# (p,k)-numerical range

## Definition

The $(p,k)$ numerical range is useful term (similarly to higher rank numerical range) to consideration some problems in quantum information theory (See for details Application of $(p,k)$ numerical range). The other useful application the $(p,k)$ numerical range we can see in [1]. In order to introduce the definition of $(p,k)$ numerical range, let $M_n$ will be the set of all matrices of dimension $n$ and by $\mathrm{U}\left(\mathbb{C}^{k},\mathbb{C}^{l}\right)$ it will be denoted the set of all isometries of dimension $k \times l$. The $(p,k)$ numerical range of the matrix $A \in M_n$ is defined as [2]

$$\Lambda_{p,k}(A) \coloneqq \left\{ B \in M_p: U^\dagger A U = B \otimes \1_k \text{ for some } U \in \mathrm{U}(\mathbb{C}^{pk}, \mathbb{C}^n) \right\}.$$

## Properties

We have the following properties of $\Lambda_{p,k}(A)$ for $A \in M_n$:

• $\Lambda_{p,k}(\alpha A + \beta \1_n)= \alpha \Lambda_{p,k} (A) +\beta \1_p \text{ for }\alpha\beta \in\mathbb{C}$.

• $\Lambda_{p,k}(U^\dagger A U) \subset \Lambda_{p,k}(A)$ for any isometry $U (\mathbb{C}^m, \mathbb{C}^n )$.

• $\Lambda_{p,k}(U^\dagger A U) = \Lambda_{p,k}(A)$ for any unitary matrix $U \in \mathrm{U}\left( \mathbb{C}^n \right)$.

• $B \in \Lambda_{p,k}(A)$ if and only if $U^\dagger B U \in \Lambda_{p,k}(A)$ for any $U \in \mathrm{U}( \mathbb{C}^p )$.

• If $B \in \Lambda_{p,k}(A)$, then $\Lambda_{n-pk+1}(A) \subset \Lambda_1(B) \subset \Lambda_k(A)$.

We can observe that the $(p,k)$ numerical range of some matrux $A$ is the generalization of higher rank numerical range and $p$-th matricial range. To be precise, $(1,k)$ numerical range is higher rank-k numerical range $\Lambda_k(A)$ and $(p,1)$ numerical range is $p$-th matricial numerical range $W(p:A)$. In the case when $p=k=1$ we obtain the standard numerical range.

In general case the study of $(p,k)$ numerical range properties is hard task for any matrices. One of the recent work shows the conditions when $\Lambda_{p,k}$ is non-empty set. Hence the following theorem tells us some properties of $(p,k)$- numerical range for Hermitian matrices.

## Theorem

The $(p,k)$ numerical range of given matrix $A \in M_n$ is non-empty set when:

• $n \geq 2(p+1)k-3$.

• There exists $U \in \mathrm{U}\left(\mathbb{C}^n\right)$ such that $U^\dagger A U=A_1 \oplus \ldots \oplus A_p$, where $\Lambda_k(A_j) \not= \emptyset$ for all $j=1,\ldots,p$.

• There exists $U \in \mathrm{U}\left(\mathbb{C}^n\right)$ such that $U^\dagger A U=A_1 \oplus \ldots \oplus A_p$, where $A_j \ge 3k-2$ for all $j=1,,p$.

• Matrix $A$ is normal and $n \geq (3k-2)p$.

## Theorem for Hermitian matrices

Let $A$ will be Hermitian matrix of dimenasion $n$ and let $pk \le n$. The set $\Lambda_{p,k}(A) \emptyset$ if and only if $\lambda_{jk}(A) \geq \lambda_{n-(p-j+1)k+1}(A) \quad \text{for } j=1,\ldots,p.$

Furthermore, for a given matrix $B \in M_n$ we can obtain $B \in \Lambda_{p,k}(A)$ if and only if:

$\lambda_{n-(p-j+1)k+1}(A) \le \lambda_j(B) \le \lambda_{jk}(A)$

for $j=1, \ldots,p$, where $\lambda_k(X)$ is $k$-th eigenvalue of $X$.

### Convexity of $(p,k)$ numerical range for Hermitian matrices

In the case when $n \geq (p+1)k$ the convexity of the set $\Lambda_{p,k}(A)$ is equivalent to $\lambda_k(A)=\lambda_{pk}(A) \, \text{ and } \, \lambda_{n-pk+1}(A)=\lambda_{n-k+1}(A).$

Proofs of above theorem we can find in [3].

1. Determine all possible $k \times k$ principal submatrices of $U^\dagger A U$ for $U ( \mathbb{C}^n )$.

2. Determine the optimal $n$ so that $\Lambda_{p,k}(A)$ is non-empty for any $A \in M_n$.

3. Find an example of normal matrix $A \in M_n$ that $\Lambda_{p,k}(A)$ is convex.

# References

1. [1]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.
2. [2]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.
3. [3]C.-K. Li, Y.-T. Poon, and N.-S. Sze, “Generalized interlacing inequalities,” Linear and Multilinear Algebra, vol. 60, no. 11-12, pp. 1245–1254, 2012, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081087.2011.619534?journalCode=glma20.