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Perturbation of unitary matrix numerical range

In our setup we consider the space $\mathrm{L}(\mathbb{C}^d)$. Imagine that the matrices are points in space $\mathrm{L}(\mathbb{C}^d)$ and the distance between them is bounded by small constant $0 < c \ll 1$. We will take two unitary matrices - matrix $U \in \mathrm{U}(\mathbb{C}^d)$ and its perturbation $V \in \mathrm{U}(\mathbb{C}^d)$ i.e. $||U-V||_\infty \le c$ by using $\infty$-Schatten norm. We want to determine the path connecting these points given by smooth curve. To do so, we fix continuous parametric (by parameter $t$) curve $U(t) \in \mathrm{U}(\mathbb{C}^d)$ for any $t \in [0,1]$ with boundary conditions $U(0) := U$ and $U(1) := V $. The most natural and also the shortest curve connecting $U$ and $V$ is geodesic [1] given by\begin{equation} t \rightarrow U \exp\left(t \mathrm{Log} \left(U^\dagger V\right)\right), \end{equation} where $\mathrm{Log}$ is the matrix function such that it changes eigenvalues $\lambda \in \lambda(U)$ into $\log(\lambda(U))$, where $-i\log(\lambda(U)) \subset (-\pi, \pi]$. We will study how the numerical range $W(U(t))$ will be changed depending on parameter $t$. Let $H := -i \mathrm{Log} \left( U^\dagger V \right)$. Let us see that $H \in \mathrm{Herm}(\mathbb{C}^d)$ and $W(H) \subset (-\pi, \pi]$ for any $U,V \in \mathrm{U}(\mathbb{C}^d)$. We can also observe that \begin{equation} \begin{split} W\left(U\exp\left(itH\right)\right) &= W\left( U \exp \left( it VDV^\dagger \right) \right) = W\left( UV \exp \left( it D \right) V^\dagger \right) \\& = W\left( V^\dagger U V \exp \left( it D \right) \right) = W\left( \widetilde{U} \exp \left( it D\right) \right) \end{split} \end{equation} where $\widetilde{U} := V^\dagger U V \in \mathrm{U}(\mathbb{C}^d)$. Hence, without loss of generality we can assume that $H$ is a diagonal matrix. Moreover, we can assume that $D \ge 0$ which follows from simple calculations \begin{equation} \begin{split} W \left( U \exp \left(it D \right) \right) &= W \left( U \exp \left(it D_{+} \right) \left( \exp \left( it \alpha \1 \right) \right) \right) = W \left( e^{it \alpha} U \exp \left(it D_{+} \right) \right) \\&= W \left( U \exp \left(it D_{+} \right) \right). \end{split} \end{equation} Let us see that the numerical range of $U(t)$ for any $t \in [0,1]$ is invariant to above calculations although the trajectory of $U(t)$ is changed. Therefore, we will consider the curve \begin{equation} t \rightarrow U \exp \left( it D_{+} \right), \end{equation} where $t \in [0,1]$ and $U \in \mathrm{U}(\mathbb{C}^d)$, $D_+ \in \mathrm{Diag}(\mathbb{C}^d)$ such that $D_+ \ge 0$. We will focus on the behavior of the spectrum of the unitary matrices $U(t)$, which will reveal the behavior of $W(U(t))$ for relatively small parameter $t$. Without loss of generality we can assume that $\tr \left( D_+ \right)=1$. Together with the fact that $D_+ \in \mathrm{Diag}(\mathbb{C}^d)$ and $D_+ \ge 0$ we can note that $D_+ = \sum_{i=1}^{d} p_i \ket{i}\bra{i}$, where $p \in \mathbb{C}^d$ is a probability vector. Let us also define the set \begin{equation} S_\lambda^M=\left\{\ket{x} \in \mathbb{C}^d: (\lambda\1_d-M)\ket{x}=0, \|\ket{x}\|_2=1\right\} \end{equation} for some matrix $M \in \mathrm{L}(\mathbb{C}^d)$ which consists of unit eigenvectors corresponding to the eigenvalue $\lambda$ of the matrix $M$. We denote by $k=r(\lambda)$ the multiplicity of eigenvalue $\lambda$ whereas by $I_{M,\lambda} \in U(\mathbb{C}^k,\mathbb{C}^d)$ we denote the isometry which columns are formed by eigenvectors corresponding to eigenvalue $\lambda$ of a such matrix $M$. Let $\lambda(t), \beta(t) \in \mathbb{C}$ for $t \ge 0$. We will write $\lambda(t) \approx \beta(t)$ for relatively small $t \geq 0$, whenever $\lambda(0)=\beta(0)$ and $\frac{\partial}{\partial t}\lambda(0)=\frac{\partial}{\partial t}\beta(0)$.


Let $U \in \mathrm{U}(\mathbb{C}^d)$ be a unitary matrix with spectral decomposition \begin{equation} U=\sum_{j=1}^d \lambda_j \ket{x_j}\bra{x_j}. \end{equation} Assume that the eigenvalue $\lambda \in \lambda(U)$ is such that $r(\lambda) = k$. Let us define a matrix $V(t)$ given by \begin{equation} V(t) = \exp(itD_+)=\sum_{i=1}^{d} e^{i p_i t} \ket{i}\bra{i} \in \mathrm{DU}(\mathbb{C}^d), \quad t \geq 0. \end{equation} Let $\lambda(t):=\lambda(UV(t)) $ and let every $\lambda_j(t) \in \lambda(t)$ corresponds to eigenvector $\ket{x_j(t)} $. Assume that $\lambda_{1}(t), \ldots, \lambda_{k}(t)$ are such eigenvalues that $\lambda_{j}(t) \to \lambda$, as $t \to 0$. Then:

1. If $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2=0$, then $\lambda$ is an eigenvalue of $UV(t)$;

2. If $|\{p_i: p_i>0\}|=l<k$, then $\lambda$ is an eigenvalue of $UV(t)$ and $r(\lambda) \ge k-l$;

3. Each eigenvalue of product $UV(t)$ moves counterclockwise or stays in the initial position as parameter $t$ increases;

4. If $k=1$, then \begin{equation*}\lambda_{1}(t) \approx \lambda \exp\left( i t \sum\limits_{i=1}^d\ p_i |\braket{i}{x_1}|^2 \right) \end{equation*} for small $t \geq 0$;

5. Let $Q:=I_{U,\lambda}^\dagger D_+ I_{U,\lambda}$ and $\lambda_1(Q) \le \lambda_2(Q) \le \ldots \le \lambda_k(Q) $. Then we have \begin{equation*} \lambda_{j}(t) \approx \lambda \exp\left( i \lambda_{j}(Q) t \right) \end{equation*} for small $t \geq 0$ and eigenvector $\ket{ x_j}$ corresponding to $\lambda_j \in \lambda(U)$ is given by \begin{equation*} \ket{x_j}=I_{U,\lambda} \ket{ v_j}, \end{equation*} where $\ket{ v_j} \in S^{Q}_{\lambda_j(Q)}$;

6. For each $j=1,\ldots,d$ we have \begin{equation*} \frac{\partial}{\partial t}\lambda_j(t)=i \lambda_j(t)\sum_{i=1}^d p_i |\braket{i}{x_j(t)}|^2.\end{equation*} Moreover, \begin{equation*} \sum_{j=1}^d \left|\frac{\partial}{\partial t}\lambda_j(t)\right|=1. \end{equation*}

Proof of this therorem you can see in [2].

Intuitively speaking, this theorem gives us equations which one can use to predict behaviour of $W(UV(t))$. Observe the postulate $(6)$ fully determines the movement of the spectrum. However, this is a theoretical statement and in practice determining the function $t \mapsto \ket{x_j(t)}$ is a numerically complex task. The postulates $(1)-(5)$ play a key role in numerical calculations of $W(UV(t))$. The most important fact comes from $(3)$ which says that all eigenvalues move in the same direction or stay in the initial position. The instantaneous velocity of a given eigenvalue in general case is given in $(5)$, while in the case of eigenvalue with multiplicity equal one, the instantaneous velocity is determined by $(4)$. We see that whenever the spectrum of the matrix $U$ is not degenerated, calculating these velocities is easy. What is more, when some eigenvalue is degenerated, the postulate $(5)$ not only gives us method to calculate the trajectory of this eigenvalue, but also determines the form of corresponding eigenvector. It is worth noting that the postulates $(4),(5)$ give us only an approximation of the velocities, so despite being useful in numerical calculations, these expressions are valid only in the neighbourhood of $t=0$. Moreover, sometimes we are able to precisely specify this velocities. This happens in the cases presented in $(1),(2)$. Whenever the calculated velocity is zero we know for sure that this eigenvalue will stay in the initial position. According to the postulate $(2)$ the same happens when the multiplicity of the eigenvalue is greater than the number of positive elements of vector $p$.

Illustration of above theorem

For each eigenvalue $\lambda(t)$ of matrix $UV(t)$ we mark its instantaneous velocity given by formula $\sum\limits_{i=1}^d\ p_i |\braket{i}{x(t)}|^2 $, where $\ket{x(t)}$ is corresponding eigenvector. Red colour denotes instantaneous velocity equal to one and blue colour corresponds to the instantaneous velocity equal to zero.

Example 1

Diagonal matrix $$U=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \ii & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\ii \end{pmatrix}.$$ We act on two subspaces with probability $p_1 = 1/3$ and $p_2 = 2/3$. Eigenvalues $-1$ and $-\ii$ stay in the initial position, because they lie on the orthogonal subspace to subspace of which we acting on.

Example 2

Random matrix $U$ of dimension $3 \times 3$. Vector of probability is equal $p = (1,0,0)$. All eigenvalues have nonzero velocity, so degeneration is impossible.

Example 3

Matrix $U$ of dimension $5\times 5$ having eigenbasis given by the Fourier matrix. Vector of probability is equal $p = (3/8,1/4,1/4,1/8,0)$. Because standard basis and Fourier basis are mutually unbiased, velocities are similar, which implies the shape changes slightly in time.

Example 4

Matrix $U$ with eigenvalues $(1,e^{\ii \pi/6}, \ii, \ii, \ii, -1, -1)$. We act on two subspaces with probability $p_1 = 1/3$ and $p_2 = 2/3$. In this case the eigenvalue $\ii$ is three fold degenerated, so it stays in the initial position.

More examples you can find in [3].

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.
2. Kukulski Ryszard, Lewandowska Paulina, Pawela Łukasz, 2020. Perturbation of the numerical range of unitary matrices. arXiv preprint arXiv: 2002.05553v1.
numerical-range/generalizations/c-perturbation-of-unitary-matrix-numerical-range.txt · Last modified: 2020/02/19 08:01 by rkukulski