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arXiv:hep-th/9409148 · Annals of the New York Academy of Sciences

The Lattice Dynamics of Completely Entangled States and Its Application to Communication Schemes

DANIEL I. FIVEL

Published Apr 1, 1995• 0 citations
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Abstract

(Presented at conference on Fundamental Problems in Physics - UMBC - June 1994) It is shown that among the orthogonal sets of EPR (completely entangled) states there is a unique basis (up to equivalence) that is a also a perfectly resolved set of coherent states with respect to a pair of complementary observables. This basis defines a lattice phase space in which quadratic Hamiltonians constructed from the observables induce site-to-site hopping at discrete time intervals. When recently suggested communication schemes\cite{BENa} are adapted to the lattice they are greatly enhanced, because the finite Heisenberg group structure allows dynamic generation of signal sequences using the quadratic Hamiltonians. We anticipate the possibility of interferometry by determining the relative phases between successive signals produced by the simplest Hamiltonians of this type, and we show that they exhibit a remarkable pattern determined by the number-theoretic Legendre symbol.

DOI
10.1111/j.1749-6632.1995.tb39010.x

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(Presented at conference on Fundamental Problems in Physics - UMBC - June 1994) It is shown that among the orthogonal sets of EPR (completely entangled) states there is a unique basis (up to equivalence) that is a also a perfectly resolved set of coherent states with respect to a pair of complementary observables. This basis defines a lattice phase space in which quadratic Hamiltonians constructed from the observables induce site-to-site hopping at discrete time intervals. When recently suggested communication schemes\cite{BENa} are adapted to the lattice they are greatly enhanced, because the finite Heisenberg group structure allows dynamic generation of signal sequences using the quadratic Hamiltonians. We anticipate the possibility of interferometry by determining the relative phases between successive signals produced by the simplest Hamiltonians of this type, and we show that they exhibit a remarkable pattern determined by the number-theoretic Legendre symbol.
(Presented at conference on Fundamental Problems in Physics - UMBC - June 1994) It is shown that among the orthogonal sets of EPR (completely entangled) states there is a unique basis (up to equivalence) that is a also a perfectly resolved set of coherent states with respect to a pair of complementary observables. This basis defines a lattice phase space in which quadratic Hamiltonians constructed from the observables induce site-to-site hopping at discrete time intervals. When recently suggested communication schemes\\cite{BENa} are adapted to the lattice they are greatly enhanced, because the finite Heisenberg group structure allows dynamic generation of signal sequences using the quadratic Hamiltonians. We anticipate the possibility of interferometry by determining the relative phases between successive signals produced by the simplest Hamiltonians of this type, and we show that they exhibit a remarkable pattern determined by the number-theoretic Legendre symbol.
The possibility to designing schemes useful for developing quantum technology devices of practical value necessitates exploiting quantum coherence effects in a scalable physical system in a feasible way. The broad aim of this thesis is to investigate the use of quantum non-equilibrium dynamics for the above, exploiting minimal control to accomplish highly coherent dynamics in a many-body system. How to harness the natural hopping dynamics of particles in a many-site lattice for controlled applications, is still an open question. Through the introduction of few impurities in the lattice potential, we devise a scheme to trigger effective tunable linear optics-like operations between arbitrary sites, that overcomes the limitations of setups based on coherent hopping dynamics, when particles are initially separated by many sites. Our scheme enables the generation of peculiar quantum interference effects as well as quantum metrology applications in a many-site lattice. We design a lattice coupling profile that enables perfect wave-packet splitting between mirror symmetric sites and leads to perfect wave-packet reconstruction, fractional revivals and perfect entanglement distribution, for arbitrary long chains. We prove that composite objects in a lattice, made of more particles initially in a lattice site, are a valuable resource for dynamically generating non-classical states between remote sites, tackling edge-localisation effects via local fields. Finally, we show how the spin independent scattering of two initially distant qubits, can be used to implement an entangling quantum gate between remote sites of a lattice. Our findings have potentially an impact on quantum information, as well as on atomic interferometry in a lattice.
We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.
Abstract Few-body physics for anyons has been intensively studied within the anyon-Hubbard model, including the quantum walk and Bloch oscillations of two-anyon states. Recently, theoretical and experimental simulations of two-anyon states in a one-dimensional lattice have been carried out by expanding the wavefunction in terms of non-orthogonal basis vectors, resulting in non-physical degrees of freedom. In the present work, we deduce finite difference equations for the two-anyon state in a one-dimensional lattice by solving the Schrödinger equation with orthogonal and complete basis vectors. Such an orthogonal scheme gives all the orthogonal physical eigenstates, while the conventional (non-orthogonal) method produces many non-physical redundant eigensolutions whose components violate the anyonic commutation relations. The dynamical property of the two-anyon states in a sufficiently large lattice is investigated and compared in both the orthogonal and conventional schemes. For initial states with two anyons at the same site or two (next-)neighboring sites, we observe the same dynamical behavior in both schemes, including the revival probability, probability density function and two-body correlation. For other initial states, the conventional scheme produces erroneous states that no longer obey the anyonic relations. The period of Bloch oscillations in the pseudo-fermionic limit has been found to be twice that in the bosonic limit, while these oscillations disappear at other statistical parameters. Our findings are vital for quantum simulations of few-body anyonic physics in lattice models.
We propose a discrete ontological reconstruction of quantum mechanics based on a graph-theoretic model with deterministic cyclic dynamics. The model is founded on the hypothesis that quantum mechanics is an effective description of reality from the perspective of an internal observer, that is, an observer who is part of the finite system (world) being observed. A central element of the model is the principle of subjective incompleteness: in general, the set of states of the system (ontological states) exceeds the set of observable states and cannot be fully distinguished by the internal observer. This leads to taking the quotient of the set of ontological states with respect to indistinguishability and naturally induces a probabilistic structure in which probabilities are determined by the cardinalities of the corresponding equivalence classes. It is shown that the cyclic structure of the dynamics, combined with this quotient construction, leads to the emergence of complex amplitudes and the Born rule. Thus, the formalism of quantum mechanics arises as an effective description of a quotient-induced deterministic dynamics. It is further shown that the geometry of the edge space of a complete graph realizes a discrete phase space with an action of the Heisenberg–Weyl group, and that in the continuum limit the model leads to the Klein–Gordon equation. Within this approach, the structure of quantum mechanics appears as a representational level of description of the quotient cyclic dynamics.