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第395行：
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第422行： + + − 其中，e ^ { i phi } </math > 是物理上有意义的相对阶段。+ 第442行：
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第473行： + + − The surface of the Bloch sphere is a [[two-dimensional space]], which represents the [[state space (physics)|state space]] of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles <math>\phi </math> and <math>\theta </math>. + − 量子比特和经典比特之间的一个重要区别是，多个量子比特可以显示量子纠缠。量子纠缠是两个或多个量子位的非局部性质，它允许一组量子位表达比传统系统更高的相关性。+
→Bloch sphere representation
此外，对于单个量子位，状态的整个阶段没有物理上可观察的结果，所以我们可以任意选择是真实的(或者在这种情况下是零) ，只留下两个自由度:
此外，对于单个量子位，状态的整个阶段没有物理上可观察的结果，所以我们可以任意选择是真实的(或者在这种情况下是零) ，只留下两个自由度:
===Bloch sphere representation===
===Bloch sphere representation布洛赫球表示===
<math>\begin{align}
<math>\begin{align}
[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The probability amplitudes for the superposition state, <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\,</math> are given by <math> \alpha = \cos\left(\frac{\theta}{2}\right) </math> and <math> \beta = e^{i \phi} \sin\left(\frac{\theta}{2}\right) </math>.]]
[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The probability amplitudes for the superposition state, <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\,</math> are given by <math> \alpha = \cos\left(\frac{\theta}{2}\right) </math> and <math> \beta = e^{i \phi} \sin\left(\frac{\theta}{2}\right) </math>.]]
[[文件：Bloch sphere.svg|量子位的拇指|[[Bloch-sphere]]表示。叠加态的概率振幅，<math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\,</math> are given by <math> \alpha = \cos\left(\frac{\theta}{2}\right) </math> and <math> \beta = e^{i \phi} \sin\left(\frac{\theta}{2}\right) </math>.]]
\beta &= e^{i \phi} \sin\frac{\theta}{2},
\beta &= e^{i \phi} \sin\frac{\theta}{2},
It might, at first sight, seem that there should be four [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle\,</math>, as <math>\alpha</math> and <math>\beta</math> are [[complex number]]s with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint {{math|{{!}}''α''{{!}}<sup>2</sup> + {{!}}''β''{{!}}<sup>2</sup> {{=}} 1}}. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of [[3-sphere#Hopf coordinates|Hopf coordinates]]:
It might, at first sight, seem that there should be four [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle\,</math>, as <math>\alpha</math> and <math>\beta</math> are [[complex number]]s with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint {{math|{{!}}''α''{{!}}<sup>2</sup> + {{!}}''β''{{!}}<sup>2</sup> {{=}} 1}}. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of [[3-sphere#Hopf coordinates|Hopf coordinates]]:
乍一看，似乎应该有四个[[自由度（物理和化学）|自由度]]在<math>|\psi\rangle=\alpha | 0\rangle+\beta | 1\rangle\，</math>，因为<math>\alpha</math>和<math>\beta</math>都是[[复数]]，每个都有两个自由度。但是，一个自由度被规范化约束{{math{{！}}''α''{{!}}<sup>2</sup>+{！}}''β''{{!}}<sup>2</sup>{{=}}1}}。这意味着，通过适当改变坐标，可以消除其中一个自由度。一种可能的选择是[[三球#Hopf坐标| Hopf坐标]]：
where <math> e^{i \phi} </math> is the physically significant relative phase.
where <math> e^{i \phi} </math> is the physically significant relative phase.
其中，<math> e^{i \phi} </math> 是物理上有意义的相对阶段。
:<math>\begin{align}
:<math>\begin{align}
Additionally, for a single qubit the overall [[phase factor|phase]] of the state {{math|''e<sup>i ψ</sup>''}} has no physically observable consequences, so we can arbitrarily choose {{math|''α''}} to be real (or {{math|''β''}} in the case that {{math|''α''}} is zero), leaving just two degrees of freedom:
Additionally, for a single qubit the overall [[phase factor|phase]] of the state {{math|''e<sup>i ψ</sup>''}} has no physically observable consequences, so we can arbitrarily choose {{math|''α''}} to be real (or {{math|''β''}} in the case that {{math|''α''}} is zero), leaving just two degrees of freedom:
此外，对于单个量子比特，{{math|''e<sup>i ψ</sup>''}}状态的整体[[相位因子|相位]]没有物理上可观察的结果，因此我们可以任意选择{{math|''α''}}为实（或者在{{math|''α''}}为零的情况下的 {{math|''β''}}），只留下两个自由度：
:<math>\begin{align}
:<math>\begin{align}
where <math> e^{i \phi} </math> is the physically significant ''relative phase''.
where <math> e^{i \phi} </math> is the physically significant ''relative phase''.
其中，<math> e^{i \phi} </math> 是物理上有意义的相对阶段。
There are various kinds of physical operations that can be performed on pure qubit states.
There are various kinds of physical operations that can be performed on pure qubit states.
The possible quantum states for a single qubit can be visualised using a [[Bloch sphere]] (see diagram). Represented on such a [[2-sphere]], a classical bit could only be at the "North Pole" or the "South Pole", in the locations where <math>|0 \rangle </math> and <math>|1 \rangle </math> are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state <math>((|0 \rangle +i|1 \rangle)/{\sqrt{2}}) </math> would lie on the equator of the sphere at the positive y-axis. In the [[classical limit]], a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.
The possible quantum states for a single qubit can be visualised using a [[Bloch sphere]] (see diagram). Represented on such a [[2-sphere]], a classical bit could only be at the "North Pole" or the "South Pole", in the locations where <math>|0 \rangle </math> and <math>|1 \rangle </math> are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state <math>((|0 \rangle +i|1 \rangle)/{\sqrt{2}}) </math> would lie on the equator of the sphere at the positive y-axis. In the [[classical limit]], a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.
单个量子位的可能量子态可以用[[布洛赫球]]来可视化（见图表）。在这样一个[[2-球体]]上表示，经典位只能位于“北极”或“南极”，分别位于<math>| 0\rangle</math>和<math>| 1\rangle</math>的位置。然而，极轴的这种特殊选择是任意的。布洛赫球表面的其余部分是经典位所无法接近的，但是一个纯量子位状态可以用表面上的任何一点来表示。例如，纯量子位态<math>（（| 0\rangle+i | 1\rangle）/{\sqrt{2}}）</math>将位于正y轴的球体赤道上。在[[经典极限]]中，一个量子位元，可以在布洛赫球上的任何地方有量子态，它可以简化为经典位元，而经典位元只能在两极找到。
The surface of the Bloch sphere is a [[two-dimensional space]], which represents the [[state space (physics)|state space]] of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles <math>\phi </math> and <math>\theta </math>.
布洛赫球的表面是一个[[二维空间]]，它代表了纯量子位态的[[状态空间（物理）|状态空间]]。这个状态空间有两个局部自由度，可以用两个角度来表示。
An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.
An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.
量子比特和经典比特的一个重要区别是多个量子比特可以表现出量子纠缠。量子纠缠是两个或多个量子比特的非局域性质，它允许一组量子比特表达比经典系统更高的相关性。
===Mixed state===
===Mixed state===