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

Properties

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

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

* $\Lambda_{p,k}(U^\dagger A U) \subset \Lambda_{p,k}(A)$ for any isometry $U \in \mathrm{U}\left( \mathbb{C}^m, \mathbb{C}^n \right)$.

* $\Lambda_{p,k}(U^\dagger A U) = \Lambda_{p,k}(A)$ 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 $U \in \mathrm{U}\left( \mathbb{C}^p \right)$.

* 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 $\dim A_j \geq 3k-2$ for all $j=1,\ldots,p$.

* 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 $pk \leq n$. The set $\Lambda_{p,k}(A) \not= \emptyset$ 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: $$\lambda_{n-(p-j+1)k+1}(A) \leq \lambda_j(B) \leq \lambda_{jk}(A) \quad \text{for } j=1,\ldots, p,$$ 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 $U \in \mathrm{U}\left( \mathbb{C}^n \right)$.

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.

1. 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.
2. 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.
3. Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2012. Generalized interlacing inequalities. Linear and Multilinear Algebra, 60, Taylor and Francis, pp.1245–1254.