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numerical-range:generalizations:application-of-higher-rank-and-p-k-numerical-range

# Application of higher rank numerical range and $(p,k)$- numerical range

In this section we present the motivation behind introduce definitions of higher rank numerical range and $(p,k)$- numerical range. Let $M_n$ will be the set of all matrices of dimension $n$. We will consider linear mapping transforming given matrix into another matrix. Such mapping can be represented by operator sum representation (Kraus representation) as $$\Phi(X) = \sum_{i} A_i X A_i$$ for some matrices $A_i$. The special linear mapping transforming state into another state is well-known as quantum channel. One would like to consider a recovery channel $R$ such that $R \circ \Phi(X) = X$ whenever $PXP=X$ for some orthogonal projection $P$. The range space of $P$ is known as a quantum error correction code of the channel $\Phi$. The task is finding $P$ with a maximum rank. For a given quantum channel $\Phi$ this problem is equivalent to existing scalars $\lambda_{i,j} \in \mathbb{C}$ such that $$PA_i^\dagger A_i P = \lambda_{i,j} P \text{ for all } 1\le i,j\le r.$$ This leads to the study higher rank numerical range.

We can also naturally extend above error correction scheme [1]. Now we consider that for a given quantum channel $\Psi$ we would like to find a recovery channel $R$ such that for each $B \in M_k$ $$R \circ \Psi \left( \left( \1_p \otimes B \right) \oplus \mathbf{0}_{n-pk} \right) = \left( A_B \otimes B \right) \oplus \mathbf{0}_{n-pk}$$ for some $A_B \in M_p$. Analogously, his problem can be is reduced to showing that such recovery channel $R$ exists if and only if there are scalars $\lambda_{ijrs} \in \mathbb{C}$ such that $$P_{kk} A_i^\dagger A_j P_{ll} = \lambda_{ijrs} P_{kl} \text{ for all } 1\le i,j\le r \text{ and } 1\le k,l \le p$$ where $P_{kl} = ( \ket{k}\bra{l} \otimes \mathbf{1}_k ) \oplus \mathbf{0}_{n-pk}$ with fixed an arbitrary orthonormal basis $\{ e_1, \ldots, e_p \}$ in $\mathbb{C}^p$. This approach we can simplify to consideration $(p,k)$ numerical range.

1. Man-Duen Choi, Nathaniel Johnston, David W Kribs, 2009. The multiplicative domain in quantum error correction. Journal of Physics A: Mathematical and Theoretical, 42, IOP Publishing, pp.245303.