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 [2018/11/04 09:56] (current) lpawela |
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- 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 (see [( :szymanski2017classification )] for details). | ||

- | - 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 |}} |

numerical-range/generalizations/joint-numerical-range.txt · Last modified: 2018/11/04 09:56 by lpawela