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 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 $R (X) = X$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 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 $B \in M_k$ 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 where $P_{kl} = ( *k ) *{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 [2]. In this case we study $m-$joint $(p,k)-$diagonal numerical range where $A_i \in M_n$ and $D_p$ is a set of $p p$ 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 $A_i=A_i^\dagger$ [2].
Theorem
Assume that $k>1$. $(A_1,\ldots, A_m) \subset H_n$ and it holds that
\[n\geq(m + 1)[^1].\]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 [2]).
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 $n \geq c^2(c^2(k−1) +k(p−1)).$
References
- [1]M.-D. Choi, N. Johnston, and D. W. Kribs, “The multiplicative domain in quantum error correction,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 24, p. 245303, 2009, [Online]. Available at: https://iopscience.iop.org/article/10.1088/1751-8113/42/24/245303/meta.
- [2]N. Cao, D. W. Kribs, C.-K. Li, M. I. Nelson, Y.-T. Poon, and B. Zeng, Higher rank matricial ranges and hybrid quantum error correction. Taylor & Francis, 2020, pp. 1–13.