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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 Both sides previous revision Previous revision 2020/02/19 10:40 plewandowska 2019/10/01 13:02 rkukulski 2019/10/01 13:01 plewandowska 2019/10/01 12:55 plewandowska 2018/11/04 09:56 lpawela 2018/11/04 09:48 lpawela 2018/10/08 08:42 plewandowska [Definition] 2018/10/08 08:22 plewandowska [Definition] 2018/06/15 16:20 lpawela 2018/06/15 16:17 lpawela 2018/06/06 12:09 lpawela 2018/06/06 12:02 lpawela 2018/05/24 13:33 lpawela 2018/05/24 13:30 lpawela 2018/05/24 10:24 lpawela 2018/05/24 10:23 lpawela 2018/05/24 10:21 lpawela created Next revision Previous revision 2020/02/19 10:40 plewandowska 2019/10/01 13:02 rkukulski 2019/10/01 13:01 plewandowska 2019/10/01 12:55 plewandowska 2018/11/04 09:56 lpawela 2018/11/04 09:48 lpawela 2018/10/08 08:42 plewandowska [Definition] 2018/10/08 08:22 plewandowska [Definition] 2018/06/15 16:20 lpawela 2018/06/15 16:17 lpawela 2018/06/06 12:09 lpawela 2018/06/06 12:02 lpawela 2018/05/24 13:33 lpawela 2018/05/24 13:30 lpawela 2018/05/24 10:24 lpawela 2018/05/24 10:23 lpawela 2018/05/24 10:21 lpawela created Last revision Both sides next revision Line 22: Line 22: == 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. Line 40: Line 41: * 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 )].