# Numerical Shadow

The web resource on numerical range and numerical shadow

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# Joint numerical range

### Definition

Consider $n$ Hermitian matrices $A_1,\ldots,A_n$ of dimension $d \times d$. Their joint numerical range (JNR) $L(A_1,\ldots,A_n)$ is defined as $$L(A_1,\ldots,A_n) = \left\{ \left( \Tr \rho A_1,\ldots, \Tr \rho A_n \right): \rho \in \Omega_d \right\}.$$ Due to taking all mixed states the joint numerical range is a convex body in a $n$-dimensional space. More information about convexity of joint numerical range we can find in .

## Kippenhahn's Theorem for joint numerical range

An algebraic curve has been associated with the joint numerical range and has been studied from the 1930s in . Let $\1_d$ denote the $d\times d$ identity matrix, define $p(x)=\det( x_0 \1_d + x_1 A_1 + \cdots + x_n A_n ),$ and consider the complex projective hypersurface $\mathcal{V}(p)=\{x\in\mathbb{P}^n \mid p(x) = 0 \}.$

#### Case $n=2$

If $n= 2$, then $\mathcal{V}(p) \subset \mathbb{P}^2$ is an algebraic curve. Then the JNR $L(A_1, A_2)$ naturally can be identified with numerical range $W(A_1 + \ii A_2)$. It was shown  that the eigenvalues of $A_1+\ii A_2$ are the foci of the affine curve of real points $T=\{ y_1+\ii y_2 \mid y_1,y_2\in\mathbb{R},(1:y_1:y_2)\in \mathcal{V}(p)^\ast\}\subset\mathbb{C},$ where $\mathcal{V}(p)^\ast$ is the dual curve to the curve $\mathcal{V}(p)$.

### Theorem (Kippenhahn)

The numerical range of $A_1+\ii A_2$ is the convex hull of $T$, in other words, $W(A_1+\ii A_2)=\text{conv}(T)$.

## Generalization

### Definition

Let $X_1,\dots,X_r$ denote the irreducible components of the hyperbolic hypersurface $\mathcal{V}(p)$. We define $T_i$ as the semi-algebraic set $T_i = \left\{ (y_1,\dots,y_n) \in (\mathbb{R}^n)^\ast \mid (1:y_1:\dots:y_n)\in (X_i^\ast)_\text{reg} \right\}, \qquad i=1,\dots,r$ and $(X^{*}_i)_{\text{reg}}$ as the set of the regular points of the dual variety $X^*$.

### Theorem

The joint numerical range $L(A_1,\ldots,A_n)$ is the convex hull of the Euclidean closure of $T_1\cup\cdots\cup T_r$, in other words, $L(A_1,\ldots,A_n) = \text{conv}\text{cl}(T_1\cup\cdots\cup T_r)$.

### Classification of JNRs

##### Two qutrit matrices

This classification comes from . Consider two Hermitian matrices $A_1$ and $A_2$ of size $3 \times 3$. Then there are four possible shapes of the JNRs

• An oval–object without any flat parts, the boundary is a sextic curve.
• Object with one flat part, a convex hull of a quatric curve.
• Convex hull of an ellipse and an outside point, which has two flat parts on the boundary.
• A triangle (when $A_1$ and $A_2$ commute). This can be further degenerated.
##### Three qutrit matrices

This classification is taken from  (see for details). Such JNRs must obey the following rules:

1. Two two-dimensional subspaces must share a common point, hence all flat parts are mutually connected.
2. Convex geometry of a three-dimensional Euclidean space supports up to four mutually intersecting ellipses.
3. If three ellipses are present in the boundary, the geometry does not allow for existence of any additional segment.
4. If two segments are present in the boundary, there exist an infinite number o other segments.

One can prove that to obtain JNR in this case, we may restrict ourselves to only pure states. Moreover, such JNRs must have no corner points. The classification is bases on flat portions in the boundary of the JNR. Each flat portion is the image of a Bloch sphere, which is either a filled ellipse or a segment.

Let $e$ denote the number of filled ellipses and $s$ the number of segments in the boundary.

All configurations permitted by these rules are realized. There exists 3-dimensional objects classified in  and experimentally constructed in  with:

• no flat parts in boundary at all $e=0$, $s=0$, • one segment in the boundary $e=0$, $s=1$, • one ellipse in the boundary $e=1$, $s=0$, • one ellipse and a segment $e=1$, $s=1$, • two ellipses in the boundry $e=2$, $s=0$,
• two ellipses and a segment $e=2$, $s=1$, • three ellipses $e=3$, $s=0$, • four ellipses $e=4$, $s=0$, Additionally in the qutrit case, if there exist of the JNR, the following configurations are possible:

• JNR is the convex hull of an ellipsoid and a point outside the ellipsoid, $e=0$, $s=\infty$, • JNR is the convex hull of an ellipse and a point outside the affine hull of the ellipse, $e=1$, $s=\infty$, ### Application

An example application of numerical range can be found in , , .

1. Chi-Kwong Li, Yiu-Tung Poon, Ya-Shu Wang, 2020. Joint numerical ranges and communtativity of matrices. arXiv preprint arXiv:2002.02768.
2. Daniel Plaumann, Rainer Sinn, Stephan Weis, 2019. Kippenhahn's Theorem for joint numerical ranges and quantum states. arXiv preprint arXiv:1907.04768.
3. F. D Murnaghan, 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, National Academy of Sciences, pp.246.
4. D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. Linear Algebra and its Applications, 252, pp.115 - 139.
5. Konrad Szymański, Stephan Weis, Karol Życzkowski, 2017. Classification of joint numerical ranges of three hermitian matrices of size three. Linear Algebra and its Applications, Elsevier.
6. Jie Xie, Aonan Zhang, Ningping Cao, Huichao Xu, Kaimin Zheng, Yiu-Tung Poon, Nung-Sing Sze, Ping Xu, Bei Zeng, Lijian Zhang, 2019. Observing geometry of quantum states in a three-level system. arXiv preprint arXiv:1909.05463.
7. Konrad Jan Szymański, Karol Życzkowski, 2019. Geometric and algebraic origins of additive uncertainty relations. Journal of Physics A: Mathematical and Theoretical, IOP Publishing.
8. Jakub Czartowski, Konrad Szymański, Bartłomiej Gardas, Yan V Fyodorov, Karol Życzkowski, 2019. Separability gap and large-deviation entanglement criterion. Physical Review A, 100, APS, pp.042326. 