Maximal violation of Mermin's inequalities

In this paper, it is proved that the maximal violation of Mermin's inequalities of $n$ 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 nn qubits

In this paper, we characterize the maximal violation of Ardehali's inequality of $n$ 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 $n$ 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 $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ 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 $\mathbb{H}$ 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 $\Omega$ is a bounded domain in $\mathbf{R}^n.$ 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