Let
\(M_{n}(\mathbb{C})\) (resp.
\(M_{n,sa}(\mathbb{C})\)) be the set of all
\(n \times n\) complex matrices (resp. all
\(n \times n\) self-adjoint matrices), endowed with the Hilbert-Schmidt scalar product
\(\langle A,B \rangle= \operatorname{Tr}[A^{*}B]\). Let
\(M_{n,+}(\mathbb {C})\) be the set of strictly positive elements of
\(M_{n}(\mathbb{C})\) and
\(M_{n,+,1}(\mathbb {C}) \subset M_{n,+}(\mathbb{C})\) be the set of strictly positive density matrices, that is,
\(M_{n,+,1}(\mathbb{C}) = \{ \rho\in M_{n}(\mathbb{C}) | \operatorname{Tr}[\rho] = 1, \rho> 0 \}\). If it is not otherwise specified, from now on we shall treat the case of faithful states, that is,
\(\rho> 0\). It is well known that the expectation of an observable
\(A \in M_{n,sa}(\mathbb{C})\) in a state
\(\rho\in M_{n,+,1}(\mathbb{C})\) is defined by
$$E_{\rho}(A) := \operatorname{Tr}(\rho A), $$
and the variance of an observable
\(A \in M_{n,sa}(\mathbb{C})\) in a state
\(\rho\in M_{n,+,1}(\mathbb{C})\) is defined by
$$V_{\rho}(A) = \operatorname{Tr} \bigl[\rho \bigl(A-E_{\rho}(A)I \bigr)^{2} \bigr]. $$
In order to represent the degree of non-commutativity between
\(\rho\in M_{n,+,1}(\mathbb{C})\) and
\(A \in M_{n,sa}(\mathbb{C})\), the Wigner-Yanase skew information
\(I_{\rho}(A)\) is defined by
$$I_{\rho}(A) = \frac{1}{2} \operatorname{Tr} \bigl[ \bigl(i \bigl[ \rho^{1/2},A \bigr] \bigr)^{2} \bigr] = \operatorname{Tr} \bigl[ \rho A^{2} \bigr]-\operatorname{Tr} \bigl[\rho ^{1/2}A \rho^{1/2}A \bigr], $$
where
\([X,Y] = XY-YX\). Furthermore the Wigner-Yanase-Dyson skew information
\(I_{\rho,\alpha}(A)\) is defined by
$$I_{\rho,\alpha}(A) = \frac{1}{2}\operatorname{Tr} \bigl[ \bigl(i \bigl[ \rho^{\alpha},A \bigr] \bigr) \bigl(i \bigl[\rho^{1-\alpha },A \bigr] \bigr) \bigr] = \operatorname{Tr} \bigl[\rho A^{2} \bigr]- \operatorname{Tr} \bigl[ \rho^{\alpha}A\rho^{1-\alpha}A \bigr] \quad\bigl( \alpha\in[0,1] \bigr). $$
The convexity of
\(I_{\rho,\alpha}(A)\) with respect to
ρ was famously shown by Lieb [
4]. The relationship between Wigner-Yanase skew information and the uncertainty relation was given by Luo and Zhang [
5] for the first time. Afterward, the relationship between Wigner-Yanase-Dyson skew information and the uncertainty relation was given by Kosaki [
6] and Yanagi
et al. [
7]. Furthermore metric adjusted skew information was defined by Hansen [
8] which is an extension of Wigner-Yanase-Dyson skew information. The relationship between metric adjusted skew information and the uncertainty relation was given by Yanagi [
9] and was generalized in Yanagi
et al. [
10] for generalized metric adjusted skew information and generalized metric adjusted correlation measures. In this paper we give some non-hermitian extensions of Heisenberg type and Schrödinger type uncertainty relations related to generalized quasi-metric adjusted skew information and generalized quasi-metric adjusted correlation measures. As a result we can obtain some results of non-hermitian uncertainty relations given by Dou and Du as corollaries of our results.