# 物理代写|量子力学代写QUANTUM MECHANICS代考|Entanglement-Breaking Channels

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## 物理代写|量子力学代写QUANTUM MECHANICS代考|Entanglement-Breaking Channels

An important class of channels is the set of entanglement-breaking channels, and we will see that both quantum-classical and classical-quantum channels are special cases of them.

DEFINITION 4.6.8 (Entanglement-Breaking Channel) An entanglementbreaking channel $\mathcal{N}^{\text {EB }}: \mathcal{L}\left(\mathcal{H}A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ is defined by the property that the channel $\operatorname{id}_R \otimes \mathcal{N}{A \rightarrow B}^{\mathrm{EB}}$ takes any state $\rho_{R A}$ to a separable state, where $R$ is a reference system of arbitrary size.

Fortunately, we do not need to check this property for all possible $\rho_{R A}$. In fact, it suffices to check whether $\left(\operatorname{id}R \otimes \mathcal{N}{A \rightarrow B}^{\mathrm{EB}}\right)\left(\Phi_{R A}\right)$ is a separable state, where $\Phi_{R A}$ is a maximally entangled state, as defined in (3.232).

EXERCISE 4.6.2 Prove that a quantum channel $\mathcal{N}{A \rightarrow B}$ is entanglementbreaking if ( $\left.\operatorname{id}_R \otimes \mathcal{N}{A \rightarrow B}\right)\left(\Phi_{R A}\right)$ is a separable state, where $\Phi_{R A}$ is a maximally entangled state. (Hint: You can use a trick similar to that which you used to solve Exercise 4.4.1. Alternatively, you can inspect the proof of Theorem 4.6.1 below.)

EXERCISE 4.6.3 Show that both a classical-quantum channel and a quantumclassical channel are entanglement-breaking-i.e., if we input the $A$ system of a bipartite state $\rho_{R A}$ to either of these channels, then the resulting state on systems $R B$ is separable.

We can prove a more general structural theorem regarding entanglementbreaking channels by exploiting its definition.

THEOREM 4.6.1 A channel is entanglement-breaking if and only if it has a Kraus representation with Kraus operators that are unit rank.

Proof We first prove the “if” part of the theorem. Suppose that the Kraus operators of a quantum channel $\mathcal{N}_{A \rightarrow B}$ are
$$\left{N_z \equiv\left|\xi_z\right\rangle_B\left\langle\left.\varphi_z\right|_A\right} .\right.$$

## 物理代写|量子力学代写QUANTUM MECHANICS代考|Quantum Instruments

The description of a quantum channel with Kraus operators gives the most general evolution that a quantum state can undergo. We may want to specialize this definition somewhat for another scenario. Suppose that we would like to determine the most general evolution where the input is a quantum system and the output consists of both a quantum system and a classical system. Such a scenario may arise in a case where Alice is trying to transmit both classical and quantum information, and Bob exploits a quantum instrument to decode both kinds of information. A quantum instrument gives such an evolution with a hybrid output.

Definition 4.6.9 (Trace Non-Increasing Map) A linear map $\mathcal{M}$ is trace nonincreasing if $\operatorname{Tr}{\mathcal{M}(X)} \leq \operatorname{Tr}{X}$ for all positive semi-definite $X \in \mathcal{L}(\mathcal{H})$, with $\mathcal{H}$ a Hilbert space.

DEFINITION 4.6.10 (Quantum Instrument) A quantum instrument consists of a collection $\left{\mathcal{E}_j\right}$ of completely positive, trace non-increasing maps such that the sum map $\sum_j \mathcal{E}_j$ is trace preserving. Let ${|j\rangle}$ be an orthonormal basis for a Hilbert space $\mathcal{H}_J$. The action of a quantum instrument on a density operator $\rho \in \mathcal{D}(\mathcal{H})$ is the following quantum channel, which features a quantum and classical output:
$$\rho \rightarrow \sum_j \mathcal{E}_j(\rho) \otimes|j\rangle\left\langle\left. j\right|_J\right.$$

# 量子力学代写

## 物理代写|量子力学代写QUANTUM MECHANICS代考|Entanglement-Breaking Channels

4.6.8 (entanglementbreaking Channel)一个entanglementbreaking Channel $\mathcal{N}^{\text {EB }}: \mathcal{L}\left(\mathcal{H}A\right) \rightarrow \mathcal{L}\left(\mathcal{H}B\right)$被定义为这样一个属性:该信道$\operatorname{id}_R \otimes \mathcal{N}{A \rightarrow B}^{\mathrm{EB}}$将任意状态$\rho{R A}$转换为一个可分离状态，其中$R$是一个任意大小的参考系统。

$$\left{N_z \equiv\left|\xi_z\right\rangle_B\left\langle\left.\varphi_z\right|_A\right} .\right.$$

## 物理代写|量子力学代写QUANTUM MECHANICS代考|Quantum Instruments

4.6.10(量子仪器)量子仪器由一组完全正的、轨迹不递增的映射组成$\left{\mathcal{E}_j\right}$，使得和映射$\sum_j \mathcal{E}_j$保持轨迹。设${|j\rangle}$是希尔伯特空间$\mathcal{H}_J$的标准正交基。量子仪器对密度算符$\rho \in \mathcal{D}(\mathcal{H})$的作用是以下量子通道，其特征是量子和经典输出:
$$\rho \rightarrow \sum_j \mathcal{E}_j(\rho) \otimes|j\rangle\left\langle\left. j\right|_J\right.$$

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