65 research outputs found
Maximal violation of Mermin's inequalities
In this paper, it is proved that the maximal violation of Mermin's
inequalities of qubits occurs only for GHZ's states and the states obtained
from them by local unitary transformations. The key point of our argument
involved here is by using the certain algebraic properties that Pauli's
matrices satisfy, which is based on the determination of local spin observables
of the associated Bell-Mermin operators.Comment: 4 page
Observable-geometric phases and quantum computation
This paper presents an alternative approach to geometric phases from the
observable point of view. Precisely, we introduce the notion of
observable-geometric phases, which is defined as a sequence of phases
associated with a complete set of eigenstates of the observable. The
observable-geometric phases are shown to be connected with the quantum geometry
of the observable space evolving according to the Heisenberg equation. They are
indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system
evolves adiabatically. It is shown that the observable-geometric phases can be
used to realize a universal set of quantum gates in quantum computation. This
scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang
\cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space
beyond the state space.Comment: 17 pages. References update, minor expande
Wave-particle duality and `bipartite' wave functions for a single particle
It is shown that `bipartite' wave functions can present a mathematical
formalism of quantum theory for a single particle, in which the associated
Schr\"{o}dinger's wave functions correspond to those `bipartite' wave functions
of product forms. This extension of Schr\"{o}dinger's form establishes a
mathematical expression of wave-particle duality and that von Neumann's entropy
is a quantitative measure of complementarity between wave-like and
particle-like behaviors. In particular, this formalism suggests that collapses
of Schr\"{o}dinger's wave functions can be regarded as the simultaneous
transition of the particle from many levels to one. Our results shed
considerable light on the basis of quantum mechanics, including quantum
measurement.Comment: 3 page
Wave equations for determining energy-level gaps of quantum systems
An differential equation for wave functions is proposed, which is equivalent
to Schr\"{o}dinger's wave equation and can be used to determine energy-level
gaps of quantum systems. Contrary to Schr\"{o}dinger's wave equation, this
equation is on `bipartite' wave functions. It is shown that those `bipartite'
wave functions satisfy all the basic properties of Schr\"{o}dinger's wave
functions. Further, it is argued that `bipartite' wave functions can present a
mathematical expression of wave-particle duality. This provides an alternative
approach to the mathematical formalism of quantum mechanics.Comment: 3 page
Maximal violation of the Ardehali's inequality of qubits
In this paper, we characterize the maximal violation of Ardehali's inequality
of qubits by showing that GHZ's states and the states obtained from them by
local unitary transformations are the unique states that maximally violate the
Ardehali's inequalities. This concludes that Ardehali's inequalities can be
used to characterize maximally entangled states of qubits, as the same as
Mermin's and Bell-Klyshko's inequalities.Comment: 5 page
Geometrical perspective on quantum states and quantum computation
We interpret quantum computing as a geometric evolution process by
reformulating finite quantum systems via Connes' noncommutative geometry. In
this formulation, quantum states are represented as noncommutative connections,
while gauge transformations on the connections play a role of unitary quantum
operations. Thereby, a geometrical model for quantum computation is presented,
which is equivalent to the quantum circuit model. This result shows a geometric
way of realizing quantum computing and as such, provides an alternative
proposal of building a quantum computer.Comment: 4 page
Quantum Finance: The Finite Dimensional Case
In this paper, we present a quantum version of some portions of Mathematical
Finance, including theory of arbitrage, asset pricing, and optional
decomposition in financial markets based on finite dimensional quantum
probability spaces. As examples, the quantum model of binomial markets is
studied. We show that this quantum model ceases to pose the paradox which
appears in the classical model of the binomial market. Furthermore, we
re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by
considering multi-period quantum binomial markets.Comment: 22 pages, revised version, submitte
Mathematical formalism of many-worlds quantum mechanics
We combine the ideas of Dirac's orthonormal representation, Everett's
relative state, and 't Hooft's ontological basis to define the notion of a
world for quantum mechanics. Mathematically, for a quantum system
with an associated Hilbert space a world of is
defined to be an orthonormal basis of The evolution of the system
is governed by Schr\"{o}dinger's equation for the worlds of it. An observable
in a certain world is a self-adjoint operator diagonal under the corresponding
basis. Moreover, a state is defined in an associated world but can be uniquely
extended to the whole system as proved recently by Marcus, Spielman, and
Srivastava. Although the states described by unit vectors in may
be determined in different worlds, there are the so-called topology-compact
states which must be determined by the totality of a world. We can apply the
Copenhagen interpretation to a world for regarding a quantum state as an
external observation, and obtain the Born rule of random outcomes. Therefore,
we present a mathematical formalism of quantum mechanics based on the notion of
a world instead of a quantum state.Comment: 7 pages. The process of measurement explaine
Dirichlet problems for stationary von Neumann-Landau wave equations
It is known that von Neumann-Landau wave equation can present a mathematical
formalism of motion of quantum mechanics, that is an extension of
Schr\"{o}dinger's wave equation. In this paper, we concern with the Dirichlet
problem of the stationary von Neumann-Landau wave equation:
{(- \triangle_x + \triangle_y) \Phi (x, y) = 0, x, y \in \Omega,
\Phi|_{\partial \Omega \times \partial \Omega} = f, where is a
bounded domain in By introducing anti-inner product spaces, we
show the existence and uniqueness of the generalized solution for the above
Dirichlet problem by functional-analytic methods.Comment: 9 page
von Neumann-Landau equation for wave functions, wave-particle duality and collapses of wave functions
It is shown that von Neumann-Landau equation for wave functions can present a
mathematical formalism of motion of quantum mechanics. The wave functions of
von Neumann-Landau equation for a single particle are `bipartite', in which the
associated Schr\"{o}dinger's wave functions correspond to those `bipartite'
wave functions of product forms. This formalism establishes a mathematical
expression of wave-particle duality and that von Neumann's entropy is a
quantitative measure of complementarity between wave-like and particle-like
behaviors. Furthermore, this extension of Schr\"{o}dinger's form suggests that
collapses of Schr\"{o}dinger's wave functions can be regarded as the
simultaneous transition of the particle from many levels to one.Comment: 4 page
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