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Register a new userHierarchy of Spin Operators, Quantum Gates, Entanglement, Tensor Product and Eigenvalues
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Added by
Willi-Hans Steeb WHS
Publication date
2015
fields
Physics
and research's language is
English
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{bf Abstract.} We show that two hierarchies of spin Hamilton operators admit the same spectrum. Both Hamilton operators play a central role for quantum gates in particular for the case spin-$frac12$ and the case spin-1. The spin-$frac12$, spin-1, spin-$frac32$ and spin-2 cases are studied in detail. Entanglement and mutually unbiased bases of the eigenvectors is discussed. Two triple Hamilton operators are also investigated. Both are also admitting the same spectrum.
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Quantum resource theory under different classes of quantum operations advances multiperspective understandings of inherent quantum-mechanical properties, such as quantum coherence and quantum entanglement. We establish hierarchies of different operations for manipulating coherence and entanglement in distributed settings, where at least one of the two spatially separated parties are restricted from generating coherence. In these settings, we introduce new classes of operations and also characterize those maximal, i.e., the resource-non-generating operations, progressing beyond existing studies on incohere
Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By maximally general is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no the tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given.
A Banach space operator $Tin B(X)$ is left polaroid if for each $lambdainhbox{iso}sigma_a(T)$ there is an integer $d(lambda)$ such that asc $(T-lambda)=d(lambda)<infty$ and $(T-lambda)^{d(lambda)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-lambda)<infty$, $(T-lambda)X$ is closed and $dim(T-lambda)^{-1}(0)<infty$ at each $lambdainhbox{iso }sigma_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $tau_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$ if and only if $0 otinhbox{iso}sigma_a(Aotimes B)$; a similar result holds for $tau_{AB}$ for finitely left polaroid $A$ and $B^*$.
Fernando Galve emph{et al.} $[Phys. Rev. Lett. textbf{110}, 010501 (2013)]$ introduced discording power for a two-qubit unitary gate to evaluate its capability to produce quantum discord, and found that a $pi/8$ gate has maximal discording power. This work analyzes the entangling power of a two-qubit unitary gate, which reflects its ability to generate quantum entanglement in another way. Based on the renowned Cartan decomposition of two-qubit unitary gates, we show that the magic power of the $pi/8$ gate produces maximal entanglement for a general value of purities for two-qubit states.
We consider the problem of correct measurement of a quantum entanglement in the two-body electron-electron scattering. An expression is derived for a spin correlation tensor of a pure two-electron state. A geometrical measure of a quantum entanglement as the distance between two forms of this tensor in entangled and separable cases is presented. We prove that this measure satisfies properties of a valid entanglement measure: nonnegativity, discriminance, normalization, non-growth under local operations and classical communication. This measure is calculated for a problem of electron-electron scattering. We prove that it does not depend on the azimuthal rotation angle of the second electron spin relative to the first electron spin before scattering. Finally, we specify how to find a spin correlation tensor and the related measure of a quantum entanglement in an experiment with electron-electron scattering.
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