Parameterization of numerical ranges

Konrad Szymański (physicist, please be patient)

RCQI Bratislava

Numerical ranges: images of quantum state spaces

standard numerical range: unit vectors $$ \cred W(X)\cblack= \{( v\cblack,\cred X \cblack v\cblack) : \\~~~ v\cblack \in\mathbb{C}^d, \lvert v\cblack\rvert=1\}.$$

joint numerical ranges: $\text{Tr}=1$ positive semidefinite matrices

($X_1, \ldots, X_k$ Hermitian)

$$\cred L(X_1, \ldots, X_k)\cblack =\\ \{(\Tr \rho\cred X_1\cblack, \ldots, \Tr \rho\cred X_k\cblack) : \Tr \rho\cblack =1, \rho\cblack\succeq 0\}\\ \cgray =\text{conv} \{((v, X_1 v),\ldots, (v, X_k v)) : \lvert v\rvert =1\}$$

$W(X_1+i X_2)\approx L(X_1, X_2)$ for Hermitian $X_1, X_2$.

Quantum information: sets of allowed expectation values

$\cred\langle X\rangle_{ v}\cblack:=( v \cblack, \cred X \cblack v\cblack )\color{gray}\in\R$ $\approx$ expectation value of $\cred X$ (assumed Hermitian)

Sum of variances $\Delta^2 \cred X\cblack +\Delta^2 \cred Y\cblack $ from $$\cred L(X,Y,X^2+Y^2)$$

Entanglement detection from exp. results

How many parameters are needed to specify $\cred W(X)$?

$\cred X\cblack\in\mathbb{C}^{d\times d}$ $\implies$ $\le 2d^2$ real parameters.
$\cred W(X)=\cblue W(S)$ for complex symmetric $\cblue S$ [2] $\implies$ $\le d^2$ real parameters.

Motivation: hypothesis involving a class of operators

Truncated Toeplitz Operators: defined by $d$ complex numbers
(at least for the most studied class: compressed shifts)
Hypothesis [1,4]: every symmetric $\cblue S\in \mathbb{C}^{d\times d}$ unitarily similar to $$\cpurple \bigoplus T_i\cblack \text{~with~TTO~} \cpurple T_i\cblack\text{?}\color{gray}\text{~~~(works for~}d\le3\text{)} $$
Theorem [2]: For every $\cred X\cblack\in\mathbb{C}^{d\times d}$ there exists a symmetric $\cblue S$ such that $\cred W(X)\cblack = \cblue W(S)$
($2. \wedge 3.$) $\implies$ every $\cred W(X)\cblack=\cblue W(\bigoplus T_i)$, at most $O(d)$ parameters.
My numerics [3]: how many parameters to $\cred W(X)$, effectively?

Numerical ranges have well behaved support functions

$$\corange h_{\cred S}\cblack (\vec y) = \max_{\cred \vec x \in S} \cred \vec x\cblack \cdot \vec y$$ $${\corange h}_{\cred L(X_1, \ldots, X_k)}(\vec n):=\corange h\cblack(\cred \vec X\cblack, \vec n):=\corange \lambda_{\max}\cblack(\overbrace{\vec n\cdot\cred\vec X\cblack}^{\sum_{i=1}^k n_i \cred X_i\cblack})$$

What changes to $\cred \vec X:=(X_1, \ldots, X_k)$ leave $\corange h_{\cred \vec X}\cblack$ unchanged?

$$D_{\cred \vec X} {\corange h}_{\vec n} [\cblue \!\!\!\!\overbrace{\vec Y}^{Y_1, \ldots, Y_k}\!\!\!\!\cblack] := \frac{\partial}{\partial \varepsilon} {\corange h}(\cred \vec X\cblack+\varepsilon \cblue\vec Y\cblack,\vec n)|_{\varepsilon=0}$$

Hellmann–Feynmann theorem

If the eigenspace to $\corange \lambda_{\max}$ of Hermitian $\cred X$ is one-dimensional: $$\cred X \cblack v_{\max}\cblack=\corange \lambda_{\max} \cblack v_{\max},$$ then $$\frac{\partial }{\partial t} \corange \lambda_{\max}\cblack (\cred X\cblack+t\cblue Y\cblack)=\color{gray}\underbrace{ \color{black}\frac{( v_{\max}\cblack,\cblue Y\cblack v_{\max}\cblack)}{( v_{\max}\cblack, v_{\max}\cblack)}}_{\langle Y\rangle_{\max}},$$

What changes to $\cred \vec X:=(X_1, \ldots, X_k)$ leave $\corange h_{\cred \vec X}\cblack: \R^k \rightarrow \R$ unchanged?

$$D_{\cred \vec X} {\corange h}_{\vec n} [\cblue \vec Y\cblack] := \frac{\partial}{\partial \varepsilon} {\corange h}(\cred \vec X\cblack+\varepsilon \cblue\vec Y\cblack,\vec n)|_{\varepsilon=0}$$ $$D_{\cred \vec X} {\corange h}_{\vec n} \cblack [\cblue \vec Y\cblack] := \cblue \langle \cblack \vec n \cdot\cblue \vec Y \rangle_{\max}\cgray=\frac{( v_{\max}, (\sum_{i=1}^k n_i Y_i) v_{\max})}{( v_{\max}, v_{\max})}$$

For what $\cblue\vec Y$ is $D_{\cred \vec X} \corange h_{\cblack \vec n}\cblack[\cblue \vec Y\cblack]$ zero for all $\vec n$?

Search for $\cblue \text{subspaces}\subset\H_d^k$ such that $D_{\cred \vec X} {\corange h}_{\vec n}\cblack[\cblue\vec Y\cblack]=0$

$$\H_d - \text{Hermitian matrices of size~}d,~\H_d^k -\text{sequences of~}k\text{~matrices}$$

For all $\vec n$!

Support function of $\cblue L(\vec X+\varepsilon \vec Y)$ the same as $\cred L(\vec X)$ (up to $O(\varepsilon^2))$

The kernel of $D_{\cred \vec X}$ $\sim$ local tangents to equivalence class $$\cred \vec X\cblack\sim\cblue \vec X'\cblack \Leftrightarrow \cred L(\vec X)\cblack=\cblue L(\vec X')?$$

Five parameters needed for 2D NR of size-2 matrices

$\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},~\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix},~\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix},~\sigma_4=\begin{pmatrix}1&0\\0&1\end{pmatrix}.$

$$\cgray W\left(\begin{pmatrix}0&2\\0&0\end{pmatrix}\right)\approx\cred L(\overbrace{\sigma_1, \sigma_2}^{\vec X:=(X_1, X_2)})\cblack,~\corange h\cblack(\cred\vec X\cblack,\!\!\!\!\!\!\!\!\!\!\overbrace{\!\!\vec n\!\!}^{(r \cos \theta, r\sin\theta)}\!\!\!\!\!\!\!\!)= \lvert r\lvert$$

$$D h_{\vec n} \left[\sigma_1,0\right] = \colcc r \cos^2 \theta$$

$$D h_{\vec n} \left[\sigma_2,0\right] =D h_{\vec n} \left[0,\sigma_1\right]= \colcs r \cos \theta \sin\theta$$

$$D h_{\vec n} \left[\sigma_3,0\right] = D h_{\vec n} \left[0,\sigma_3\right]= 0$$

$$D h_{\vec n} \left[\sigma_4,0\right] = \colc r\cos\theta$$

$$D h_{\vec n} \left[0,\sigma_2\right] = \colss r \sin^2\theta$$

$$D h_{\vec n} \left[0,\sigma_4\right] = \cols r\sin\theta$$

Five linearly independent functions $\sim$ five parameters.

Search for subspaces $\subset\H_d^k$ such that $D_{\cred \vec X} \corange h_{\vec n}\cblack[\cblue\vec Y\cblack]=0$

for all $\vec n$ $\mapsto$ (weaker) for some $\vec n_1, \ldots, \vec n_p$.

$$\Delta_{\cred \vec X} \cblack[\cblue\vec Y\cblack]:=(D_{\cred \vec X} \corange h_{\cblack\vec n_1} \cblack[\cblue\vec Y\cblack], \ldots, D_{\cred\vec X} \corange h_{\cblack\vec n_p} \cblack[\cblue\vec Y\cblack])$$ is a linear operator (from $\H_d^k$ to $\R^p$)

Hypothetically: $\cblue \ker \Delta$ $\approx~$local tangents to equivalence class $$\color{gray}\overbrace{\cred \vec X\cblack}^{(X_1, \ldots, X_k)}\!\!\!\!\!\!\!\!\cblack\sim \!\!\!\!\!\!\color{gray}\underbrace{\cblue\vec X'}_{(X'_1, \ldots, X'_k)}\!\!\!\!\!\!\!\cblack \Leftrightarrow \cred L(X_1, \ldots, X_k)\cblack=\cblue L(X'_1, \ldots, X'_k)?$$

Local dimension $p(d,k)$ of $\cred L(X_1, \ldots, X_k)$ for $ \cred \vec X \cblack \in(\H_d)^k$

(numerical image dimension estimation; $X_1, \ldots, X_k$ sampled from Gaussian Unitary Ensemble)

$p(d,k)$	$k=2$	$k=3$	$k=4$	$k=5$	$k=6$	$k=7$
$d=2$	5	9	13	17	21	25
$d=3$	9	19	28	37	46
$d=4$	14	33	49	65	81
$d=5$	20	51	76	101
$d=6$	27	73	109	145
$d=7$	35	99	148	197
$d=8$	44	129	193

Conjecture: $\cblue p(d,k=2)\cblack=\frac{d(d+3)}2,~\cblue p(d,k>2)\cblack=(k-1)d^2+1$.

Conclusion and references

Conjecture: Are all symmetric matrices representable by simple sums of TTOs [1]?

Conjecture: Is the number of parameters needed to fully define $\cred L(X_1, \ldots, X_k)$ for $\cred \vec X \cblack \in \H^k_d$ $$\cblue p(d,k=2)\cblack=\frac{d(d+3)}2,~\cblue p(d,k>2)\cblack=(k-1)d^2+1?$$

Numerical evidence [3] suggests so! $\H_d$ are Hermitian matrices of size $d$.

Open question: Why is it different for $k=2$ and $k>2$?

Symmetric real-offdiagonal matrices?

R. O'Loughlin, Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces , LAA 2023
J. W. Helton and I. Spitkovsky, The possible shapes of numerical ranges, OAM 2012
K. Szymański, https://quantumstat.es/note/JNR-and-parameters (research notes)
S. Garcia, D. Poore, W. Ross, Unitary equivalence to a truncated Toeplitz operator: analytic symbols