# 物理代写|热力学代写thermodynamics代考|PHYS2712

## 物理代写|热力学代写thermodynamics代考|Control, Score, and Constraint

Here, we consider the general question: How to find a system dynamics that optimizes a desired pertinent quantity? One can define a control problem in terms of control parameters, a score $P$ that measures the success of control optimization, and a constraint $E$ (e.g., the energy allotted for control), as detailed below.

The real control parameters $f_l(1 \leq l \leq N)$ form an $N$-dimensional vector $\boldsymbol{f}$. In the case of time-dependent control, the $f_l(\tau)$ parameterize the system Hamiltonian or the unitary evolution operator as
$$\hat{H}{\mathrm{S}}=\hat{H}{\mathrm{S}}[f(\tau)], \quad \hat{U}(\tau)=\mathrm{T} e^{-(i / \hbar) \int_0^\tau d \tau^{\prime} \hat{H}\left(\tau^{\prime}\right)} \equiv \hat{U}[\boldsymbol{f}(\tau)] .$$
Such parameterization of $\hat{U}$ circumvents the complication of time-ordered integration of its exponent. The evolution operator $\hat{U}(\tau)$ thus obtained can then be used to calculate the system Hamiltonian
$$\hat{H}_{\mathrm{S}}(\tau)=i \hbar\left[\frac{\partial}{\partial \tau} \hat{U}(\tau)\right] \hat{U}^{\dagger}(\tau) .$$

The score $P$ that measures the success of controlling the quantity of interest can be written as a real-valued functional of the system state $\hat{\varrho}(t)$ at the time $t_{\mathrm{f}}$ when the control ends. This score may be, for example, the maximal fidelity of a given pure state $|\Psi\rangle$ under bath-induced decoherence, $F_{|\Psi\rangle}=\langle\Psi|\hat{\varrho}| \Psi\rangle$. Alternatively, the score may be the maximal concurrence $\mathrm{Co}$
$$C o_{\left|\Psi_{\mathrm{A} B}\right\rangle}=\left[2\left(1-\operatorname{Tr}{\hat{A}}^2\right)\right]^{1 / 2}, \quad \hat{\varrho}{\mathrm{A}}=\operatorname{Tr}{\mathrm{B}}\left|\Psi{\mathrm{AB}}\right\rangle\left\langle\Psi_{\mathrm{AB}}\right|,$$
which is a measure of the entanglement of a bipartite state $\left|\Psi_{\mathrm{AB}}\right\rangle$. More generally, the score may be the maximum value of any real-valued function $P[\hat{\varrho}(t)]$ in the time interval $\left[0, t_{\mathrm{f}}\right]$,
$$P=\max {t \in\left[0, t_t\right]} P[\hat{\varrho}(t)] .$$ The choice of a constraint that is generally required to ensure the existence of a physical solution for the control is dictated by the most critical source of error. A possible constraint is the average speed with which the controls change, $$E=\hbar \int_0^t d \tau \dot{\boldsymbol{f}}^2(\tau),$$ which depends on the control spectral bandwidth. Another choice is the meansquared modulation energy, $$E=\hbar^{-1} \int_0^{t_f} d \tau\left\langle(\Delta \hat{H})^2(\tau)\right\rangle{\mathrm{id}},$$

## 物理代写|热力学代写thermodynamics代考|Fixed Time Approach

Our goal is to achieve, by means of classical control fields, a time dependence of the system Hamiltonian within the interval $\left[0, t_{\mathrm{f}}\right]$ that sets the score $P\left(t_{\mathrm{f}}\right)=P\left[\hat{\varrho}\left(t_{\mathrm{f}}\right)\right]$ at a desired value in the presence of the bath. This should be the optimal (maximal or minimal) value of the score $P\left(t_{\mathrm{f}}\right)$. If the initial system state $\hat{\varrho}(0)$ is given, then the change in the score $\Delta P=P\left(t_f\right)-P(0)$ due to the effects of control and the bath can be used instead of $P\left(t_{\mathrm{f}}\right)$ as the score. To first order in the Taylor expansion of the score change as a function of the state change $\Delta \hat{\varrho}=\hat{\varrho}\left(t_f\right)-\hat{\varrho}(0)$ in a chosen basis, we have
$$\Delta P \approx \sum_{m, n} \frac{\partial P}{\partial \varrho_{m n}} \Delta \varrho_{m n}=\operatorname{Tr}(\hat{P} \Delta \hat{\varrho}) .$$
Here, the coefficients

$$\left.\frac{\partial P}{\partial \varrho_{m n}}\right|{t=0} \equiv(\hat{P}){n m}$$
are the matrix elements (in the chosen basis) of a Hermitian operator $\hat{P}$, which is the gradient of $P[\hat{\varrho}(t)]$ with respect to $\hat{\varrho}$ at $t=0$ :
$$\hat{P}=\left.\left(\nabla_{\hat{e}} P\right)\right|{t=0}=\left.(\partial P / \partial \hat{\varrho})\right|{t=0}=0 .$$
The operator $\hat{P}$ contains the complete information on the controlled variable. The transposition in (12.11a) allows us to express the sum over the entries (the Hadamard matrix product) in (12.10) as a trace of the operator product $\hat{P} \Delta \hat{\varrho}$. Equation (12.10) holds when $\Delta \hat{\varrho}$ and $\Delta P$ are small, which implies weak system-bath coupling. The score change $\Delta P$ then quantifies the bath effects and not the internal system dynamics.

If $P$ is the state purity, $P=\operatorname{Tr}\left(\hat{\varrho}^2\right)$, then (12.11a) is proportional to the state, $\hat{P}=2 \hat{\varrho}(0)$

## 物理代写|热力学代写thermodynamics代考|Control, Score, and Constraint

$$\hat{H} \mathrm{~S}=\hat{H} \mathrm{~S}[f(\tau)], \quad \hat{U}(\tau)=\mathrm{T} e^{-(i / \hbar) \int_0^\tau d \tau^{\prime} \hat{H}\left(\tau^{\prime}\right)} \equiv \hat{U}[\boldsymbol{f}(\tau)] .$$

$$\hat{H}S(\tau)=i \hbar\left[\frac{\partial}{\partial \tau} \hat{U}(\tau)\right] \hat{U}^{\dagger}(\tau) .$$ 分数 $P$ 衡量控制感兴趣数量的成功程度可以写成系统状态的实值泛函 $\varrho(t)$ 当时 $t{\mathrm{f}}$ 当控制结束时。例如，该 分数可以是给定纯状态的最大保真度 $|\Psi\rangle$ 在浴引起的退相干下， $F_{\Psi\rangle}=\langle\Psi|\hat{\varrho}| \Psi\rangle$. 或者，分数可以是最大 并发Co
$$C o_{\left|\Psi_{A B}\right\rangle}=\left[2\left(1-\operatorname{Tr} \hat{A}^2\right)\right]^{1 / 2}, \quad \hat{\varrho} \mathrm{A}=\operatorname{Tr} \mathrm{B}|\Psi \mathrm{AB}\rangle\left\langle\Psi_{\mathrm{AB}}\right|,$$

$$P=\max t \in\left[0, t_t\right] P[\hat{\varrho}(t)] .$$

$$E=\hbar \int_0^t d \tau \dot{f}^2(\tau),$$

$$E=\hbar^{-1} \int_0^{t_f} d \tau\left\langle(\Delta \hat{H})^2(\tau)\right\rangle \mathrm{id}$$

## 物理代写|热力学代写thermodynamics代考|Fixed Time Approach

$$\Delta P \approx \sum_{m, n} \frac{\partial P}{\partial \varrho_{m n}} \Delta \varrho_{m n}=\operatorname{Tr}(\hat{P} \Delta \hat{\varrho}) .$$

$$\frac{\partial P}{\partial \varrho_{m n}} \mid t=0 \equiv(\hat{P}) n m$$

$$\hat{P}=\left(\nabla_{\hat{\epsilon}} P\right)|t=0=(\partial P / \partial \hat{\varrho})| t=0=0 .$$

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