Integrability in the mesoscopic dynamics

Sowa, Artur

doi:10.1016/j.geomphys.2004.11.005

Cited by 9 publications

(21 citation statements)

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“…We note that the columns of K * thus satisfy the original Schrödinger equation (2). In addition, if K satisfies this equation, then ρ = K * K follows its canonical dynamics.…”

Section: 2mentioning

confidence: 90%

“…From a new standpoint, we discuss certain theoretical concepts originally introduced in [1], [2] and subsequently applied in [3]- [5]. Our methods can be used to model systems operating in the quantum Hall effect (QHE) regime and its functional vicinity.…”

Section: Synopsismentioning

confidence: 99%

“…This is a functional that generates the dynamics in (2). Indeed, splitting the inner product into the real and imaginary parts leads to a Euclidean product and a symplectic form:…”

Section: Synopsismentioning

confidence: 99%

“…Again, the ρ corresponding to eigenstates are diagonalized simultaneously with H. We note that for the solution to exist, ν must dominate all the eigenvalues of H. We can make sense of Eq. (31) even when H is a differential operator [2] by simply restricting the domain of ρ. In this case, referring to (1), we obtain…”

Section: Eigenstates As a Consequence Of Nonlinearitymentioning

confidence: 99%

“…This conclusion is based on the surprising observation that Eq. (28) can be solved almost explicitly in the sense that the solutions can be expressed via solutions of the linear Schrödinger equation [2]. In fact, this can be done even when the spaces H and H are infinite-dimensional.…”

Section: Correlations During Time Evolutionmentioning

confidence: 99%

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Quantum entanglement in composite systems

Sowa

2009

Theor Math Phys

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We discuss a certain class of models of mesoscopic quantum phenomena that result from associating a particular type of Hamiltonian dynamics with the entangled states of composite systems. Using such models, we can concisely describe the phenomenology of a two-dimensional electron gas, in particular, the quantum Hall effect. We show how such models (if they are regarded as reflecting actual physical principles and not only the phenomenology) can be tested by a quantum teleportation-type experiment. The so-called mesoscopic models stipulate that the dynamics of a two-dimensional gas coupled to an ambient field is captured by a functional of the type Tr[Hρ] + β Tr f (ρ), where ρ is the density operator and H is a single-particle Hamiltonian. We use the proposed approach to demonstrate that a suitable quantum teleportation experiment can provide information about the analytic function f . This leads us to view the composite system of an electron gas and an ambient field as a natural quantum computer. Keywords: composite system, nonlocal model, quantum entanglement, quantum Hall effect SynopsisFrom a new standpoint, we discuss certain theoretical concepts originally introduced in [1], [2] and subsequently applied in [3]-[5]. Our methods can be used to model systems operating in the quantum Hall effect (QHE) regime and its functional vicinity. Moreover, they seem to indicate the significance of certain universal principles for models of nanosystems and for nanosystem physics in general. The main novelty in our approach is, perhaps, using a dynamical variable that can be considered the square root of the density matrix. Here, we emphasize that the structure can be interpreted in the framework of quantum information theory with quantum entanglement playing a pivotal role. This aspect of the considered structure is discussed systematically for the first time.We start by discussing the elementary case of a noninteracting electron gas, which sets the stage for subsequently discussing the main theme, i.e., the interacting gas. This allows interpreting quantum Hall systems in the framework of natural quantum computation. Assuming that the Hilbert spaces and operators are finite-dimensional does not seriously limit the conceptual framework of our discussion. We generally work in this framework and abandon it only when necessary. The few exceptions, where we must refer to differential operators and functional spaces, are noted explicitly. Noninteracting systems: Linear mechanics A basic description of an ensemble of noninteracting electrons.We start by reviewing the basics of quantum dynamics. Our discussion in the following sections is based on a discerning analysis of these concepts. The basic structure is only relevant for describing an ensemble of electrons that do not interact among themselves and do not interact with other systems (with ambient fields). Let H be a space

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“…We note that the columns of K * thus satisfy the original Schrödinger equation (2). In addition, if K satisfies this equation, then ρ = K * K follows its canonical dynamics.…”

Section: 2mentioning

confidence: 90%

Section: Synopsismentioning

confidence: 99%

“…This is a functional that generates the dynamics in (2). Indeed, splitting the inner product into the real and imaginary parts leads to a Euclidean product and a symplectic form:…”

Section: Synopsismentioning

confidence: 99%

Section: Eigenstates As a Consequence Of Nonlinearitymentioning

confidence: 99%

Section: Correlations During Time Evolutionmentioning

confidence: 99%

See 3 more Smart Citations