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In quantum mechanics, probability vectors are generalized to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. This article focuses on the quantum state vector formalism for simplicity.

In quantum mechanics, probability vectors are generalized to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. This article focuses on the quantum state vector formalism for simplicity.

在量子力学中，概率向量被推广到'''<font color="#ff8000"> 密度算子Density operators</font>'''。这是技术上严格的量子逻辑门的数学基础，但中间量子态向量形式通常首先介绍，因为它在概念上比较简单。为了简单起见，本文着重讨论量子态向量形式。

在量子力学中，概率向量被推广到'''<font color="#ff8000"> 密度算子Density operators</font>'''。这是技术上严格的'''<font color="#ff8000"> 量子逻辑门的数学基础</font>'''，但中间量子态向量形式通常首先介绍，因为它在概念上比较简单。为了简单起见，本文着重讨论量子态向量形式。

In general, the coefficients <math display="inline">\alpha</math> and <math display="inline">\beta</math> are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state <math display="inline">|\psi\rangle</math> is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine if the state is <math display="inline">|0\rangle</math> or <math display="inline">|1\rangle</math> (this is known as a computational basis measurement), the zero state would be observed with probability <math display="inline">|\alpha|^2</math> and the one state with probability <math display="inline">|\beta|^2</math>. The numbers <math display="inline">\alpha</math> and <math display="inline">\beta</math> are called quantum amplitudes.

In general, the coefficients <math display="inline">\alpha</math> and <math display="inline">\beta</math> are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state <math display="inline">|\psi\rangle</math> is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine if the state is <math display="inline">|0\rangle</math> or <math display="inline">|1\rangle</math> (this is known as a computational basis measurement), the zero state would be observed with probability <math display="inline">|\alpha|^2</math> and the one state with probability <math display="inline">|\beta|^2</math>. The numbers <math display="inline">\alpha</math> and <math display="inline">\beta</math> are called quantum amplitudes.

一般来说，系数 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>都是复数。在这种情况下，一个量子比特的信息被称为被编码到量子存储器中。状态<math display="inline">|\psi\rangle</math>本身不是一个概率向量，但可以通过一个测量操作与一个概率向量相连。如果量子内存被测量以确定其状态是否为 <math display="inline">|0\rangle</math> 或<math display="inline">|1\rangle</math>(这被称为计算基础测量) ，那么零状态将被观察到概率 <math display="inline">|\alpha|^2</math>和概率 <math display="inline">|\beta|^2</math> 。数字 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>被称为量子幅值。

一般来说，系数 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>都是'''<font color="#ff8000"> 复数</font>'''。在这种情况下，一个量子比特的信息被称为被编码到量子存储器中。状态<math display="inline">|\psi\rangle</math>本身不是一个概率向量，但可以通过一个测量操作与一个概率向量相连。如果量子内存被测量以确定其状态是否为 <math display="inline">|0\rangle</math> 或<math display="inline">|1\rangle</math>(这被称为计算基础测量) ，那么零状态将被观察到概率 <math display="inline">|\alpha|^2</math>和概率 <math display="inline">|\beta|^2</math> 。数字 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>被称为'''<font color="#ff8000"> 量子幅值Quantum amplitudes</font>'''。

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

这种单比特量子存储器的状态可以通过量子逻辑门来控制，类似于用经典逻辑门来控制经典存储器。对经典和量子计算都重要的一个门是'''<font color="#ff8000"> 非门NOT gate</font>'''，它可以用矩阵表示

这种单比特量子存储器的状态可以通过'''<font color="#ff8000"> 量子逻辑门</font>'''来控制，类似于用'''<font color="#ff8000"> 经典逻辑门</font>'''来控制经典存储器。对经典和量子计算都重要的一个门是'''<font color="#ff8000"> 非门NOT gate</font>'''，它可以用矩阵表示

<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus <math display="inline">X|0\rangle = |1\rangle</math> and <math display="inline">X|1\rangle = |0\rangle</math>.

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus <math display="inline">X|0\rangle = |1\rangle</math> and <math display="inline">X|1\rangle = |0\rangle</math>.

在数学上，这种逻辑门应用于量子态矢量是用矩阵乘法模型来建模的。因此 <math display="inline">X|0\rangle = |1\rangle</math> 和 <math display="inline">X|1\rangle = |0\rangle</math>。

在数学上，这种逻辑门应用于'''<font color="#ff8000">量子态矢量</font>'''是用矩阵乘法模型来建模的。因此 <math display="inline">X|0\rangle = |1\rangle</math> 和 <math display="inline">X|1\rangle = |0\rangle</math>。

As a mathematical consequence of this definition, <math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, and <math display="inline">CNOT|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

As a mathematical consequence of this definition, <math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, and <math display="inline">CNOT|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

作为这个定义的数学推论，<math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, 和<math display="inline">CNOT|11\rangle = |10\rangle</math>。换句话说，当且仅当第一个量子位处于状态 <math display="inline">|1\rangle</math> 时，CNOT 对第二个量子位应用 NOT 门(<math display="inline">X</math>)。如果第一个量子位是 <math display="inline">|0\rangle</math>，则对任何一个量子位都不做处理。

作为这个定义的数学推论，<math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, 和<math display="inline">CNOT|11\rangle = |10\rangle</math>。换句话说，当且仅当第一个量子位处于状态 <math display="inline">|1\rangle</math> 时，CNOT 对第二个量子位应用 非NOT 门(<math display="inline">X</math>)。如果第一个量子位是 <math display="inline">|0\rangle</math>，则对任何一个量子位都不做处理。