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

Perturbation of unitary matrix numerical range

We are given an arbitrary unitary matrix $U$. The numerical range $W(U)$ is convex hull of eigenvalues of matrix $U$, $W(U)=\text{conv}(\lambda(U))$. If $V$ denotes unitary matrix arbitrary close to matrix $U$ then the numerical range $W(V)$ should differs from $W(U)$ slightly. It comes from the fact, that the function $U \to \lambda(U)$, that for given unitary matrix returns vector of its eigenvalues is continuous. The question is; can we actually predict how small changes impact changes in numerical range?

In the first setup we take two unitary matrices - matrix $U \in U_d$ and its perturbation $V \in U_d$ - for example for given constant $0<c< \! \! <1$, we have $\| U - V \| \le c$. Then we fix continuous parametric curve $U(t) \in U_d$ for $t \in [0,1]$, that connects matrices $U=U(0)$ and $V=U(1)$. Which one should we take?

We start our considerations by taking the most natural curve between $U$ and $V$, which is the shortest one - geodesic. The geodesic between unitary matrices is well-known [1] and in our case it is given by formula $$ t \to U \exp(itH(U^\dagger V)),$$ where function $H$ for arbitrary unitary matrix $U$ has the form $H(U)=-i \text{Log}(U) \in H_d$, with convection, that $\lambda(H(U)) \subset (-\pi, \pi]$.

We can simplyfy formulation of our problem to investigate $W(U \exp(itH))$, where $\|H\|_{\infty}\le \pi$. Without loss of generality we can assume that $H$ is a diagonal matrix. Because, tha global phase does not matter ($W(U) \sim W(\lambda U)$) and we put special interest on $t$ arbitrary small, we take such matrix $H$, for which $\lambda(H)$ is a probability vector.

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 $V(t) = \sum_{i=1}^{d} e^{i p_i t} \ket{i}\bra{i} \in U_d$ for $t \geq 0$. 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 $t \geq 0$,

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

  • Solutions $\ket{x_1}$ of $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2$ and $\ket{x_k}$ of $\max\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2$ are orthogonal.
  • If $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2=0$ 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 $\dim(S_\lambda^{UV(t)}) \geq k-l.$
1. Jorge Antezana, Gabriel Larotonda, Alejandro Varela, 2014. Optimal paths for symmetric actions in the unitary group. Communications in Mathematical Physics, 328, Springer, pp.481–497.
numerical-range/generalizations/c-perturbation-of-unitary-matrix-numerical-range.txt · Last modified: 2019/07/14 08:40 by rkukulski