Any unitary operation $U \in su(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (thus $U = e^{-iHt}$), which corresponds to a unitary operator generated by $H$ for an evolution time $t$. For the time-dependent case, $A$ becomes more complex and is expressed in terms of a time-ordered exponential.

My question is about the matrix element of $A$, denoted as $A_{pq}$. Assuming I can always find a Hamiltonian $H$ (or $H(t)$) that can generate any unitary operator $U \in su(N)$, does this imply that each matrix element $A_{pq}$ of $A$ must possess a unique functional dependence on $H$ (or $H(t)$)?

The origin of my question lies in this Science report paper about quantum optimal control landscapes. Particularly, the question is essentially related to the transition from Eq.(2) to (3) there. The main argument they present is as follows: First, consider a time-dependent Hamiltonian $H(t)$ that depends on a control field $C(t)$. The unitary operator $U$ is generated by this Hamiltonian, indicating that $U$ is also a functional of $C(t)$. The cost function, which is the functional derivative of the transition probability from state $\lvert i \rangle$ to $\lvert f \rangle$ (i.e., $\lvert \langle i \rvert U \lvert f \rangle \rvert^2 = \lvert U_{if} \rvert^2$) with respect to $C(t)$, is denoted as \begin{equation} \frac{\delta \lvert U_{if}\rvert^2}{\delta C(t)}. \end{equation} The paper then makes use of the fact that $U = e^{iA}$ where $A$ is an arbitrary Hermitian matrix, and therefore, $U_{if} = \langle i \rvert U \rvert f \rangle = \langle i \rvert e^{iA} \lvert f \rangle$, leading to the expression

\begin{equation} \frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = \sum_{p,q} \frac{\partial \lvert U_{if} \rvert^2}{\partial A_{pq}} \frac{\delta A_{pq}}{\delta C(t)}. \end{equation}

The argument they then rely on is that each matrix element $A_{pq}[C(t)]$ of $A$ must be independently addressable, implying that each $A_{pq}[C(t)]$ should have a unique functional dependence on $C(t)$ based on the "controllability assumption" they introduced earlier. Therefore, they claimed that the set of functions $\{\delta A_{pq}/\delta C(t)\}$ are linearly independent for all $p,q,$ and $t$. As I understood from Eq.(1), the controllability assumption essentially states that there exists at least one control field $C(t)$ such that $\lvert U_{if} \rvert^2 = 1$ corresponds to $\frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = 0$ for all time $t \in [0, T]$. However, I don't fully understand how such arguments put together to get such conclusion. I'd be greatly appreciate it if anyone could provide any insight and explanation. I know my question tends to compress a lot of things all at once, so please feel free to leave any question or comment for any clarification.

Any unitary operation $U \in U(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (thus $U = e^{-iHt}$), which corresponds to a unitary operator generated by $H$ for an evolution time $t$. For the time-dependent case, $A$ becomes more complex and is expressed in terms of a time-ordered exponential.

My question is about the matrix element of $A$, denoted as $A_{pq}$. Assuming I can always find a Hamiltonian $H$ (or $H(t)$) that can generate any unitary operator $U \in U(N)$, does this imply that each matrix element $A_{pq}$ of $A$ must possess a unique functional dependence on $H$ (or $H(t)$)?

The origin of my question lies in this Science report paper about quantum optimal control landscapes. Particularly, the question is essentially related to the transition from Eq.(2) to (3) there. The main argument they present is as follows: First, consider a time-dependent Hamiltonian $H(t)$ that depends on a control field $C(t)$. The unitary operator $U$ is generated by this Hamiltonian, indicating that $U$ is also a functional of $C(t)$. The cost function, which is the functional derivative of the transition probability from state $\lvert i \rangle$ to $\lvert f \rangle$ (i.e., $\lvert \langle i \rvert U \lvert f \rangle \rvert^2 = \lvert U_{if} \rvert^2$) with respect to $C(t)$, is denoted as \begin{equation} \frac{\delta \lvert U_{if}\rvert^2}{\delta C(t)}. \end{equation} The paper then makes use of the fact that $U = e^{iA}$ where $A$ is an arbitrary Hermitian matrix, and therefore, $U_{if} = \langle i \rvert U \rvert f \rangle = \langle i \rvert e^{iA} \lvert f \rangle$, leading to the expression

\begin{equation} \frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = \sum_{p,q} \frac{\partial \lvert U_{if} \rvert^2}{\partial A_{pq}} \frac{\delta A_{pq}}{\delta C(t)}. \end{equation}

The argument they then rely on is that each matrix element $A_{pq}[C(t)]$ of $A$ must be independently addressable, implying that each $A_{pq}[C(t)]$ should have a unique functional dependence on $C(t)$ based on the "controllability assumption" they introduced earlier. Therefore, they claimed that the set of functions $\{\delta A_{pq}/\delta C(t)\}$ are linearly independent for all $p,q,$ and $t$. As I understood from Eq.(1), the controllability assumption essentially states that there exists at least one control field $C(t)$ such that $\lvert U_{if} \rvert^2 = 1$ corresponds to $\frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = 0$ for all time $t \in [0, T]$. However, I don't fully understand how such arguments put together to get such conclusion. I'd be greatly appreciate it if anyone could provide any insight and explanation. I know my question tends to compress a lot of things all at once, so please feel free to leave any question or comment for any clarification.

Any unitary operation $U \in su(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (thus $U = e^{-iHt}$), which corresponds to a unitary operator generated by $H$ for an evolution time $t$. For the time-dependent case, $A$ becomes more complex and is expressed in terms of a time-ordered exponential.

My question is about the matrix element of $A$, denoted as $A_{pq}$. Assuming I can always find a Hamiltonian $H$ (or $H(t)$) that can generate any unitary operator $U \in su(N)$, does this imply that each matrix element $A_{pq}$ of $A$ must possess a unique functional dependence on $H$ (or $H(t)$)?

The origin of my question lies in this Science report paper about quantum optimal control landscapes. Particularly, the question is essentially related to the transition from Eq.(2) to (3) there. The main argument they present is as follows: First, consider a time-dependent Hamiltonian $H(t)$ that depends on a control field $C(t)$. The unitary operator $U$ is generated by this Hamiltonian, indicating that $U$ is also a functional of $C(t)$. The cost function, which is the functional derivative of the transition probability from state $\lvert i \rangle$ to $\lvert f \rangle$ (i.e., $\lvert \langle i \rvert U \lvert f \rangle \rvert^2 = \lvert U_{if} \rvert^2$) with respect to $C(t)$, is denoted as \begin{equation} \frac{\delta \lvert U_{if}\rvert^2}{\delta C(t)}. \end{equation} The paper then makes use of the fact that $U = e^{iA}$ where $A$ is an arbitrary Hermitian matrix, and therefore, $U_{if} = \langle i \rvert U \rvert f \rangle = \langle i \rvert e^{iA} \lvert f \rangle$, leading to the expression

\begin{equation} \frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = \sum_{p,q} \frac{\partial \lvert U_{if} \rvert^2}{\partial A_{pq}} \frac{\delta A_{pq}}{\delta C(t)}. \end{equation}

The argument they then rely on is that each matrix element $A_{pq}[C(t)]$ of $A$ must be independently addressable, implying that each $A_{pq}[C(t)]$ should have a unique functional dependence on $C(t)$ based on the "controllability assumption" they introduced earlier. Therefore, they claimed that the set of functions $\{\delta A_{pq}/\delta C(t)\}$ are linearly independent for all $p,q,$ and $t$. As I understood from Eq.(1), the controllability assumption essentially states that there exists at least one control field $C(t)$ such that $\lvert U_{if} \rvert^2 = 1$ corresponds to $\frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = 0$ for all time $t \in [0, T]$. I don't fully understand how such arguments put together. I'd be greatly appreciate it if anyone could provide any insight and explanation. I know my question tends to compress a lot of things all at once, so please feel free to leave any question or comment for any clarification.

Any unitary operation $U \in su(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (thus $U = e^{-iHt}$), which corresponds to a unitary operator generated by $H$ for an evolution time $t$. For the time-dependent case, $A$ becomes more complex and is expressed in terms of a time-ordered exponential.

My question is about the matrix element of $A$, denoted as $A_{pq}$. Assuming I can always find a Hamiltonian $H$ (or $H(t)$) that can generate any unitary operator $U \in su(N)$, does this imply that each matrix element $A_{pq}$ of $A$ must possess a unique functional dependence on $H$ (or $H(t)$)?

The origin of my question lies in this Science report paper about quantum optimal control landscapes. Particularly, the question is essentially related to the transition from Eq.(2) to (3) there. The main argument they present is as follows: First, consider a time-dependent Hamiltonian $H(t)$ that depends on a control field $C(t)$. The unitary operator $U$ is generated by this Hamiltonian, indicating that $U$ is also a functional of $C(t)$. The cost function, which is the functional derivative of the transition probability from state $\lvert i \rangle$ to $\lvert f \rangle$ (i.e., $\lvert \langle i \rvert U \lvert f \rangle \rvert^2 = \lvert U_{if} \rvert^2$) with respect to $C(t)$, is denoted as \begin{equation} \frac{\delta \lvert U_{if}\rvert^2}{\delta C(t)}. \end{equation} The paper then makes use of the fact that $U = e^{iA}$ where $A$ is an arbitrary Hermitian matrix, and therefore, $U_{if} = \langle i \rvert U \rvert f \rangle = \langle i \rvert e^{iA} \lvert f \rangle$, leading to the expression

\begin{equation} \frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = \sum_{p,q} \frac{\partial \lvert U_{if} \rvert^2}{\partial A_{pq}} \frac{\delta A_{pq}}{\delta C(t)}. \end{equation}

The argument they then rely on is that each matrix element $A_{pq}[C(t)]$ of $A$ must be independently addressable, implying that each $A_{pq}[C(t)]$ should have a unique functional dependence on $C(t)$ based on the "controllability assumption" they introduced earlier. Therefore, they claimed that the set of functions $\{\delta A_{pq}/\delta C(t)\}$ are linearly independent for all $p,q,$ and $t$. As I understood from Eq.(1), the controllability assumption essentially states that there exists at least one control field $C(t)$ such that $\lvert U_{if} \rvert^2 = 1$ corresponds to $\frac{\delta \lvert U_{if} \rvert^2}{\delta C(t)} = 0$ for all time $t \in [0, T]$. However, I don't fully understand how such arguments put together to get such conclusion. I'd be greatly appreciate it if anyone could provide any insight and explanation. I know my question tends to compress a lot of things all at once, so please feel free to leave any question or comment for any clarification.