# 物理代写|热力学代写thermodynamics代考|MQS Formation Conditions

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## 物理代写|热力学代写thermodynamics代考|MQS Formation Conditions

The condition (8.81) for MQS formation is satisfied if $\Delta_{\mathrm{L}}\left(\tau_{\mathrm{MQS}}\right) \gg \gamma\left(\tau_{\mathrm{MQS}}\right)$.
For the satisfaction of condition (8.81), $\tau_{\mathrm{MQS}}$ should exceed the non-Markovian timescale, as explained in this section. At sufficiently low temperatures, $\gamma(t)$ is drastically reduced in the Markovian limit $\left(t \gg t_{\mathrm{c}}\right)$ as opposed to its fast initial non-Markovian increase, where $t_{\mathrm{c}}$, the correlation (memory) time of the bath, is the inverse width of $G_{\mathrm{s}}(\omega)$. Namely,
$$\gamma\left(t \ll t_{\mathrm{c}}\right) \gg \gamma(t \rightarrow \infty)=\gamma$$
[Figs. 8.7(c, d) and 8.8]. The reason for this trend is that $\gamma(t)$ initially has contributions from all the bath modes, $\int G_{\mathrm{s}}(\omega) d \omega$, but subsequently decreases, as the bath-mode excitations at different frequencies cause the mode states to go out of phase as they approach the Markovian regime. On the other hand, $\Delta_{\mathrm{L}}(t)$ increases in the course of the transition from the non-Markovian to the Markovian regime, so that its long-time value satisfies
$$\left|\Delta_{\mathrm{L}}(t \rightarrow \infty)\right| \gg\left|\Delta_{\mathrm{L}}\left(t \ll t_{\mathrm{c}}\right)\right|$$
Hence, it is beneficial to have $\tau_{\mathrm{MQS}}$ longer than the bath correlation time $t_{\mathrm{c}}$, so that MQS formation encounters a much lower $\gamma$, and much higher $\Delta_{\mathrm{L}}$, than their non-Markovian counterparts, consistently with (8.81).

## 物理代写|热力学代写thermodynamics代考|Quantum Measurements and Pointer Bases

In the standard (von Neumann) model, a quantum measurement of a system observable is performed indirectly by coupling the system $\mathrm{S}$ with a “meter” system M and then measuring the latter. Zurek generalized this model by examining what happens when the meter observable (pointer) differs from the “standard pointer,” which commutes with the state of the meter. Yet this generalization has given rise to widespread unfounded statements on the fundamentals of quantum measurement theory, as summarized below.
(a) The measured observable of $\mathrm{S}$ is uniquely determined by the measured observable of $\mathrm{M}$.
(b) Decoherence “dynamically selects” the pointer basis, whereas other pointer bases cannot be used.

(c) The meter decoherence “dynamically selects” those meter observables that can be measured.

Statement (c) is a consequence of (a) and (b). Below it is shown that statement (a) is not correct, and neither are statements (b) and (c).

Here we consider only the coupling of the quantum system to a quantum meter, both isolated from the environment. The standard (von Neumann) treatment of quantum measurements is then as follows.

Suppose that the observable to be measured is represented in its basis of (orthonormal) eigenstates $\left|S_n\right\rangle$, as
$$\hat{S}=\sum_n a_n\left|S_n\right\rangle\left\langle S_n\right|$$
with
$$\left\langle S_m \mid S_n\right\rangle=\delta_{m n}$$
whereas the initial state of the system $S$ is
$$\left|\psi_{\mathrm{s}}\right\rangle=\sum_n c_n\left|S_n\right\rangle$$
The system-meter interaction is turned on in the time interval $\left(0, \tau_{\mathrm{M}}\right)$. This interaction correlates the initial factorized state of $\mathrm{S}$ and $\mathrm{M}$, which then obeys the Schmidt decomposition,
$$\left|\psi_{\mathrm{SM}}(0)\right\rangle=\sum_n c_n\left|S_n\right\rangle \otimes|M\rangle \rightarrow\left|\psi_{\mathrm{SM}}\left(\tau_{\mathrm{M}}\right)\right\rangle=\sum_n c_n\left|S_n\right\rangle \otimes\left|P_n\right\rangle$$
where the meter states also satisfy orthonormality,
$$\left\langle P_m \mid P_n\right\rangle=\delta_{m n}$$

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|MQS Formation Conditions

MQS 形成的条件 (8.81) 满足如果
$\Delta_{\mathrm{L}}\left(\tau_{\mathrm{MQS}}\right) \gg \gamma\left(\tau_{\mathrm{MQS}}\right)$.

$$\gamma\left(t \ll t_{\mathrm{c}}\right) \gg \gamma(t \rightarrow \infty)=\gamma$$
[图。8.7 (c，d) 和 8.8]。出现这种趋势的原因是 $\gamma(t)$ 最初有来自所有沐浴模式的贡献， $\int G_s(\omega) d \omega$ ，但随后 减少，因为不同频率的浴模式激发导致模式状态在接近 马尔可夫制度时异相。另一方面， $\Delta_{\mathrm{L}}(t)$ 在从非马尔可 夫制度到马尔可夫制度的过渡过程中增加，因此其长期 价值满足
$$\left|\Delta_{\mathrm{L}}(t \rightarrow \infty)\right| \gg\left|\Delta_{\mathrm{L}}\left(t \ll t_{\mathrm{c}}\right)\right|$$

## 物理代写|热力学代写thermodynamics代考|Quantum Measurements and Pointer Bases

(a) 测量的可观测值 $\mathrm{S}$ 由测量的可观测值唯一确定 $M$.
(b) 退相干“动态选择”指针基，而不能使用其他指针基。
(c) 仪表退相干“动态选择”那些可以测量的仪表观测值。

$$\hat{S}=\sum_n a_n\left|S_n\right\rangle\left\langle S_n\right|$$

$$\left\langle S_m \mid S_n\right\rangle=\delta_{m n}$$

$$\left|\psi_{\mathrm{s}}\right\rangle=\sum_n c_n\left|S_n\right\rangle$$

$$\left|\psi_{\mathrm{SM}}(0)\right\rangle=\sum_n c_n\left|S_n\right\rangle \otimes|M\rangle \rightarrow\left|\psi_{\mathrm{SM}}\left(\tau_{\mathrm{M}}\right)\right\rangle=\sum_n$$

$$\left\langle P_m \mid P_n\right\rangle=\delta_{m n}$$

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