## 数学代写|信息论作业代写information theory代考|Quantum operations

Quantum operations are formulated in terms of the quantum state or density operator description of a quantum mechanical system. Rigorously, a quantum operation is a linear, completely positive and trace-nondecreasing map from the set of quantum states of one system into that of another.

To define quantum operations, we recall from Definition 1.8 .2 that $₹(\mathbb{H})$ is the $\mathrm{Ba}$ nach space of trace-class operators $\mathbf{T}$ on the complex Hilbert space $\mathbb{H}$ under the tracenorm $|\mathbf{T}|_1$ defined by $|\mathbf{T}|_1:=\operatorname{tr}[|\mathbf{T}|]=\operatorname{tr}\left[\sqrt{\mathbf{T}^* \mathbf{T}}\right]$.
A linear map $\Phi: \mathcal{T}\left(\mathbb{H}_A\right) \rightarrow \mathcal{T}\left(\mathrm{H}_B\right)$ is said to be trace nonincreasing if
$$\operatorname{tr}[\Phi(\rho)] \leq \operatorname{tr}[\rho], \quad \forall \rho \in \widetilde{T}\left(\mathbb{H}_A\right)$$
and $\Phi: \mathfrak{T}\left(\mathbb{H}_A\right) \rightarrow \tau\left(\mathbb{H}_B\right)$ is said to be trace preserving if
$$\operatorname{tr}[\Phi(\rho)]=\operatorname{tr}[\rho], \quad \forall \rho \in \widetilde{T}\left(\mathbb{H}_A\right)$$

Definition 5.1.1. A linear completely positive trace-nonincreasing map
$$\Phi: \mathcal{S}\left(\mathbb{H}A\right) \rightarrow \mathcal{S}\left(\mathbb{H}_B\right)$$ is called a quantum operation from system $A$ to system $B$. Definition 5.1.2. A linear completely positive trace-nonincreasing map $$\Phi: \tau{+}\left(\mathbb{H}A\right) \rightarrow \tau{+}\left(\mathbb{H}_B\right)$$
is called an extended quantum operation from system $A$ to system $B$.
The collection of quantum operations from $A$ to $B$ will be denoted by $\mathfrak{Q Q}(A, B)$ and the collection of extended quantum operations from system $A$ to system $B$ will be denoted by $\operatorname{EQO}(A, B)$. When $\mathbb{H}_A=\mathbb{H}_B, \mathfrak{Q O}(A, B)$ and $\mathfrak{E Q O}(A, B)$ will be written as $\mathfrak{Q O}(A)$ and $\mathfrak{E Q O}(A)$, respectively.

Trace-preserving and trace-nonincreasing positive linear maps between $\tau_{+}\left(\mathbb{H}A\right)$ and $\widetilde{T}{+}\left(\mathbb{H}_B\right)$ can be considered noncommutative analogs of Markov and sub-Markov maps in the classical probability theory.

## 数学代写|信息论作业代写information theory代考|Quantum channels

Roughly speaking, a quantum channel is a specific type of map from one quantum system to another, which will be described mathematically in Schrodinger and Heisenberg pictures in the next two subsections. In the Schrodinger picture, the quantum channels will be presented as being the resulting transformation of a quantum state of $\mathrm{H}_A$ after a contact and an evolution with some environment to a quantum state of $\mathbb{H}_B$. As usual, for all quantum evolutions there is a dual picture, an Heisenberg picture, where the evolution is seen from the point of view of observables instead of states.

To explore the relationship between the Shrodinger picture and Heisenberg picture of a quantum system represented by a separable Hilbert space $\mathbb{H}$, consider the Banach spaces of trace-class operators $\mathfrak{T}(\mathbb{H})$ under trace-norm $|\cdot|_1$ and the Banach space of bounded linear operators $\mathfrak{B}(\mathbb{H})$ under the operator norm $|\cdot|_{\infty}$. It has been shown in Proposition 2.3.14 that $\mathfrak{T}(\mathbb{H})$ is a predual of $\mathfrak{B}(\mathbb{H})$ via the bilinear relation $\langle\langle\cdot \cdot\rangle\rangle: \mathfrak{B}(\mathbb{H}) \times \mathfrak{I}(\mathbb{H}) \rightarrow \mathbb{C}$ defined by
$$\langle\langle\mathbf{a}, \mathbf{T}\rangle\rangle=\operatorname{tr}[\mathbf{a} \mathbf{T}], \quad \forall \mathbf{a} \in \mathfrak{B}(\mathbb{H}) \text { and } \forall \mathbf{T} \in \Im(\mathbb{H})$$
In other words, $\mathfrak{T}^*(\mathbb{H})$, the topological dual of $\mathfrak{T}(\mathbb{H})$, equals $\mathfrak{B}(\mathbb{H})$. Moreover, for any $\mathbf{T} \in \mathfrak{B}(\mathbb{H})$, its operator norm $|\mathbf{T}|_{\infty}$ is related to its trace-class norm $|\mathbf{T}|_1$ via the following relation: $$|\mathbf{T}|_{\infty}=\sup {|\rho|_1 \leq 1}|\mathbf{T} \rho|_1 .$$ In particular, if $\mathbf{T}$ is such that $|\mathbf{T}|_1=1$, then $|\mathbf{T}|{\infty}=|\mathbf{T}|_1=1$.

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## 数学代写|信息论作业代写information theory代考|Quantum operations

$$\operatorname{tr}[\Phi(\rho)] \leq \operatorname{tr}[\rho], \quad \forall \rho \in \widetilde{T}\left(\mathbb{H}_A\right)$$

$$\operatorname{tr}[\Phi(\rho)]=\operatorname{tr}[\rho], \quad \forall \rho \in \widetilde{T}\left(\mathbb{H}_A\right)$$

$$\Phi: \mathcal{S}(\mathbb{H} A) \rightarrow \mathcal{S}\left(\mathbb{H}_B\right)$$

$$\Phi: \tau+(\mathbb{H} A) \rightarrow \tau+\left(\mathbb{H}_B\right)$$

Trace-preserving 和 trace-nonincreasing 之间的正线性映射 $\tau_{+}(\mathbb{H} A)$ 和 $\widetilde{T}+\left(\mathbb{H}_B\right)$ 可以 被认为是经典概率论中马尔可夫和子马尔可夫映射的非交换类比。

## 数学代写|信息论作业代写information theory代考|Quantum channels

$\langle\langle\cdot \cdot\rangle\rangle: \mathfrak{B}(\mathbb{H}) \times \mathfrak{I}(\mathbb{H}) \rightarrow \mathbb{C}$ 被定义为
$$\langle\langle\mathbf{a}, \mathbf{T}\rangle\rangle=\operatorname{tr}[\mathbf{a} \mathbf{T}], \quad \forall \mathbf{a} \in \mathfrak{B}(\mathbb{H}) \text { and } \forall \mathbf{T} \in \mathfrak{I}(\mathbb{H})$$

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