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numerical-range:generalizations:restricted-numerical-range:product-numerical-range

# Product numerical range

## Definition

Product numerical range $W^\otimes(A)$ of a square matrix $A$ of size $N = N_1 N_2$ is a subset of the complex plane defined as

$$\begin{split} W^\otimes(A)=\left\{z \in \mathbb{C}:\ z= \left( \bra{\psi} \otimes \bra{\phi} \right) A \left( \ket{\psi} \otimes \ket{\phi} \right), \ket{\psi}\in\mathbb{C}^{N_1}, \ket{\phi}\in\mathbb{C}^{N_2}\\ \braket{\psi}{\psi}=1, \braket{\phi}{\phi}=1 \right\}. \end{split}$$

In general, this definition can be written as: product numerical range $W^\otimes(A)$ of a square matrix $A$ of size $N = \prod_{i=1}^K N_i$ is a subset of the complex plane defined as

$$\begin{split} W^\otimes(A)=\left\{z \in \mathbb{C}:\ z= \left( \bigotimes_{i=1}^K \bra{\psi_i} \right) A \left( \bigotimes_{i=1}^K \ket{\psi_i} \right), \ket{\psi_i}\in\mathbb{C}^{N_i}, \\ \braket{\psi_i}{\psi_i}=1 \text{ for }\ i=1,\ldots,K \right\}. \end{split}$$

## Examples

#### Example 1

Consider a family of operators with a tensor product structure

$$Y(r_1,r_2) = X_1 \otimes X_2 = \left( \begin{array}{cc} 1 & \ 2 r_1\\ 0 & 1 \end{array} \right) \otimes \left( \begin{array}{cc} 1 & \ 2r_2\\ 0 & 1 \end{array} \right).$$

Example numerical ranges dependent on $r_1$ and $r_2$ are shown below.

$(r_1, r_2)$ equal to $(1,1)$ (cardioid)

$(r_1, r_2)$ equal to $(0.7,1)$ (limacon of Pascal)

$(r_1, r_2)$ equal to $(0.5,1.2)$ (Cartesian oval)

#### Example 2

$(r_1, r_2)$ equal to $(0.5,1.2)$ (Cartesian oval)

Figure: Product numerical range [1] with respect to four qubit space given by equation $$W^\otimes(A)=\{z:\ z= \left( \bigotimes_{i=1}^4 \langle\psi_i| \right) A \left( \bigotimes_{i=1}^4 |\psi_i\rangle \right), \text{ for } i=1,\ldots,4\ \langle\psi_i|\psi_i\rangle=1 \text{ and } |\psi_i\rangle\in\mathbb{C}^2 \}$$ of matrix $$$A ={\rm diag} \bigl( e^{\frac{i \pi }{4}},i,i,e^{\frac{3 i \pi }{4}},-1,e^{-\frac{3 i \pi }{4}}, e^{-\frac{i \pi }{4}},1,-1,e^{-\frac{i \pi }{4}},e^{-\frac{3 i \pi }{4}},1, e^{\frac{3 i \pi }{4}},-i,-i,e^{\frac{i \pi }{4}} \bigr)$$.$

## References

1. Puchała, Z., Gawron, P., Miszczak, J.A., Skowronek, Ł., Choi, M.D., Życzkowski, K., 2011. Product numerical range in a space with tensor product structure. Linear Algebra and its Applications, 434, Elsevier, pp.327–342.