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