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

#### Doug I. Jones

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• Statistical Computing 统计计算
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## 物理代写|量子力学代写QUANTUM MECHANICS代考|Isometric Evolution

A quantum channel admits a purification as well. We motivate this idea with a simple example.
Example: Isometric Extension of the Bit-Flip Channel
Consider the bit-flip channel from (4.330) —it applies the identity operator with some probability $1-p$ and applies the bit-flip Pauli operator $X$ with probability $p$. Suppose that we input a qubit system $A$ in the state $|\psi\rangle$ to this channel. The ensemble corresponding to the state at the output has the following form:
$${{1-p,|\psi\rangle},{p, X|\psi\rangle}},$$
and the density operator of the resulting state is
$$(1-p)|\psi\rangle\langle\psi|+p X| \psi\rangle\langle\psi| X$$
The following state is a purification of the above density operator (you should quickly check that this relation holds):
$$\sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$
We label the original system as $A$ and label the purification system as $E$. In this context, we can view the purification system as the environment of the channel.
There is another way for interpreting the dynamics of the above bit-flip channel. Instead of determining the ensemble for the channel and then purifying, we can say that the channel directly implements the following map from the system $A$ to the larger joint system $A E$ :
$$|\psi\rangle_A \rightarrow \sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$
We see that any $p \in(0,1)$, i.e., any amount of noise in the channel, can lead to entanglement of the input system with the environment $E$. We then obtain the noisy dynamics of the channel by discarding (tracing out) the environment system $E$.

## 物理代写|量子力学代写QUANTUM MECHANICS代考|An Isometry is Part of a Unitary on a Larger System

We can view the dynamics in (5.21) as an interaction between an initially pure environment and the qubit state $|\psi\rangle$. So, an equivalent way to implement an isometric mapping is with a two-step procedure. We first assume that the environment of the channel is in a pure state $|0\rangle_E$ before the interaction begins. The joint state of the qubit $|\psi\rangle$ and the environment is
$$|\psi\rangle_A|0\rangle_E$$
These two systems then interact according to a unitary operator $V_{A E}$. We can specify two columns of the unitary operator (we make this more clear in a bit) by means of the isometric mapping in (5.21):
$$V_{A E}|\psi\rangle_A|0\rangle_E=\sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$
In order to specify the full unitary $V_{A E}$, we must also specify how the map behaves when the initial state of the qubit and the environment is
$$|\psi\rangle_A|1\rangle_E$$
We choose the mapping to be as follows so that the overall interaction is unitary:
$$V_{A E}|\psi\rangle_A|1\rangle_E=\sqrt{p}|\psi\rangle_A|0\rangle_E-\sqrt{1-p} X|\psi\rangle_A|1\rangle_E$$
EXERCISE 5.2.2 Check that the operator $V_{A E}$, defined by (5.24) and (5.26), is unitary by determining its action on the computational basis $\left{|00\rangle_{A E},|01\rangle_{A E},|10\rangle_{A E},|11\rangle_{A E}\right}$ and showing that all of the outputs for each of these inputs form an orthonormal basis.
EXERCISE 5.2.3 Verify that the matrix representation of the full unitary operator $V_{A E}$, defined by (5.24) and (5.26), is
$$\left[\begin{array}{cccc} \sqrt{1-p} & \sqrt{p} & 0 & 0 \ 0 & 0 & \sqrt{p} & -\sqrt{1-p} \ 0 & 0 & \sqrt{1-p} & \sqrt{p} \ \sqrt{p} & -\sqrt{1-p} & 0 & 0 \end{array}\right],$$
by considering the matrix elements $\left\langle\left. i\right|_A\left\langle\left. j\right|_E V \mid k\right\rangle_A \mid l\right\rangle_E$.

# 量子力学代写

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

$${{1-p,|\psi\rangle},{p, X|\psi\rangle}},$$

$$(1-p)|\psi\rangle\langle\psi|+p X| \psi\rangle\langle\psi| X$$

$$\sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$

$$|\psi\rangle_A \rightarrow \sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$

## 物理代写|量子力学代写QUANTUM MECHANICS代考|An Isometry is Part of a Unitary on a Larger System

$$|\psi\rangle_A|0\rangle_E$$

$$V_{A E}|\psi\rangle_A|0\rangle_E=\sqrt{1-p}|\psi\rangle_A|0\rangle_E+\sqrt{p} X|\psi\rangle_A|1\rangle_E$$

$$|\psi\rangle_A|1\rangle_E$$

$$V_{A E}|\psi\rangle_A|1\rangle_E=\sqrt{p}|\psi\rangle_A|0\rangle_E-\sqrt{1-p} X|\psi\rangle_A|1\rangle_E$$

$$\left[\begin{array}{cccc} \sqrt{1-p} & \sqrt{p} & 0 & 0 \ 0 & 0 & \sqrt{p} & -\sqrt{1-p} \ 0 & 0 & \sqrt{1-p} & \sqrt{p} \ \sqrt{p} & -\sqrt{1-p} & 0 & 0 \end{array}\right],$$

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