# Numerical Shadow

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# 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 \begin{equation} \Phi(X) = \sum_{i} A_i X A_i \end{equation} 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 \begin{equation} PA_i^\dagger A_i P = \lambda_{i,j} P \text{ for all } 1\le i,j\le r. \end{equation} This leads to the study higher rank numerical range.

We can also naturally extend above error correction scheme . Now we consider that for a given quantum channel $\Psi$ we would like to find a recovery channel $R$ such that for each $B \in M_k$ \begin{equation} 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} \end{equation} 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 \begin{equation} 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 \end{equation} 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.

# Application in Hybrid Quantum Error Correction

The non-emptiness of $(p,k)$ numerical range plays crucial role in hybrid (classical and quantum) error correction code schemes . In this case we study $m-$joint $(p,k)-$diagonal numerical range \begin{equation} \widetilde{\Lambda}_{p,k}(A_1,\ldots,A_m) := \left\{ (D_1,\ldots,D_m) \subset D_p: \, U^\dagger A_i U = D_i \otimes \mathbf{1}_k \text{ for some } U \in \mathrm{U}\left(\mathbb{C}^{pk},\mathbb{C}^{n} \right) \right\}, \end{equation} where $A_i \in M_n$ and $D_p$ is a set of $p \times p$ diagonal matrices.

The important property of $\widetilde{\Lambda}_{p,k}$ which is studied is its non-emptiness. Here we present a sufficient condition in the case, when $A_i=A_i^\dagger$ .

#### Theorem

Assume that $k>1$. $(A_1,\ldots, A_m) \subset H_n$ and it holds that $$n\geq(m + 1)((m + 1)(k-1) +k(p-1)).$$ Then $\widetilde{\Lambda}_{p,k}(A_1,\ldots,A_m) \not= \emptyset.$

This theorem provides simply check for given quantum channel whenever there exists a hybrid error correcting code (for more details see ).

#### Theorem

Let $\Phi$ be a quantum channel acting on the space of $M_n$, which Choi rank is equal to $c$. Then $\Phi$ has a hybrid error correcting code of dimensions $(p,k)$ if $$n \geq c^2(c^2(k-1) +k(p-1)).$$

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.
2. Ningping Cao, David W Kribs, Chi-Kwong Li, Mike I Nelson, Yiu-Tung Poon, Bei Zeng, 2020. Higher rank matricial ranges and hybrid quantum error correction. Linear and Multilinear Algebra, Taylor \& Francis, pp.1–13. 