of the zeros, is discussed. Detailed analysis of the paths for periodic systems, is presented. A periodic system which has the displacement operator to a real power t, as time evolution operator, is studied. Several numerical examples, which elucidate these ideas, are presented.

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arXiv2015

Mobius operators and non-additive quantum probabilities in the Birkhoff-von Neumann lattice

The properties of quantum probabilities are linked to the geometry of quantum mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Mobius (or non-additivity) operators which are defined through Mobius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice Lambda , causes deviations from the law of the total probability (which is central in Kolmogorov's probability theory). Projectors which quantify the lack of distributivity in Lambda , and also deviations from the law of the total probability, are introduced. All these operators, are observables and they can be measured experimentally. Constraints for the Mobius operators, which are based on the properties of the Birkhoff-von Neumann lattice (which in the case of finite quantum systems is a modular lattice), are derived.Application of this formalism in the context of coherent states, generalizes coherence to multi-dimensional structures.

math-ph↗

arXiv2015

Analytic representations with Theta functions for systems on Z(d) and on S

An analytic representation with Theta functions on a torus, for systems with variables in Z(d), is considered. Another analytic representation with Theta functions on a strip, for systems with positions in a circle S and momenta in Z, is also considered. The reproducing kernel formalism for these two systems is studied. Wigner and Weyl functions in this language, are also studied

math-ph↗

arXiv2014

Interpolation between phase space quantities with bifractional displacement operators

Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G, that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(?, ?; ??, ??), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q-functions, and bifractional P-functions. The physical meaning of these quantities is discussed.

quant-ph↗

arXiv2014

Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces

The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

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arXiv2014

Lower and upper probabilities in the distributive lattice of subsystems

The set of subsystems of a finite quantum system (with variables in Z(n)) together with logical connectives, is a distributive lattice. With regard to this lattice, the (where P(m) is the projector to) obeys a supermodularity inequality, and it is interpreted as a lower probability in the sense of the Dempster-Shafer theory, and not as a Kolmogorov probability. It is shown that the basic concepts of the Dempster-Shafer theory (lower and upper probabilities and the Dempster multivaluedness) are pertinent to the quantum formalism of finite systems.

quant-ph↗

arXiv2013

The complete Heyting algebra of subsystems and contextuality

The finite set of subsystems of a finite quantum system with variables in ${\mathbb Z}(n)$, is studied as a Heyting algebra. The physical meaning of the logical connectives is discussed. It is shown that disjunction of subsystems is more general concept than superposition. Consequently the quantum probabilities related to projectors in the subsystems, are incompatible with associativity of the join in the Heyting algebra, unless if the variables belong to the same chain. This leads to contextuality, which in the present formalism has as contexts, the chains in the Heyting algebra. Logical Bell inequalities, which contain `Heyting factors', are discussed. The formalism is also applied to the infinite set of all finite quantum systems, which is appropriately enlarged in order to become a complete Heyting algebra.

quant-ph↗

arXiv2013

Quantum mechanics on profinite groups and partial order

Inverse limits and profinite groups are used in a quantum mechanical context. Two cases are considered. A quantum system with positions in the profinite group ${\mathbb Z}_p$ and momenta in the group ${\mathbb Q}_p/{\mathbb Z}_p$; and a quantum system with positions in the profinite group ${\hat {\mathbb Z}}$ and momenta in the group ${\mathbb Q}/{\mathbb Z}$. The corresponding Schwatz-Bruhat spaces of wavefunctions and the Heisenberg-Weyl groups are discussed. The sets of subsystems of these systems are studied from the point of view of partial order theory. It is shown that they are directed-complete partial orders. It is also shown that they are topological spaces with $T_0$ topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and $T_0$ topologies, in a quantum mechanical context, is discussed.

math-ph↗

arXiv2012

Complementarity and phases in SU(3)

Phase operators and phase states are introduced for irreducible representations of the Lie algebra su(3) using a polar decomposition of ladder operators. In contradistinction with su(2), it is found that the su(3) polar decomposition does not uniquely determine a Hermitian phase operator. We describe two possible ways of proceeding: one based in imposing SU(2) invariance and the other based on the idea of complementarity. The generalization of these results to SU(n) is sketched.

math-ph↗

arXiv2012

Partial order and a $T_0$-topology in a set of finite quantum systems

A `whole-part' theory is developed for a set of finite quantum systems $Σ(n)$ with variables in ${\mathbb Z}(n)$. The partial order `subsystem' is defined, by embedding various attributes of the system $Σ(m)$ (quantum states, density matrices, etc) into their counterparts in the supersystem $Σ(n)$ (for $m|n$). The compatibility of these embeddings is studied. The concept of ubiquity is introduced for quantities which fit with this structure. It is shown that various entropic quantities are ubiquitous. The sets of various quantities become $T_0$-topological spaces with the divisor topology, which encapsulates fundamental physical properties. These sets can be converted into directed-complete partial orders (dcpo), by adding `top elements'. The continuity of various maps among these sets is studied.

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