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numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range

# Perturbation of unitary matrix numerical range

## Theorem

Let $U \in U_d$ be unitary matrix of dimension $d$ and denote $S_y^M=\{\ket{x}: (y\mathbb{1}_d-M)\ket{x}=0, \|x\|=1\}$ for some matrix $M$. Assume that $\lambda$ is eigenvalue of $U$, $p=\{p_i\}_{i=1}^d$ is a probability vector and define matrix $V(t) = \sum_{i=1}^{d} e^{i p_i t} \ket{v_i}\bra{v_i} \in U_d$ for $t \geq 0$, where vectors $\{\ket{v_i}\}$ forms orthonormal basis. Then:

• Each eigenvalue of product $UV(t)$ moves counterclockwise as $t \rightarrow 2 \pi$ or stays in initial state $t=0$
• If $\dim(S_\lambda^U)=k$ and eigenvalues of $UV(t)$, for which initial position was $\lambda$ are $\{\lambda_{t,j}\}_{j=1}^k$, then for small enough $t \geq 0$,

$$\lambda_{t,1} \approx \lambda \exp\left( i t \min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{v_i}{x}|^2 \right),$$ $$\lambda_{t,k} \approx \lambda \exp\left( i t \max\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{v_i}{x}|^2 \right),$$

• Solutions $\ket{x_1}$ of $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{v_i}{x}|^2$ and $\ket{x_k}$ of $\max\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{v_i}{x}|^2$ are orthogonal.
• If $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{v_i}{x}|^2=0$ then $\lambda_{t,1}= \lambda$
• If $\dim(S_\lambda^U)=k$ and $|\{p_i: p_i>0\}|=l<k$, then $\lambda$ is eigenvalue of $UV(t)$ and $\dim(S_\lambda^{UV(t)}) \geq k-l.$