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# 物理代写|量子力学代写Quantum mechanics代考|Axiomatic Approach to Quantum Evolutions

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## 物理代写|量子力学代写Quantum mechanics代考|Axiomatic Approach to Quantum Evolutions

We now discuss a powerful approach to understanding quantum physical evolutions called the axiomatic approach. Here we make three physically reasonable assumptions that any quantum evolution should satisfy and then prove that these axioms imply mathematical constraints on the form of any quantum physical evolution.

All of the constraints we impose are motivated by the reasonable requirement for the output of the evolution to be a quantum state (density operator) if the input to the evolution is a quantum state (density operator). An important assumption to clarify at the outset is that we are viewing a quantum physical evolution as a “black box,” meaning that Alice can prepare any state that she wishes before the evolution begins, including pure states or mixed states. Critically, we even allow her to input one share of an entangled state. This is a standard assumption in quantum information theory, but one could certainly question whether this assumption is reasonable. If we do accept this criterion as physically reasonable, then the Choi-Kraus representation theorem for quantum evolutions follows as a consequence.

NOTATION 4.4.1 (Density Operators and Linear Operators) Let $\mathcal{D}(\mathcal{H})$ denote the space of density operators acting on a Hilbert space $\mathcal{H}$, let $\mathcal{L}(\mathcal{H})$ denote the space of square linear operators acting on $\mathcal{H}$, and let $\mathcal{L}\left(\mathcal{H}_A, \mathcal{H}_B\right)$ denote the space of linear operators taking a Hilbert space $\mathcal{H}_A$ to a Hilbert space $\mathcal{H}_B$.

Throughout this development, we let $\mathcal{N}$ denote a map which takes density operators in $\mathcal{D}\left(\mathcal{H}_A\right)$ to those in $\mathcal{D}\left(\mathcal{H}_B\right)$. In general, the respective input and output Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ need not be the same. Implicitly, we have already stated a first physically reasonable requirement that we impose on $\mathcal{N}$, namely, that $\mathcal{N}\left(\rho_A\right) \in \mathcal{D}\left(\mathcal{H}_B\right)$ if $\rho_A \in \mathcal{D}\left(\mathcal{H}_A\right)$. Extending this requirement, we demand that $\mathcal{N}$ should be convex linear when acting on $\mathcal{D}\left(\mathcal{H}_A\right)$ :
$$\mathcal{N}\left(\lambda \rho_A+(1-\lambda) \sigma_A\right)=\lambda \mathcal{N}\left(\rho_A\right)+(1-\lambda) \mathcal{N}\left(\sigma_A\right)$$
where $\rho_A, \sigma_A \in \mathcal{D}\left(\mathcal{H}_A\right)$ and $\lambda \in[0,1]$.

## 物理代写|量子力学代写Quantum mechanics代考|Unique Specification of a Quantum Channel

We emphasize again that any linear map $\mathcal{N}: \mathcal{L}\left(\mathcal{H}A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ is specified completely by its action $\mathcal{N}{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|A\right)\right.$ on an operator of the form $|i\rangle\left\langle\left. j\right|_A\right.$ where $\left{|i\rangle_A\right}$ is some orthonormal basis. Thus, two linear maps $\mathcal{N}{A \rightarrow B}$ and $\mathcal{M}{A \rightarrow B}$ are equal if they have the same effect on all operators of the form $|i\rangle\langle j|$ : $$\mathcal{N}{A \rightarrow B}=\mathcal{M}{A \rightarrow B} \quad \Leftrightarrow \quad \forall i, j \quad \mathcal{N}{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|A\right)=\mathcal{M}{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|A\right)\right.\right.$$ As a consequence, there is an interesting way to test whether two quantum channels are equal to each other. Let us now consider a maximally entangled qudit state $|\Phi\rangle{R A}$ where
$$|\Phi\rangle_{R A}=\frac{1}{\sqrt{d}} \sum_{i=0}^{d-1}|i\rangle_R|i\rangle_A,$$
and $d$ is the dimension of each system $R$ and $A$. The density operator $\Phi_{R A}$ corresponding to $|\Phi\rangle_{R A}$ is as follows:
$$\Phi_{R A}=\frac{1}{d} \sum_{i, j=0}^{d-1}|i\rangle\left\langle\left. j\right|R \otimes \mid i\right\rangle\left\langle\left. j\right|_A\right.$$ Let us now send the $A$ system of $\Phi{R A}$ through a quantum channel $\mathcal{N}$ :
$$\left(\mathrm{id}R \otimes \mathcal{N}{A \rightarrow B}\right)\left(\Phi_{R A}\right)=\frac{1}{d} \sum_{i, j=0}^{d-1}|i\rangle\left\langlej | _ { R } \otimes \mathcal { N } _ { A \rightarrow B } \left(|i\rangle\left\langle\left. j\right|A\right) .\right.\right.$$ The resulting state completely characterizes the quantum channel $\mathcal{N}$ because the following map translates between the state in (4.226) and the operators $\mathcal{N}{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|A\right)\right.$ in $(4.223)$ : $$d\left\langle\left. i\right|_R\left(\operatorname{id}_R \otimes \mathcal{N}{A \rightarrow B}\right)\left(\Phi_{R A}\right) \mid j\right\rangle_R=\mathcal{N}_{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|_A\right)\right.$$

## 物理代写|量子力学代写Quantum mechanics代考|Axiomatic Approach to Quantum Evolutions

$$\mathcal{N}\left(\lambda \rho_A+(1-\lambda) \sigma_A\right)=\lambda \mathcal{N}\left(\rho_A\right)+(1-\lambda) \mathcal{N}\left(\sigma_A\right)$$

## 物理代写|量子力学代写Quantum mechanics代考|Unique Specification of a Quantum Channel

$d$是每个系统的维数$R$和$A$。$|\Phi\rangle_{R A}$对应的密度算子$\Phi_{R A}$如下:
$$\Phi_{R A}=\frac{1}{d} \sum_{i, j=0}^{d-1}|i\rangle\left\langle\left. j\right|R \otimes \mid i\right\rangle\left\langle\left. j\right|A\right.$$现在让我们通过量子通道$\mathcal{N}$发送$\Phi{R A}$的$A$系统: $$\left(\mathrm{id}R \otimes \mathcal{N}{A \rightarrow B}\right)\left(\Phi{R A}\right)=\frac{1}{d} \sum_{i, j=0}^{d-1}|i\rangle\left\langlej | _ { R } \otimes \mathcal { N } _ { A \rightarrow B } \left(|i\rangle\left\langle\left. j\right|A\right) .\right.\right.$$所得到的状态完全表征了量子通道$\mathcal{N}$，因为下面的映射在(4.226)中的状态和$(4.223)$中的操作符$\mathcal{N}{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|A\right)\right.$之间转换: $$d\left\langle\left. i\right|R\left(\operatorname{id}_R \otimes \mathcal{N}{A \rightarrow B}\right)\left(\Phi{R A}\right) \mid j\right\rangle_R=\mathcal{N}_{A \rightarrow B}\left(|i\rangle\left\langle\left. j\right|_A\right)\right.$$

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