numerical-range:generalizations:joint-numerical-range

Consider $k$ Hermitian matrices $A_1,\ldots,A_k$ of dimension $d \times d$. Their joint numerical range (JNR) $L(A_1,\ldots,A_k)$ is defined as $$ L(A_1,\ldots,A_k) = \left\{ \left( \Tr \rho A_1,\ldots, \Tr \rho A_k \right): \rho \in \Omega_d \right\}. $$ Due to taking all mixed states the joint numerical range is a convex body in a $k$-dimensional space. More information about convexity of joint numerical range we can find in [1].

An algebraic curve has been associated with the numerical range and has been studied from the 1930s on [2]. Let $\1$ denote the $d\times d$ identity matrix, \[ p(x)=\det( x_0 \1 + x_1 A_1 + \cdots + x_n A_n ), \] and consider the complex projective hypersurface \[ \mathcal{V}(p)=\{x\in\mathbb{P}^n \mid p(x) = 0 \}. \] It was shown that the eigenvalues of $A_1+\ii A_2$ are the foci of the affine curve of real points \[ T=\{ y_1+\ii y_2 \mid y_1,y_2\in\mathbb{R},(1:y_1:y_2)\in \mathcal{V}(p)^\ast\}\subset\mathbb{C}. \] Kippenhahn recognized the meaning of the convex hull of $T$.

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

Let we denote $T_i$ 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 $X^*$.

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

This classification comes from [3]. 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.

This classification is taken from [4] (see for details). Such JNRs must obey the following rules

- In this case we may restrict ourselves to only pure states.
- Any flat part in the boundary is the image of the Bloch sphere - two-dimensional subspace of a the sapce of $3 \times 3$ Hermitian matrices without corner points for

all configurations of Figure 2. We are unaware of earli

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

All configurations permitted by these rules are realized. Let us denote by $e$ the number of ellipses in the boundary and by $s$ the number of segments. There exist object 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, $e=1$, $s=\infty$,

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.
D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. *Linear Algebra and its Applications*, 252, pp.115 - 139.

4.
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.

5.
Konrad Jan Szymański, Karol Życzkowski, 2019. Geometric and algebraic origins of additive uncertainty relations. *Journal of Physics A: Mathematical and Theoretical*, IOP Publishing.

6.
Jakub Czartowski, Konrad Szymański, Bartłomiej Gardas, Yan V Fyodorov, Karol Życzkowski. *Separability gap and large deviation entanglement criterion.* APS.

numerical-range/generalizations/joint-numerical-range.txt · Last modified: 2020/05/16 13:03 by plewandowska