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numerical-range:generalizations:joint-numerical-range

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numerical-range:generalizations:joint-numerical-range [2018/11/04 09:48]
lpawela
numerical-range:generalizations:joint-numerical-range [2019/10/01 13:02]
rkukulski
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 == Three qutrit matrices == == Three qutrit matrices ==
-This classification is taken from [( :​szymanski2017classification )]. Such JNRs must obey the following rules+This classification is taken from [( :​szymanski2017classification )] (see for details). Such JNRs must obey the following rules
  
-  - In this case we may restrict ourselves to only pure states ​(see [( :​szymanski2017classification )] for details)+  - 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+  - 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.   - 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.   - Convex geometry of a three-dimensional Euclidean space supports up to four mutually intersecting ellipses.
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   * three ellipses $e=3$, $s=0$, {{ :​numerical-range:​generalizations:​img_5886.png?​nolink&​200 |}}   * three ellipses $e=3$, $s=0$, {{ :​numerical-range:​generalizations:​img_5886.png?​nolink&​200 |}}
   * four ellipses $e=4$, $s=0$, {{ :​numerical-range:​generalizations:​img_5957.png?​nolink&​200 |}}   * four ellipses $e=4$, $s=0$, {{ :​numerical-range:​generalizations:​img_5957.png?​nolink&​200 |}}
 +
 +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, $e=0$, $s=\infty$, {{ :​numerical-range:​generalizations:​img_5944.png?​nolink&​200 |}}
 +  * JNR is the convex hull of an ellipse and a point outside the affine hull of the ellipse, $e=1$, $s=\infty$, {{ :​numerical-range:​generalizations:​img_5944bis.png?​nolink&​200 |}}
 +
 +====Application====
 +An example application of numerical range can be found in [( :​szymanski2019geometric )], [( :​czartowskiseparability )].
numerical-range/generalizations/joint-numerical-range.txt · Last modified: 2020/02/19 10:40 by plewandowska