numerical-range:generalizations:joint-numerical-range

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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.

This classification comes from [1]. 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 [2]. Such JNRs must obey the following rules

- In this case we may restrict ourselves to only pure states (see [2] 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 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, $e=1$, $s=\infty$,

An example application of numerical range can be found in [3].

1.
D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. *Linear Algebra and its Applications*, 252, pp.115 - 139.

2.
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.

3.
Konrad Jan Szymański, Karol Życzkowski, 2019. Geometric and algebraic origins of additive uncertainty relations. *Journal of Physics A: Mathematical and Theoretical*, IOP Publishing.

numerical-range/generalizations/joint-numerical-range.1569934511.txt.gz · Last modified: 2019/10/01 12:55 by plewandowska