1 Introduction
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 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 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 where \([X,Y] = XY-YX\). Furthermore the Wigner-Yanase-Dyson skew information \(I_{\rho,\alpha}(A)\) is defined by 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.
$$E_{\rho}(A) := \operatorname{Tr}(\rho A), $$
$$V_{\rho}(A) = \operatorname{Tr} \bigl[\rho \bigl(A-E_{\rho}(A)I \bigr)^{2} \bigr]. $$
$$I_{\rho}(A) = \frac{1}{2} \operatorname{Tr} \bigl[ \bigl(i \bigl[ \rho^{1/2},A \bigr] \bigr)^{2} \bigr] = \operatorname{Tr} \bigl[ \rho A^{2} \bigr]-\operatorname{Tr} \bigl[\rho ^{1/2}A \rho^{1/2}A \bigr], $$
$$I_{\rho,\alpha}(A) = \frac{1}{2}\operatorname{Tr} \bigl[ \bigl(i \bigl[ \rho^{\alpha},A \bigr] \bigr) \bigl(i \bigl[\rho^{1-\alpha },A \bigr] \bigr) \bigr] = \operatorname{Tr} \bigl[\rho A^{2} \bigr]- \operatorname{Tr} \bigl[ \rho^{\alpha}A\rho^{1-\alpha}A \bigr] \quad\bigl( \alpha\in[0,1] \bigr). $$
2 Dou-Du’s non-hermitian uncertainty relations
Definition 1
For \(A,B \in M_{n}(\mathbb{C})\), \(\rho\in M_{n,+,1}(\mathbb{C})\), we define the following.
(1)
\([A,B]^{0} = \frac{1}{2}([A,B]+[A^{*},B^{*}])\), where \([A,B] = AB-BA\).
(2)
\(\{A,B \}^{0} = \frac{1}{2}(\{A,B \}+\{A^{*},B^{*} \})\), where \(\{ A,B \} = AB+BA\).
(3)
\(|\operatorname{Var}_{\rho}|(A) = \operatorname{Tr}[\rho A_{0}^{*}A_{0}]\), where \(A_{0} = A-\operatorname{Tr}[\rho A]I\).
(4)
\(|\operatorname{Var}_{\rho}^{0}|(A) = \frac{1}{2}(|\operatorname{Var}_{\rho}|(A)+|\operatorname{Var}_{\rho}|(A^{*}))\).
Anzeige
Theorem 1
(Dou-Du [2])
For
\(A,B \in M_{n}(\mathbb{C})\), \(\rho\in M_{n,+,1}(\mathbb{C})\), we have the following.
(1)
\(|\operatorname{Var}_{\rho}^{0}|(A) |\operatorname{Var}_{\rho}^{0}|(B) \geq\frac {1}{4}|\operatorname{Tr}[\rho[A,B]]|^{2}\).
(2)
\(|\operatorname{Var}_{\rho}^{0}|(A) |\operatorname{Var}_{\rho}^{0}|(B) \geq\frac {1}{4}|\operatorname{Tr}[\rho\{A_{0},B_{0} \}]|^{2}\).
(3)
\(|\operatorname{Var}_{\rho}^{0}|(A) |\operatorname{Var}_{\rho}^{0}|(B) \geq\frac {1}{4}|\operatorname{Tr}[\rho[A,B]^{0}]|^{2}+\frac{1}{4}| \operatorname{Tr}[\rho\{ A_{0},B_{0} \}^{0} ]|^{2}\).
(4)
\(|U_{\rho}|(A) |U_{\rho}|(B) \geq\frac{1}{4}|\operatorname{Tr}[\rho [A,B]^{0}]|^{2}\), where
$$\begin{aligned}& |U_{\rho}|(A) =\sqrt{ \bigl(\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A) \bigr)^{2} - \bigl(\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A)-|I_{\rho}|(A) \bigr)^{2}},\\& |I_{\rho}|(A) = \frac{1}{2}\operatorname{Tr} \bigl[ \bigl(i \bigl[ \rho^{1/2},A^{*} \bigr] \bigr) \bigl(i \bigl[\rho^{1/2},A \bigr] \bigr) \bigr]. \end{aligned}$$
3 Quantum Fisher information
A function \(f:(0,+\infty) \rightarrow\mathbb{R}\) is said operator monotone if, for any \(n \in\mathbb{N}\), and \(A, B \in M_{n,+}(\mathbb{C})\) such that \(0 \leq A \leq B\), the inequalities \(0 \leq f(A) \leq f(B)\) hold. An operator monotone function is said symmetric if \(f(x) = xf(x^{-1})\) and normalized if \(f(1) = 1\). The following definitions were given by Hansen and Gibilisco, etc.
Definition 2
\(\mathcal{F}_{\mathrm{op}}\) is the class of functions \(f:(0,+\infty) \rightarrow (0,+\infty)\) satisfying
(1)
\(f(1) = 1\),
(2)
\(tf(t^{-1}) = f(t)\),
(3)
f is operator monotone.
Anzeige
For \(f \in\mathcal{F}_{\mathrm{op}}\) define \(f(0) = \lim_{x \rightarrow 0}f(x)\). We introduce the sets of regular and non-regular functions and notice that trivially \(\mathcal{F}_{\mathrm{op}} = \mathcal{F}_{\mathrm{op}}^{r} \cup \mathcal{F}_{\mathrm{op}}^{n}\).
$$\mathcal{F}_{\mathrm{op}}^{r} = \bigl\{ f \in\mathcal{F}_{\mathrm{op}} | f(0) \neq0 \bigr\} , \qquad\mathcal{F}_{\mathrm{op}}^{n} = \bigl\{ f \in \mathcal{F}_{\mathrm{op}} | f(0) = 0 \bigr\} $$
Definition 3
For \(f \in\mathcal{F}_{\mathrm{op}}^{r}\) we define
$$\tilde{f}(x) = \frac{1}{2} \biggl[ (x+1)-(x-1)^{2} \frac{f(0)}{f(x)} \biggr],\quad x > 0. $$
Proposition 1
([11])
The correspondence
\(f \rightarrow\tilde{f}\)
is a bijection between
\(\mathcal{F}_{\mathrm{op}}^{r}\)
and
\(\mathcal{F}_{\mathrm{op}}^{n}\).
Example 1
$$\begin{aligned}& f_{RLD}(x) = \frac{2x}{x+1},\qquad f_{BKN}(x) = \frac{x-1}{\log x}, \\& f_{SLD}(x) = \frac{x+1}{2}, \qquad\tilde{f}_{SLD}(x) = \frac{2x}{x+1}, \\& f_{WY}(x) = \biggl( \frac{\sqrt{x}+1}{2} \biggr)^{2}, \qquad\tilde {f}_{WY}(x) = \sqrt{x}, \\& f_{WYD}(x) = \alpha(1-\alpha)\frac{(x-1)^{2}}{(x^{\alpha}-1)(x^{1-\alpha }-1)}, \quad\alpha\in(0,1), \\& \tilde{f}_{WYD}(x) = \frac{1}{2} \bigl\{ x+1- \bigl(x^{\alpha}-1 \bigr) \bigl(x^{1-\alpha}-1 \bigr) \bigr\} , \\& \frac{2x}{x+1} < \sqrt{x} < \frac{x-1}{\log x} < f_{WYD} < \biggl( \frac {\sqrt{x}+1}{2} \biggr)^{2} < \frac{x+1}{2} \quad(x \neq1). \end{aligned}$$
In the Kubo-Ando theory of matrix means one associates a mean to each operator monotone function \(f \in\mathcal{F}_{\mathrm{op}}\) by the formula where \(A, B \in M_{n,+}(\mathbb{C})\).
$$m_{f}(A,B) = A^{1/2}f \bigl(A^{-1/2}BA^{-1/2} \bigr)A^{1/2}, $$
By using the notion of matrix means one may define the class of monotone metrics (also called quantum Fisher information) by the following formula: where \(L_{\rho}(A) = \rho A\), \(R_{\rho}(A) = A \rho\).
$$\langle A,B \rangle_{\rho,f} = \operatorname{Tr} \bigl[A \cdot m_{f}(L_{\rho},R_{\rho})^{-1}(B) \bigr], $$
4 Generalized quasi-metric adjusted skew information and correlation measure
Definition 4
Let \(g, f \in\mathcal{F}_{\mathrm{op}}^{r}\) satisfy for some \(k > 0\). We define
$$g(x) \geq k \frac{(x-1)^{2}}{f(x)} $$
$$ \Delta_{g}^{f}(x) = g(x) - k\frac{(x-1)^{2}}{f(x)} \in \mathcal{F}_{\mathrm{op}}. $$
(1)
By Lemma 5.2 in [12], \({-k \frac {(x-1)^{2}}{f(x)}}\) is operator concave. Then \(\Delta_{g}^{f}(x)\) is also operator concave. Since \(\Delta_{g}^{f}(x) > 0\) for \((0,\infty)\), \(\Delta_{g}^{f}(x)\) is operator monotone.
Definition 5
For \(A, B \in M_{n}(\mathbb{C})\) and \(\rho\in M_{n,+,1}(\mathbb{C})\), we define the following quantities: The quantity \(|I_{\rho}^{(g,f)}|(A)\) and \(|\operatorname{Corr}_{\rho}^{(g,f)}|(A,B)\) are called generalized quasi-metric adjusted skew information and generalized quasi-metric adjusted correlation measures, respectively.
$$\begin{aligned}& \bigl|\operatorname{Corr}_{\rho}^{(g,f)}\bigr|(A,B) = k \bigl\langle i[\rho,A],i[\rho,B] \bigr\rangle _{\rho,f},\\& \bigl|I_{\rho}^{(g,f)}\bigr|(A) = \bigl|\operatorname{Corr}_{\rho}^{(g,f)}\bigr|(A,A),\\& \bigl|C_{\rho}^{f}\bigr|(A,B) = \operatorname{Tr} \bigl[A^{*} m_{f}(L_{\rho},R_{\rho})B \bigr],\qquad \bigl|C_{\rho }^{f}\bigr|(A) = \bigl|C_{\rho}^{f}\bigr|(A,A),\\& \bigl|U_{\rho}^{(g,f)}\bigr|(A) = \sqrt{ \bigl(\bigl|C_{\rho}^{g}\bigr|(A)+\bigl|C_{\rho}^{\Delta _{g}^{f}}\bigr|(A) \bigr) \bigl(\bigl|C_{\rho}^{g}\bigr|(A)-\bigl|C_{\rho}^{\Delta_{g}^{f}}\bigr|(A) \bigr)}. \end{aligned}$$
Then we have the following proposition.
Proposition 2
For
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\), we have the following relations:
(1)
\(|I_{\rho}^{(g,f)}|(A) = |I_{\rho}^{(g,f)}|(A_{0}) = |C_{\rho }^{g}|(A_{0})-|C_{\rho}^{\Delta_{g}^{f}}|(A_{0})\),
(2)
\(|J_{\rho}^{(g,f)}|(A) = |C_{\rho}^{g}|(A_{0})+|C_{\rho}^{\Delta _{g}^{f}}|(A_{0})\),
(3)
\(|U_{\rho}^{(g,f)}|(A) = \sqrt{|I_{\rho}^{(g,f)}|(A) \cdot |J_{\rho}^{(g,f)}|(A)}\),
(4)
\(|\operatorname{Corr}_{\rho}^{(g,f)}|(A,B) = |\operatorname{Corr}_{\rho}^{(g,f)}|(A_{0},B_{0})\), where
\(A_{0} = A-\operatorname{Tr}[\rho A]I\)
and
\(B_{0} = B-\operatorname{Tr}[\rho B]I\).
Theorem 2
(Schrödinger type)
For
\(f \in\mathcal{F}_{\mathrm{op}}^{r}\), we have
where
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\).
$$\bigl|I_{\rho}^{(g,f)}\bigr|(A) \cdot\bigl|I_{\rho}^{(g,f)}\bigr|(B) \geq\bigl|\bigl|\operatorname{Corr}_{\rho }^{(g,f)}\bigr|(A,B)\bigr|^{2}, $$
Proof of Theorem 1
By Schwarz’s inequality we have Then we have □
$$\langle A,A \rangle_{\rho,f} \langle B,B \rangle_{\rho,f} \geq\bigl| \langle A,B \rangle_{\rho,f} \bigr|^{2}. $$
$$\begin{aligned} & \bigl|I_{\rho}^{(g,f)}\bigr|(A) \cdot\bigl|I_{\rho}^{(g,f)}\bigr|(B) \\ &\quad = \bigl|\operatorname{Corr}_{\rho}^{(g,f)}\bigr|(A,A) \cdot\bigl|\operatorname{Corr}_{\rho}^{(g,f)}\bigr|(B,B) \\ &\quad \geq \bigl|\bigl|\operatorname{Corr}_{\rho}^{(g,f)}\bigr|(A,B)\bigr|^{2}. \end{aligned}$$
Theorem 3
(Heisenberg type)
For
\(f \in\mathcal{F}_{\mathrm{op}}^{r}\), if
for some
\(\ell> 0\), then we have
where
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\).
$$ g(x)+\Delta_{g}^{f}(x) \geq\ell f(x) $$
(2)
$$ \bigl|U_{\rho}^{(g,f)}\bigr|(A) \cdot\bigl|U_{\rho}^{(g,f)}\bigr|(B) \geq k \ell\bigl|\operatorname{Tr} \bigl(\rho [A,B] \bigr)\bigr|^{2}, $$
(3)
We need the following lemma in order to prove Theorem 3.
Lemma 1
Proof
By (1) and (2),
Then by (4) and (5), we have □
$$\begin{aligned}& m_{\Delta_{g}^{f}}(x,y) = m_{g}(x,y)-k\frac{(x-y)^{2}}{m_{f}(x,y)}, \end{aligned}$$
(4)
$$\begin{aligned}& m_{g}(x,y)+m_{\Delta_{g}^{f}}(x,y) \geq\ell m_{f}(x,y). \end{aligned}$$
(5)
$$\begin{aligned} & m_{g}(x,y)^{2}-m_{\Delta_{g}^{f}}(x,y)^{2} \\ &\quad = \bigl\{ m_{g}(x,y)-m_{\Delta_{g}^{f}}(x,y) \bigr\} \bigl\{ m_{g}(x,y)+m_{\Delta_{g}^{f}}(x,y) \bigr\} \\ & \quad\geq \frac{k(x-y)^{2}}{m_{f}(x,y)}\ell m_{f}(x,y) \\ &\quad = k\ell(x-y)^{2}. \end{aligned}$$
Proof of Theorem 2
Let \(\{ |{\phi}_{1} \rangle,|{\phi}_{2} \rangle,\ldots,|{\phi}_{n} \rangle \}\) be a basis of eigenvectors of ρ, corresponding to the eigenvalues \(\{ \lambda_{1} ,\lambda_{2} ,\ldots, \lambda_{n} \}\). We put \(a_{jk} = \langle{\phi}_{j} |A_{0}|{\phi}_{k} \rangle\), \(b_{jk} = \langle{\phi}_{j} |B_{0}|{\phi}_{k} \rangle\). Then we have and We remark that and By Lemma 1 we have Similarly we have Therefore □
$$\begin{aligned} & \quad \bigl|I_{\rho}^{(g,f)}\bigr|(A) \\ &\quad = \bigl|C_{\rho}^{g}\bigr|(A_{0}) -\bigl|C_{\rho}^{{\Delta}_{g}^{f}}\bigr|(A_{0}) \\ &\quad = \operatorname{Tr} \bigl[A_{0}^{*} m_{f}(L_{\rho},R_{\rho})A_{0} \bigr] -\operatorname{Tr} \bigl[A_{0}^{*} m_{{\Delta }_{g}^{f}}(L_{\rho},R_{\rho})A_{0} \bigr] \\ &\quad = \sum_{i,j} \bigl\{ m_{f}( \lambda_{i}, \lambda_{j}) -m_{{\Delta }_{g}^{f}}( \lambda_{i}, \lambda_{j}) \bigr\} |a_{ij}|^{2} \end{aligned}$$
$$\bigl|J_{\rho}^{(g,f)}\bigr|(A) = \sum_{i,j} \bigl\{ m_{f}(\lambda_{i}, \lambda_{j}) +m_{{\Delta}_{g}^{f}}(\lambda_{i}, \lambda_{j}) \bigr\} |a_{ij}|^{2}. $$
$$\operatorname{Tr} \bigl(\rho[A,B] \bigr) = \sum_{i} \sum_{j} (\lambda_{i} - \lambda_{j}){a_{ij}} {b_{ji}} $$
$$\bigl|\operatorname{Tr} \bigl(\rho[A,B] \bigr)\bigr| \leq\sum _{i} \sum_{j} | \lambda_{i} - \lambda_{j}| |a_{ij}||b_{ji}|. $$
$$\begin{aligned} & k \ell\bigl|\operatorname{Tr} \bigl[\rho[A,B] \bigr]\bigr|^{2} \\ & \quad\leq \biggl\{ \sum_{i} \sum _{j} \sqrt{kl}|\lambda_{i}-\lambda _{j}||a_{ij}||b_{ji}| \biggr\} ^{2} \\ &\quad \leq \biggl\{ \sum_{i} \sum _{j} \bigl( m_{g}(\lambda_{i}, \lambda _{j})^{2}-m_{\Delta_{g}^{f}}(\lambda_{i}, \lambda_{j})^{2} \bigr)^{1/2} |a_{ij}||b_{ji}| \biggr\} ^{2} \\ & \quad\leq \biggl\{ \sum_{i} \sum _{j} \bigl( m_{g}(\lambda_{i}, \lambda _{j})-m_{\Delta_{g}^{f}}(\lambda_{i}, \lambda_{j}) \bigr)|a_{ij}|^{2} \biggr\} \biggl\{ \sum_{i} \sum _{j} \bigl( m_{g}(\lambda_{i}, \lambda_{j})+m_{\Delta _{g}^{f}}(\lambda_{i}, \lambda_{j}) \bigr)|b_{ji}|^{2} \biggr\} \\ &\quad = I_{\rho}^{(g,f)}(A) J_{\rho}^{(g,f)}(B). \end{aligned}$$
$$k \ell\bigl|\operatorname{Tr} \bigl[\rho[A,B] \bigr]\bigr|^{2} \leq I_{\rho}^{(g,f)}(B)J_{\rho}^{(g,f)}(A). $$
$$\bigl|U_{\rho}^{(g,f)}\bigr|(A) \cdot\bigl|U_{\rho}^{(g,f)}\bigr|(B) \geq k \ell\bigl|\operatorname{Tr} \bigl(\rho [A,B] \bigr)\bigr|^{2}. $$
Example 2
When we can show positivity: and Then In particular, for \(\alpha= 1/2\),
$$\begin{aligned}& g(x) = \frac{x+1}{2},\qquad f(x) = \alpha(1-\alpha)\frac {(x-1)^{2}}{(x^{\alpha}-1)(x^{1-\alpha}-1)}\quad (0 < \alpha< 1),\\& k = \frac{f(0)}{2},\qquad \ell= 2, \end{aligned}$$
$$\Delta_{g}^{f}(x) = g(x)-k\frac{(x-1)^{2}}{f(x)} = \frac{1}{2} \bigl(x^{\alpha }+x^{1-\alpha} \bigr) \geq0 $$
$$\begin{aligned} g(x)+\Delta_{g}^{f}(x)-\ell f(x) & = \frac{1}{2(x^{\alpha}-1)(x^{1-\alpha}-1)} \bigl\{ \bigl(x^{2\alpha }-1 \bigr) \bigl(x^{2(1-\alpha)}-1 \bigr)-4 \alpha(1-\alpha) (x-1)^{2} \bigr\} \\ & \geq 0. \end{aligned}$$
$$\begin{aligned} \bigl|I_{\rho}^{(f,g)}\bigr|(A) =& \bigl|I_{\rho}^{(f,g)}\bigr|(A_{0}) = \frac {1}{2}\operatorname{Tr} \bigl[\rho A_{0}A_{0}^{*} \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[\rho A_{0}^{*}A_{0} \bigr]\\ &{} -\frac{1}{2}\operatorname{Tr} \bigl[\rho^{\alpha}A_{0} \rho^{1-\alpha}A_{0}^{*} \bigr]-\frac {1}{2}\operatorname{Tr} \bigl[\rho^{1-\alpha}A_{0}^{*}\rho^{\alpha}A_{0} \bigr]. \end{aligned}$$
$$\begin{aligned} \bigl|I_{\rho}^{(f,g)}\bigr|(A) =& \bigl|I_{\rho}^{(f,g)}\bigr|(A_{0}) = \frac{1}{2}\operatorname{Tr} \bigl[\rho A_{0} A_{0}^{*} \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[\rho A_{0}^{*} A_{0} \bigr] -\operatorname{Tr} \bigl[ \rho^{1/2}A_{0}\rho^{1/2}A_{0}^{*} \bigr]. \end{aligned}$$
In this case we can give some results by Dou-Du as corollaries.
Corollary 1
(Dou-Du (4))
For
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\),
$$\begin{aligned} & |U_{\rho}|(A) \cdot|U_{\rho}|(B) \\ & \quad\geq \frac{1}{4}\bigl|\operatorname{Tr} \bigl[\rho[A,B] \bigr]\bigr|^{2} \geq\frac{1}{4}\bigl| \operatorname{Im} \operatorname{Tr} \bigl[\rho [A,B] \bigr]\bigr|^{2} \\ &\quad = \frac{1}{4} \biggl\vert \frac{1}{2}\operatorname{Tr} \bigl[ \rho[A,B] \bigr]-\frac{1}{2}\overline {\operatorname{Tr} \bigl[\rho[A,B] \bigr]} \biggr\vert ^{2} \\ &\quad = \frac{1}{4} \biggl\vert \frac{1}{2}\operatorname{Tr} \bigl[ \rho[A,B] \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[\rho \bigl[A^{*},B^{*} \bigr] \bigr] \biggr\vert ^{2} \\ &\quad = \frac{1}{4} \biggl\vert \operatorname{Tr} \biggl[\rho \frac{[A,B]+[A^{*},B^{*}]}{2} \biggr] \biggr\vert ^{2} \\ &\quad = \frac{1}{4}\bigl|\operatorname{Tr} \bigl[\rho[A,B]^{0} \bigr]\bigr|^{2}. \end{aligned}$$
Corollary 2
(Dou-Du (1), (2))
For
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\),
(1)
\(|V_{\rho}|(A) \cdot|V_{\rho}|(B) \geq |U_{\rho}|(A) \cdot|U_{\rho}|(B) \geq\frac{1}{4}|\operatorname{Tr}[\rho[A,B]]|^{2}\).
(2)
\(|V_{\rho}|(A) \cdot|V_{\rho}|(B) \geq\frac {1}{4}|\operatorname{Tr}[\rho\{ A_{0},B_{0} \}]|^{2}\).
Proof
(2)
For \(A, B \in M_{n}(\mathbb{C})\) and \(f(x) = \frac{x+1}{2}\), we define an inner product on \(M_{n}(\mathbb{C})\) by
$$\langle A,B \rangle= \operatorname{Tr} \bigl[A_{0}^{*} m_{f}(L_{\rho},R_{\rho})B_{0} \bigr]. $$
$$\bigl| \langle A,B \rangle\bigr|^{2} \leq\langle A,A \rangle \langle B,B \rangle. $$
$$\begin{aligned} \bigl|\langle A,B \rangle\bigr|^{2} = & \biggl\vert \operatorname{Tr} \biggl[A_{0}^{*} \frac{L_{\rho }+R_{\rho}}{2}B_{0} \biggr] \biggr\vert ^{2} \\ = & \biggl\vert \frac{1}{2}\operatorname{Tr} \bigl[A_{0}^{*} \rho B_{0} \bigr]+ \frac{1}{2}\operatorname{Tr} \bigl[A_{0}^{*}B_{0}\rho \bigr] \biggr\vert ^{2} \\ = & \biggl\vert \frac{1}{2}\operatorname{Tr} \bigl[\rho B_{0}A_{0}^{*} \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[ \rho A_{0}^{*}B_{0} \bigr] \biggr\vert ^{2} \\ = & \frac{1}{4} \bigl|\operatorname{Tr} \bigl[\rho \bigl\{ A_{0}^{*},B_{0} \bigr\} \bigr]\bigr|^{2} \end{aligned}$$
$$\begin{aligned} \langle A,A \rangle = & \operatorname{Tr} \biggl[A_{0}^{*} \frac{L_{\rho}+R_{\rho}}{2}A_{0} \biggr] \\ = & \frac{1}{2}\operatorname{Tr} \bigl[A_{0}^{*}\rho A_{0} \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[A_{0}^{*} A_{0} \rho \bigr] \\ = & \frac{1}{2}\operatorname{Tr} \bigl[\rho A_{0}A_{0}^{*} \bigr]+\frac{1}{2}\operatorname{Tr} \bigl[\rho A_{0}^{*}A_{0} \bigr] \\ = & \frac{1}{2}\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A). \end{aligned}$$
$$\bigl|\operatorname{Tr} \bigl[\rho \bigl\{ A_{0}^{*},B_{0} \bigr\} \bigr]\bigr|^{2} \leq\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A) \cdot\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(B). $$
$$\bigl|\operatorname{Tr} \bigl[\rho\{A_{0},B_{0} \} \bigr]\bigr|^{2} \leq\bigl|\operatorname{Var}_{\rho}^{0}\bigr| \bigl(A^{*} \bigr) \cdot\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(B). $$
5 Remark
Dou-Du’s result (3)
For \(A, B \in M_{n}(\mathbb{C})\) and \(\rho\in M_{n,+,1}(\mathbb{C})\),
$$ \bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A) \cdot\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(B) \geq\frac{1}{4}\bigl|\operatorname{Tr} \bigl[\rho [A,B]^{0} \bigr]\bigr|^{2}+\frac{1}{4}\bigl|\operatorname{Tr} \bigl[\rho\{ A_{0},B_{0} \}^{0} \bigr]\bigr|^{2}. $$
(6)
Corollary 3
For
\(A, B \in M_{n}(\mathbb{C})\)
and
\(\rho\in M_{n,+,1}(\mathbb{C})\),
We cannot compare the RHS of (6) with the RHS of (7). In fact, let
Since the RHS of
\((\mbox{6}) = 0\)
and the RHS of
\((\mbox{7}) = \frac{9}{16}\), we have the RHS of (6) < the RHS of (7). On the other hand let
Since the RHS of
\((\mbox{6}) = 1\)
and the RHS of
\((\mbox{7}) = 0\), we have the RHS of (6) > the RHS of (7).
$$ \bigl|\operatorname{Var}_{\rho}^{0}\bigr|(A) \cdot\bigl|\operatorname{Var}_{\rho}^{0}\bigr|(B) \geq|U_{\rho}|(A) \cdot |U_{\rho}|(B) \geq\frac{1}{4}| \operatorname{Tr} \bigl[\rho[A,B] \bigr]|^{2}. $$
(7)
$$\rho= \begin{pmatrix} \frac{3}{4} & 0 \\ 0 & \frac{1}{4} \end{pmatrix}, \qquad A = \begin{pmatrix} i & 1 \\ -1 & i \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}. $$
$$\rho= \begin{pmatrix} \frac{3}{4} & 0 \\ 0 & \frac{1}{4} \end{pmatrix},\qquad A = \begin{pmatrix} i & 1 \\ -1 & i \end{pmatrix},\qquad B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. $$
6 Conclusion
The results (1), (4) of Dou-Du are given as corollaries of Theorem 3. Result (2) is proved by Schwarz’s inequality. Also is shown, that (3) cannot be compared with our result.
Acknowledgements
The first author (KY) was partially supported by JSPS KAKENHI Grant Number 26400119.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.