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

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

Suppose we are given a density operator $\rho_A$ on a system $A$. Every such density operator has a purification, as defined below and depicted in Figure 5.1:

DEFINITION 5.1.1 (Purification) A purification of a density operator $\rho_A \in$ $\mathcal{D}\left(\mathcal{H}A\right)$ is a pure bipartite state $|\psi\rangle{R A} \in \mathcal{H}R \otimes \mathcal{H}_A$ on a reference system $R$ and the original system $A$, with the property that the reduced state on system $A$ is equal to $\rho_A$ : $$\rho_A=\operatorname{Tr}_R\left{|\psi\rangle\left\langle\left.\psi\right|{R A}\right} .\right.$$
Suppose that a spectral decomposition for the density operator $\rho_A$ is as follows:
$$\rho_A=\sum_x p_X(x)|x\rangle\left\langle\left. x\right|A\right.$$ We claim that the following state $|\psi\rangle{R A}$ is a purification of $\rho_A$ :
$$|\psi\rangle_{R A} \equiv \sum_x \sqrt{p_X(x)}|x\rangle_R|x\rangle_A$$
where the set $\left{|x\rangle_R\right}_x$ of vectors is some set of orthonormal vectors for the reference system $R$. The next exercise asks you to verify this claim.

EXERCISE 5.1.1 Show that the state $|\psi\rangle_{R A}$, as defined in (5.3), is a purification of the density operator $\rho_A$, with a spectral decomposition as given in (5.2).

## 物理代写|量子力学代写QUANTUM MECHANICS代考|Interpretation of Purifications

The purification idea has an interesting physical interpretation: we can think of the noisiness inherent in a particular quantum system as being due to entanglement with some external reference system to which we do not have access. That is, we can think that the density operator $\rho_A$ arises from the entanglement of the system $A$ with the reference system $R$ and from our lack of access to the system $R$.

Stated another way, the purification idea gives us a fundamentally different way to interpret noise. The interpretation is that any noise on a local system is due to entanglement with another system to which we do not have access. This interpretation extends to the noise from a noisy quantum channel. We can view this noise as arising from the interaction of the system that we possess with an external environment over which we have no control.

The global state $|\psi\rangle_{R A}$ is a pure state, but a reduced state $\rho_A$ is not a pure state in general because we trace over the reference system to obtain it. A reduced state $\rho_A$ is pure if and only if the global state $|\psi\rangle_{R A}$ is a pure product state.

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