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numerical-range:generalizations:higher-order-numerical-range

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numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:31]
lpawela
numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:56]
lpawela
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 as a tool to handle compression problems in quantum information theory. Since then as a tool to handle compression problems in quantum information theory. Since then
 their theory and applications have been advanced with remarkable enthusiasm. The their theory and applications have been advanced with remarkable enthusiasm. The
-sequence of papers ​ADD CITE!, for example, led to a striking+sequence of papers ​[( :​choi2007higher )], [( :​choi2008geometry )], [( :​woerdeman2008higher )], [( :​li2008canonical )], for example, led to a striking
 extension of the classical Toeplitz--Hausdorff theorem (convexity of $W(M)$): **all** the extension of the classical Toeplitz--Hausdorff theorem (convexity of $W(M)$): **all** the
 $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections
 of conveniently computable half--planes in $\mathbb{C}$. of conveniently computable half--planes in $\mathbb{C}$.
-Among the many more recent papers concerning the $\Lambda_k(M)$,​ let us mention ​CITE!! ​[LPS2009,​GLW2010].+Among the many more recent papers concerning the $\Lambda_k(M)$,​ let us mention [( :​li2009condition )] and [( :​gau2010higher )].
  
-Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and \.Zyczkowski ​(see [CK\.Z2006a,​CK\.Z2006b])+Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and Życzkowski ​(see [( :​choi2006higher )])
 defined the rank--$k$ numerical range of $M$ as defined the rank--$k$ numerical range of $M$ as
 \[ \[
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 where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard
 to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that
-there is some $k$--dimensional ​ subspace $S$ of $\mathbb{C}^N$ such that $(Mu,​u)=\lambda$ for \textbf{all}+there is some $k$--dimensional ​ subspace $S$ of $\mathbb{C}^N$ such that $(Mu,​u)=\lambda$ for **all**
 unit vectors in $S$. In particular, we see that unit vectors in $S$. In particular, we see that
 \[ \[
 W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad. W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad.
 \] \]
numerical-range/generalizations/higher-order-numerical-range.txt · Last modified: 2013/05/08 12:56 by lpawela