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numerical-range:generalizations:joint-numerical-range [2019/10/01 13:02]
rkukulski
numerical-range:generalizations:joint-numerical-range [2020/06/30 07:28] (current)
plewandowska
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 L(A_1,​\ldots,​A_k) = \left\{ \left( \Tr \rho A_1,\ldots, \Tr \rho A_k \right): \rho \in \Omega_d \right\}. 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.+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 [( :​li2020joint )]. 
 + 
 +===== Kippenhahn Theorem'​s for joint numerical range ===== 
 +An algebraic curve has been associated with the numerical range and has been 
 +studied from the 1930s on [( :​plaumann2019kippenhahn )]. 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 $T$. 
 + 
 + 
 +==== Theorem ====  
 +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 $X^*$. 
 + 
 +==== Theorem ====  
 +The joint numerical range $W$ is the convex hull of the Euclidean closure of  
 +$T_1\cup\cdots\cup T_r$, in other words, $W = \text{conv}\text{cl}(T_1\cup\cdots\cup T_r)$. 
 + 
 + 
  
 ==== Classification of JNRs ==== ==== Classification of JNRs ====
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   - If two segments are present in the boundary, there exist an infinite number o other segments.   - 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:+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 exists ​object with [( :​xie2019observing )]:
   * no flat parts in boundary at all $e=0$, $s=0$, {{ :​numerical-range:​generalizations:​img_5903.png?​nolink&​200 |}}   * no flat parts in boundary at all $e=0$, $s=0$, {{ :​numerical-range:​generalizations:​img_5903.png?​nolink&​200 |}}
   * one segment in the boundary $e=0$, $s=1$, {{ :​numerical-range:​generalizations:​img_5935.png?​nolink&​200 |}}   * one segment in the boundary $e=0$, $s=1$, {{ :​numerical-range:​generalizations:​img_5935.png?​nolink&​200 |}}
Line 47: Line 85:
  
 ====Application==== ====Application====
-An example application of numerical range can be found in [( :​szymanski2019geometric )], [( :​czartowskiseparability )].+An example application of numerical range can be found in [( :​szymanski2019geometric )], [( :​czartowskiseparability ​)], [( :​plaumann2019kippenhahn ​)].
numerical-range/generalizations/joint-numerical-range.1569934964.txt.gz · Last modified: 2019/10/01 13:02 by rkukulski