Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information

The quantum uncertainty can also be characterized by skew information. The Wigner-Yanase (WY) information and Wigner-Yanase-Dyson (WYD) skew information associated to a quantum state $\rho$ and an observable $A$ have been defined in [23]. The WYD skew information has been further extended to the generalized Wigner-Yanase-Dyson (GWYD) skew information [24]. The relationship between WY skew information and the uncertainty relation has been originally established by Luo and Zhang [25], and various types of uncertainty relations based on the WY skew information, WYD skew information and GWYD skew information have been presentd [26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 28].

By considering state-channel interaction, in [41] Luo and Sun defined a quantity $\mathrm{I}_{\rho}(\Phi)$ and its dual one $\mathrm{J}_{\rho}(\Phi)$, and explored the complementarity relation between them. Wu, Zhang and Fei introduced the non-Hermitian extension of the GWYD skew information and generalized the complementarity relation to a more general case in [42, 43].

On the other hand, another generalization of the WYD skew information, the weighted Wigner-Yanase-Dyson (WWYD) skew information, has been introduced in [44]. As its non-Hermitian extension, the modified weighted Wigner-Yanase-Dyson (MWWYD) skew information has been defined and investigated in [45]. Recently, by using the convex combination, instead of the arithmetic mean of $\rho^{\alpha}$ and $\rho^{1-\alpha}$, the two-parameter extension of the Wigner-Yanase skew information has been formulated [46].

Recently, the sum uncertainty relations based on the variance and WY skew information have attracted considerable attention [47, 48, 49, 50]. In [47, 48] Chen and Fei proposed some uncertainty inequalities in terms of the sum of variances, standard deviations and the WY skew information for arbitrary $N$ mutually noncommutative observables, respectively. After that, Zhang, Gao and Yan [49] established a tighter uncertainty relation via WY skew information for arbitrary $N$ mutually noncommutative observables, which extend the results in [48]. Zhang and Fei [50] further improved the results in [49] and proposed new tighter bounds than the existing ones. Cai [51] generalized the sum uncertainty relations for WY skew information introduced in [48] to an arbitrary metric-adjusted skew information version. Ren, Li, Ye and Li [52] proposed tighter sum uncertainty relations than the ones in [51].

In [53] Fu, Sun and Luo established the uncertainty relations for two quantum channels based on the WY skew information for arbitrary operators. Afterwards, Zhang, Gao and Yan [49] generalized the uncertainty relations for two quantum channels to arbitrary $N$ quantum channels. Zhang, Wu and Fei [54] further generalized the results in [49] and proposed new bounds which are tighter than the existing ones. Cai [51] confirmed that the results in [53] also hold for all metric-adjusted skew information.

The remainder of this paper is structured as follows. In Section 2, we recall some basic concepts and propose the definitions of $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson ($(\alpha,\beta,\gamma)$ WWYD) skew information and $(\alpha,\beta,\gamma)$ modified weighted Wigner-Yanase-Dyson ($(\alpha,\beta,\gamma)$ MWWYD) skew information. In Section 3, we present uncertainty inequalities for arbitrary $N$ mutually noncommutative observables in terms of the $(\alpha,\beta,\gamma)$ WWYD skew information. Especially, we show that when $\alpha=\beta=\frac{1}{2}$, i.e., the $(\alpha,\beta,\gamma)$ WWYD skew information reduce to the WY skew information, the lower bounds of our inequalities improve the existing ones by a detailed example. In Section 4, we explore the $(\alpha,\beta,\gamma)$ MWWYD skew information-based sum uncertainty relations for quantum channels. Some concluding remarks are given in Section 5.

For a quantum state $\rho\in\mathcal{D(H)}$ and an observable $A\in\mathcal{S(H)}$, the Wigner-Yanase (WY) skew information [23] is defined by

$$\mathrm{I}_{\rho}(A):=-\frac{1}{2}\mathrm{Tr}\left([\sqrt{\rho},A]^{2}\right)=\frac{1}{2}\|[\sqrt{\rho},A]\|^{2},$$

(2)

where $\|\cdot\|$ denotes the Hilbert Schmidt norm, $\|T\|=\sqrt{\mathrm{Tr}T^{{\dagger}}T}$. $\mathrm{I}_{\rho}(A)$ is generalized by Dyson to

$$\mathrm{I}_{\rho}^{\alpha}(A):=-\frac{1}{2}\mathrm{Tr}([\rho^{\alpha},A][\rho^{1-\alpha},A]),\,\,~{}0\leq\alpha\leq 1,$$

(3)

which is now called the Wigner-Yanase-Dyson (WYD) skew information[23]. $\mathrm{I}_{\rho}^{\alpha}(A)$ is further generalized to [24]

$$\mathrm{I}_{\rho}^{\alpha,\beta}(A)=-\frac{1}{2}\mathrm{Tr}([\rho^{\alpha},A][\rho^{\beta},A]\rho^{1-\alpha-\beta}),~{}~{}~{}\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,$$

(4)

which is termed as generalized Wigner-Yanase-Dyson (GWYD) skew information.

Another generalization of the WYD skew information is given in [44],

$\displaystyle\mathrm{K}_{\rho}^{\alpha}(A)=-\frac{1}{2}\mathrm{Tr}\left(\left[\frac{\rho^{\alpha}+\rho^{1-\alpha}}{2},A\right]^{2}\right)=\frac{1}{2}\left\|\left[\frac{\rho^{\alpha}+\rho^{1-\alpha}}{2},A\right]\right\|^{2}~{},\,\,0\leq\alpha\leq 1,$

(5)

which is called the weighted Wigner-Yanase-Dyson (WWYD) skew information. The authors in [45] proposed the modified weighted Wigner-Yanase-Dyson (MWWYD) skew information, which is the non-Hermitian extension of the WWYD skew information.

By replacing the arithmetic mean of $\rho^{\alpha}$ and $\rho^{1-\alpha}$ with their convex combination, the two-parameter extension of Wigner-Yanase skew information has been introduced,

	$\displaystyle\mathrm{K}_{\rho,\gamma}^{\alpha}(A)=$	$\displaystyle-\frac{1}{2}\mathrm{Tr}\left([(1-\gamma)\rho^{\alpha}+\gamma\rho^{1-\alpha},A]^{2}\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\left\|\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{1-\alpha},A\right]\right\|^{2}~{},~{}\,\,0\leq\alpha\leq 1~{},\,\,0\leq\gamma\leq 1.$		(6)

For convenience, we call it the $(\alpha,\gamma)$ weighted Wigner-Yanase-Dyson ($(\alpha,\gamma)$ WWYD) skew information in this paper. Note that Eq. (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) reduces to Eq. (5) and Eq. (2) when $\gamma=\frac{1}{2}$ and $\alpha=\frac{1}{2}$, respectively.

For a quantum state $\rho\in\mathcal{D(H)}$ and an observable $A\in\mathcal{S(H)}$, we define the $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson ($(\alpha,\beta,\gamma)$ WWYD) skew information as

	$\displaystyle\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A)=$	$\displaystyle-\frac{1}{2}\mathrm{Tr}([(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A]^{2}\rho^{1-\alpha-\beta})$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A\right]\right\|^{2},~{}~{}\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,0\leq\gamma\leq 1.$		(7)

Note that Eq. (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) reduces to Eq. (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) when $\beta=1-\alpha$.

We also define the $(\alpha,\beta,\gamma)$ modified weighted Wigner-Yanase-Dyson ($(\alpha,\beta,\gamma)$ MW
WYD) skew information with respect to a quantum state $\rho\in\mathcal{D(H)}$ and an arbitrary operator $E\in\mathcal{B(H)}$ (not necessarily Hermitian),

	$\displaystyle\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(E)=$	$\displaystyle-\frac{1}{2}\mathrm{Tr}([(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},E^{{\dagger}}][(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},E]\rho^{1-\alpha-\beta})$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\left\|\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},E\right]\rho^{\frac{1-\alpha-\beta}{2}}\right\|^{2},~{}~{}~{}\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,0\leq\gamma\leq 1,$		(8)

which is the non-Hermitian extension of the $(\alpha,\beta,\gamma)$ WWYD skew information. Note that Eq. (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) reduces to Eq. (10) in [43] when $\gamma=\frac{1}{2}$.

Following the idea in [41], we further define the $(\alpha,\beta,\gamma)$ MWWYD skew information of $\rho$ with respect to a channel $\Phi$ as

$$\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(\Phi)=\sum_{i=1}^{n}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(E_{i}),$$

(9)

where $\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,0\leq\gamma\leq 1$, and $E_{i}(i=1,2,\cdots,n)$ are Kraus operators of the channel $\Phi$, i.e., $\Phi(\rho)=\sum_{i=1}^{n}E_{i}\rho E_{i}^{{\dagger}}$.

In particular, for $N=2$ from Theorem 1 we have the following Corollary 1.

Corollary 1 For arbitrary two noncommutative observables $A$ and $B$, we have

$$\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A)+\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(B)\geq\mathrm{max}\frac{1}{2}\{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A+B),\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A-B)\},$$

(11)

where $\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1$.

Note that (11) of Corollary 1 reduces to the formula (3) of Theorem 1 in [48] when $\alpha=\beta=\frac{1}{2}$.

Theorem 2 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N\geq 2$), we have

$$\sum_{i=1}^{N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}\geq\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N}A_{i}\right)}~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1,$$

(12)

and

$$\sum_{i=1}^{N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}\geq\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N-1}A_{i}-A_{N}\right)}~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1.$$

(13)

Proof By using the norm inequality, we obtain

	$\displaystyle\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N}A_{i}\right)}=$	$\displaystyle\frac{1}{\sqrt{2}}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},\sum_{i=1}^{N}A_{i}\right]\right\|$
	$\displaystyle\leq$	$\displaystyle\frac{1}{\sqrt{2}}\sum_{i=1}^{N}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{i}\right]\right\|$
	$\displaystyle=$	$\displaystyle\sum_{i=1}^{N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1,$

and

	$\displaystyle\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N-1}A_{i}-A_{N}\right)}=$	$\displaystyle\frac{1}{\sqrt{2}}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},\sum_{i=1}^{N-1}A_{i}-A_{N}\right]\right\|$
	$\displaystyle\leq$	$\displaystyle\frac{1}{\sqrt{2}}\sum_{i=1}^{N}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{i}\right]\right\|$
	$\displaystyle=$	$\displaystyle\sum_{i=1}^{N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1.\Box$

Setting $N=2$ in Theorem 2, we have the following Corollary 2.

Corollary 2 For arbitrary two noncommutative observables $A$ and $B$, we have

$$\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A)}+\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(B)}\geq\mathrm{max}\left\{\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A+B)},\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A-B)}\right\}~{},$$

(14)

where $\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1$.

Note that (14) of Corollary 2 reduces to (5) of Theorem 2 in [48] when $\alpha=\beta=\frac{1}{2}$.

Theorem 3 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N>2$), we have

	$\displaystyle\sum_{i=1}^{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})$	$\displaystyle\geq$	$\displaystyle\frac{1}{N-2}\left[\sum_{1\leq i<j\leq N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}+A_{j})-\frac{1}{(N-1)^{2}}\right.$		(15)
			$\displaystyle\left.\left(\sum_{1\leq i<j\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}+A_{j})}\right)^{2}\right],$

where $\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1$.

Proof Employing the following inequality [47],

	$\displaystyle\sum_{i=1}^{N}\|u_{i}\|^{2}\geq$	$\displaystyle\frac{1}{N-2}\left[\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|^{2}-\frac{1}{(N-1)^{2}}\left(\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|\right)^{2}\right]$
	$\displaystyle\geq$	$\displaystyle\frac{1}{2(N-1)}\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|^{2},$

and setting $u_{i}=\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{i}\right]$ and $u_{j}=\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}\right]$, we obtain (15). $\Box$

Note that (15) of Theorem 3 reduces to (12) of Theorem 4 in [48] when $\alpha=\beta=\frac{1}{2}$.

Theorem 4 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N>2$), we have

	$\displaystyle\sum_{i=1}^{N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}\geq\frac{1}{N-2}\left[\sum_{1\leq i<j\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}+A_{j})}\right.$
	$\displaystyle\left.-\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N}A_{i}\right)}\right]~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1.$		(16)

Proof By using the following inequality [47, 55, 56],

	$\displaystyle\sum_{i=1}^{N}\|u_{i}\|\geq$	$\displaystyle\frac{1}{N-2}\left(\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|-\left\|\sum_{i=1}^{N}u_{i}\right\|\right)$
	$\displaystyle\geq$	$\displaystyle\left\|\sum_{i=1}^{N}u_{i}\right\|,$

with $u_{i}=\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{i}\right]$ and $u_{j}=\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}\right]$, we obtain (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information). $\Box$

Note that (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) of Theorem 4 reduces to (14) of Theorem 5 in [48] when $\alpha=\beta=\frac{1}{2}$. Moreover, from the proof of Theorem 4, it can be seen that the right hand side of (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) is tighter than the right hand side of (12).

Theorem 5 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N\geq 2$), we have

	$\displaystyle\sum_{i=1}^{N}{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}\geq\frac{1}{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{i=1}^{N}A_{i}\right)+\frac{2}{N^{2}(N-1)}$
	$\displaystyle\left(\sum_{1\leq i<j\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}-A_{j})}\right)^{2}~{},$		(17)

where $\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1$.

Proof According to the lemma in [49], we have

	$\displaystyle\frac{2}{N^{2}(N-1)}\left(\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|\right)^{2}\leq$	$\displaystyle\frac{1}{N}\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|^{2}$
	$\displaystyle=$	$\displaystyle\sum_{i=1}^{N}\|u_{i}\|^{2}-\frac{1}{N}\left\|\sum_{i=1}^{N}u_{i}\right\|^{2},$

that is,

$\displaystyle\sum_{i=1}^{N}\|u_{i}\|^{2}\geq\frac{1}{N}\left\|\sum_{i=1}^{N}u_{i}\right\|^{2}+$

$\displaystyle\frac{2}{N^{2}(N-1)}\left(\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|\right)^{2}.$

By substituting $u_{i}$ and $u_{j}$ with $\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{i}\right]$ and $\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\right.$
$\left.\gamma\rho^{\beta},A_{j}\right]$, respectively, we obtain (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information). $\Box$

Note that (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) of Theorem 5 reduces to (12) of Theorem 1 in [49] when $\alpha=\beta=\frac{1}{2}$.

Theorem 6 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N\geq 2$), we have

	$\displaystyle\sum_{i=1}^{N}{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}$	$\displaystyle\geq$	$\displaystyle\frac{1}{2(N-1)}\left[\frac{2}{N(N-1)}\left(\sum_{1\leq i<j\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}+A_{j})}\right)^{2}\right.$		(18)
			$\displaystyle\left.+\sum_{1\leq i<j\leq N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}-A_{j})\right]~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1,$

and

	$\displaystyle\sum_{i=1}^{N}{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i})}$	$\displaystyle\geq$	$\displaystyle\frac{1}{2(N-1)}\left[\frac{2}{N(N-1)}\left(\sum_{1\leq i<j\leq N}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}-A_{j})}\right)^{2}\right.$		(19)
			$\displaystyle\left.+\sum_{1\leq i<j\leq N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{i}+A_{j})\right]~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1.$

Proof By using the following equality,

$\displaystyle 2(N-1)\sum_{i=1}^{N}\|u_{i}\|^{2}=\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|^{2}+\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|^{2}$

and the Cauchy-Schwarz inequality, we obtain

$\displaystyle\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|^{2}\geq\frac{2}{N(N-1)}\left(\sum_{1\leq i<j\leq N}\|u_{i}+u_{j}\|\right)^{2},$

and

$\displaystyle\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|^{2}\geq\frac{2}{N(N-1)}\left(\sum_{1\leq i<j\leq N}\|u_{i}-u_{j}\|\right)^{2},$

respectively. Therefore, we have

$\displaystyle\sum_{i=1}^{N}\|u_{i}\|^{2}\geq\frac{1}{2(N-1)}\left[\frac{2}{N(N-1)}\left(\sum_{1\leq i<j\leq N}\|u_{i}\pm u_{j}\|\right)^{2}+\sum_{1\leq i<j\leq N}\|u_{i}\mp u_{j}\|^{2}\right].$

The inequalities (18) and (19) follow by replacing $u_{i}$ and $u_{j}$ with $\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\right.$
$\left.\gamma\rho^{\beta},A_{i}\right]$ and $\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}\right]$, respectively. $\Box$

As a special case, when $\alpha=\beta=\frac{1}{2}$, (18) and (19) of Theorem 6 reduce to (12) and (13) of Theorem 2 in [50], respectively. Note also that Theorem 5 and Theorem 6 are identical when $N=2$.

Theorem 7 For arbitrary $N$ mutually noncommutative observables $A_{1},A_{2},\cdots,A_{N}$ ($N\geq 2$) and a quantum state $\rho$, let $G$ be an $N\times N$ matrix with entries $G_{jk}=\mathrm{Tr}(X_{j}X_{k}^{\dagger})$, where $X_{j}=i\rho^{\frac{1-\alpha-\beta}{2}}[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}]/\|\rho^{\frac{1-\alpha-\beta}{2}}[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}]\|$, $i=\sqrt{-1}$. We have

$$\sum_{j=1}^{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{j})\geq\frac{1}{\lambda_{max}(G)}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{j=1}^{N}A_{j}\right)~{},\,\,\alpha,\beta\geq 0,~{}\alpha+\beta\leq 1,~{}0\leq\gamma\leq 1,$$

(20)

where $\lambda_{max}(G)$ denotes the maximal eigenvalue of $G$.

Proof It is obvious that $G$ is a positive semi-definite matrix. Noting that

		$\displaystyle\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}\left(\sum_{j=1}^{N}A_{j}\right)$
	$\displaystyle=$	$\displaystyle-\frac{1}{2}\mathrm{Tr}\left(\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},\sum_{j=1}^{N}A_{j}\right]^{2}\rho^{1-\alpha-\beta}\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\sum_{j,k}\mathrm{Tr}\left[\left(i\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}\right]\right)\left(i\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{k}\right]\rho^{\frac{1-\alpha-\beta}{2}}\right)\right]$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\sum_{j,k}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{j}\right]\right\|G_{jk}\left\|\rho^{\frac{1-\alpha-\beta}{2}}\left[(1-\gamma)\rho^{\alpha}+\gamma\rho^{\beta},A_{k}\right]\right\|$
	$\displaystyle=$	$\displaystyle\sum_{j,k}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{j})}G_{jk}\sqrt{\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{k})}$
	$\displaystyle\leq$	$\displaystyle\lambda_{\mathrm{max}}(G)\sum_{j=1}^{N}\mathrm{K}_{\rho,\gamma}^{\alpha,\beta}(A_{j}),$

we prove that (20) holds. $\Box$

In particular, the Theorem 6 in [48] is a special case of our Theorem 7 when $\alpha=\beta=\frac{1}{2}$.

When $\alpha=\beta=\frac{1}{2}$, the $(\alpha,\beta,\gamma)$ WWYD skew information reduces to the WY skew information. We next compare our uncertainty relations with the existing ones. For $\alpha=\beta=\frac{1}{2}$, we denote by $LB_{0},LB_{1},LB_{2}$ and $LB_{3}$ the right hand sides of (15), (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information), (18) and (19), respectively.

Example 1 Given a qubit state $\rho=\frac{1}{2}(\mathbf{1}+\mathbf{r}\cdot\bm{\sigma})$, where $\mathbf{r}=(x,y,z)$ is the Bloch vector satisfying $|\mathbf{r}|\leq 1$, $\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ with $\sigma_{j}$ $(j=1,2,3)$ the Pauli matrices, and $\mathbf{r}\cdot\bm{\sigma}=\sum^{3}_{j=1}r_{j}\sigma_{j}$. The eigenvalues of $\rho$ are $\lambda_{1,2}=(1\mp\sqrt{t})/2$, where $t=|\mathbf{r}|^{2}$.
The sum of the skew information of three Pauli operators is given by

$$\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{1})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{2})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{3})=2(1-\sqrt{1-t}).$$

(21)

From (18) and (15) we have, respectively,

$$\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{1})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{2})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{3})\geq(1-\sqrt{1-t})\left(1+\frac{xy+xz+yz}{2t}\right)+\frac{1}{12}\beta^{2}$$

(22)

and

$$\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{1})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{2})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{3})\geq(1-\sqrt{1-t})\left(4-\frac{2(xy+xz+yz)}{t}\right)-\frac{1}{4}\beta^{2},$$

(23)

where

$$\alpha=\sqrt{1-\sqrt{1-t}}\left(\sqrt{1+\frac{z^{2}+2xy}{t}}+\sqrt{1+\frac{y^{2}+2xz}{t}}+\sqrt{1+\frac{x^{2}+2yz}{t}}\right).$$

(24)

(19) and (Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information) give rise to

$$\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{1})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{2})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{3})\geq(1-\sqrt{1-t})\left(1-\frac{xy+xz+yz}{2t}\right)+\frac{1}{12}\alpha^{2}$$

(25)

and

$$\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{1})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{2})+\mathrm{K}_{\rho,\gamma}^{\frac{1}{2},\frac{1}{2}}(\sigma_{3})\geq\frac{2}{3}(1-\sqrt{1-t})\left(1-\frac{xy+xz+yz}{t}\right)+\frac{1}{9}\alpha^{2},$$

(26)

respectively, where

$$\beta=\sqrt{1-\sqrt{1-t}}\left(\sqrt{1+\frac{z^{2}-2xy}{t}}+\sqrt{1+\frac{y^{2}-2xz}{t}}+\sqrt{1+\frac{x^{2}-2yz}{t}}\right).$$

(27)

Comparing the lower bound $LB_{2}$ ($LB_{3}$) on the right hand of the inequality (18) ((19)) with the bound $LB_{0}$ ($LB_{1}$) on the right hand of inequality (15) ((Sum uncertainty relations based on $(\alpha,\beta,\gamma)$ weighted Wigner-Yanase-Dyson skew information)), we obtain $LB_{2}-LB_{0}=(1-\sqrt{1-t})\gamma$ and $LB_{3}-LB_{1}=(1-\sqrt{1-t})\gamma_{1}$, respectively, where

$$\gamma=-3+\frac{5(xy+xz+yz)}{2t}+\frac{1}{3}\left(\sqrt{1+\frac{z^{2}-2xy}{t}}+\sqrt{1+\frac{y^{2}-2xz}{t}}+\sqrt{1+\frac{x^{2}-2yz}{t}}\right)^{2}$$

(28)

and

$$\gamma_{1}=\frac{1}{3}+\frac{xy+xz+yz}{6t}-\frac{1}{36}\left(\sqrt{1+\frac{z^{2}+2xy}{t}}+\sqrt{1+\frac{y^{2}+2xz}{t}}+\sqrt{1+\frac{x^{2}+2yz}{t}}\right)^{2}.$$

(29)

Note that $1-\sqrt{1-t}>0$. Denote $x=\sqrt{t}\sin\theta\cos\varphi$, $y=\sqrt{t}\sin\theta\sin\varphi$ and $z=\sqrt{t}\cos\theta$ with $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$. We have

	$\displaystyle\gamma$	$\displaystyle=$	$\displaystyle-3+\frac{5}{2}(\sin^{2}\theta\sin\varphi\cos\varphi+\sin\theta\cos\theta\cos\varphi+\sin\theta\cos\theta\sin\varphi)$		(30)
			$\displaystyle+\frac{1}{3}\left(\sqrt{1+\cos^{2}\theta-2\sin^{2}\theta\sin\varphi\cos\varphi}+\sqrt{1+\sin^{2}\theta\sin^{2}\varphi-2\sin\theta\cos\theta\cos\varphi}\right.$
			$\displaystyle\left.+\sqrt{1+\sin^{2}\theta\cos^{2}\varphi-2\sin\theta\cos\theta\sin\varphi}\right)^{2}>0,$

and

	$\displaystyle\gamma_{1}$	$\displaystyle=$	$\displaystyle\frac{1}{3}+\frac{1}{6}(\sin^{2}\theta\sin\varphi\cos\varphi+\sin\theta\cos\theta\cos\varphi+\sin\theta\cos\theta\sin\varphi)$		(31)
			$\displaystyle-\frac{1}{36}\left(\sqrt{1+\cos^{2}\theta+2\sin^{2}\theta\sin\varphi\cos\varphi}+\sqrt{1+\sin^{2}\theta\sin^{2}\varphi+2\sin\theta\cos\theta\cos\varphi}\right.$
			$\displaystyle\left.+\sqrt{1+\sin^{2}\theta\cos^{2}\varphi+2\sin\theta\cos\theta\sin\varphi}\right)^{2}>0,$

see Fig. 1.