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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bloch vector ： ウィキペディア英語版 Bloch sphere In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, this is simply the complex projective line ℂℙ1. This is the Bloch sphere. The Bloch sphere is a unit 2-sphere, with each pair of antipodal points corresponding to mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors $|0\backslash rangle$ and $|1\backslash rangle$, respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states.〔 〕〔http://www.quantiki.org/wiki/Bloch_sphere〕 The Bloch sphere may be generalized to an ''n''-level quantum system but then the visualization is less useful. In optics, the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. 6 common polarization types exist and are called Jones Vectors. The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space ℂ2 to the Bloch sphere is the Hopf fibration. == Definition == Given an orthonormal basis, any pure state $|\backslash psi\backslash rangle$ of a two-level quantum system can be written as a superposition of the basis vectors $|0\; \backslash rangle$ and $|1\; \backslash rangle$, where the coefficient or amount of each basis vector is a complex number. Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of $|0\; \backslash rangle$ to be real and non-negative. We also know from quantum mechanics that the total probability of the system has to be one: $\backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; =\; 1$, or equivalently $||\backslash psi\; \backslash rangle|^2\; =\; 1$. Given this constraint, we can write $|\backslash psi\backslash rangle$ using the following representation: :$|\backslash psi\backslash rangle\; =\; \backslash cos\backslash left(\backslash tfrac\backslash right)\; |0\; \backslash rangle\; \backslash ,\; +\; \backslash ,\; e^\; \backslash sin\backslash left(\backslash tfrac\backslash right)\; |1\; \backslash rangle\; =$ \cos\left(\tfrac\right) |0 \rangle \, + \, ( \cos \phi + i \sin \phi) \, \sin\left(\tfrac\right) |1 \rangle , where $0\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash pi$ and . Except in the case where $|\backslash psi\backslash rangle$ is one of the ket vectors $|0\; \backslash rangle$ or $|1\; \backslash rangle$, the representation is unique. The parameters $\backslash theta\; \backslash ,$ and $\backslash phi\; \backslash ,$, re-interpreted as spherical coordinates, specify a point :$\backslash vec\; =\; (\backslash sin\; \backslash theta\; \backslash cos\; \backslash phi,\; \backslash ;\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash phi,\; \backslash ;\; \backslash cos\; \backslash theta)$ on the unit sphere in $\backslash mathbb^$. For mixed states, one needs to consider the density operator. Any two-dimensional density operator $\backslash rho$ can be expanded using the identity $I$ and the Hermitian, traceless Pauli matrices $\backslash vec$: :$\backslash rho\; =\; \backslash frac\backslash left(I\; +\backslash vec\; \backslash cdot\; \backslash vec\; \backslash right)$, where $\backslash vec\; \backslash in\; \backslash mathbb^3$ is called the Bloch vector of the system. It is this vector that indicates the point within the sphere that corresponds to a given mixed state. The eigenvalues of $\backslash rho$ are given by $\backslash frac\backslash left(1\; \backslash pm\; |\backslash vec|\backslash right)$. As density operators must be positive-semidefinite, we have $|\backslash vec|\; \backslash le\; 1$. For pure states we must have :$\backslash mathrm(\backslash rho^2)\; =\; \backslash frac\backslash left(1\; +|\backslash vec|^2\; \backslash right)\; =\; 1\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; |\backslash vec|\; =\; 1$ in accordance with the previous result. Hence the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bloch sphere」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.