1. Introduction
As one of the cornerstones of quantum mechanics, the uncertainty principle reveals the insights that distinguish quantum theory from classical theory. Heisenberg [1] originally proposed the uncertainty principle in 1927, which indicates that the position and momentum of a particle cannot be precisely determined simultaneously. Robertson [2] generalized the variance-based uncertainty relation for position and momentum to arbitrary two observables A and B.
$\begin{eqnarray}{\rm{\Delta }}A{\rm{\Delta }}B\geqslant \displaystyle \frac{1}{2}| \langle \psi | [A,B]| \psi \rangle | ,\end{eqnarray}$
where [A, B] = AB − BA and ${\rm{\Delta }}M=\sqrt{\langle \psi | {M}^{2}| \psi \rangle -\langle \psi | M| \psi {\rangle }^{2}}$ is the standard deviation of the observable M with respect to a fixed state ∣ψ⟩.The uncertainty principle has attracted sustained attention. A host of methods have been presented to characterize the uncertainty [3–16]. The skew information is one of the typical ways to describe the uncertainty principle. The skew information of an observable A with respect to a quantum state ρ is given by [17].
$\begin{eqnarray}{{\rm{I}}}_{\rho }(A)=-\displaystyle \frac{1}{2}\mathrm{Tr}\left({\left[\sqrt{\rho },A\right]}^{2}\right)=\displaystyle \frac{1}{2}{\parallel \left[\sqrt{\rho },A\right]\parallel }^{2},\end{eqnarray}$
which is called the Wigner–Yanase (WY) skew information. Later, Dyson proposed a one-parameter extension of the WY skew information, called Wigner–Yanase–Dyson (WYD) skew information. In [18] the WYD skew information is further generalized to the generalized Wigner–Yanase–Dyson (GWYD) skew information. The corresponding definitions of the above-mentioned skew information for arbitrary (not necessarily Hermitian) operators have also been introduced [9, 10, 19, 20]. The uncertainty relation related to WY skew information was initially introduced by Luo [21]. Recently, uncertainty relations based on various generalized skew information have been explored intensely in [19, 20, 22–30].By considering the arithmetic mean of ρα and ρ1−α, Furuichi et al [31] introduced another one-parameter skew information, 3 ) was presented in [32] for an arbitrary operator E, 4 ) reduces to equation (14 ) in [9], and equation (3 ) reduces to equation (2 ) when $\alpha =\tfrac{1}{2}$, respectively.
$\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho }^{\alpha }(A) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}\left({\left[\displaystyle \frac{{\rho }^{\alpha }+{\rho }^{1-\alpha }}{2},A\right]}^{2}\right)\\ & = & \displaystyle \frac{1}{2}{\parallel \left[\displaystyle \frac{{\rho }^{\alpha }+{\rho }^{1-\alpha }}{2},A\right]\parallel }^{2},\,0\leqslant \alpha \leqslant 1,\end{array}\end{eqnarray}$
which is called the weighted Wigner–Yanase–Dyson (WWYD) skew information [20]. A generalization of equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho }^{\alpha }(E) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}\left(\left[\displaystyle \frac{{\rho }^{\alpha }+{\rho }^{1-\alpha }}{2},{E}^{\dagger }\right]\left[\displaystyle \frac{{\rho }^{\alpha }+{\rho }^{1-\alpha }}{2},E\right]\right)\\ & = & \displaystyle \frac{1}{2}{\parallel \left[\displaystyle \frac{{\rho }^{\alpha }+{\rho }^{1-\alpha }}{2},E\right]\parallel }^{2},\,0\leqslant \alpha \leqslant 1,\end{array}\end{eqnarray}$
which is termed as the modified weighted Wigner–Yanase–Dyson (MWWYD) skew information in [20]. Note that equation (Recently, the two-parameter extension of equation (3 ) was introduced in [33],
$\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho ,\gamma }^{\alpha }(A) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}\left({[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{1-\alpha },A]}^{2}\right)\\ & = & \displaystyle \frac{1}{2}{\parallel \left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{1-\alpha },A\right]\parallel }^{2},\\ & & 0\leqslant \alpha \leqslant 1\,,\,0\leqslant \gamma \leqslant 1,\end{array}\end{eqnarray}$
which is called the (α, γ) weighted Wigner–Yanase–Dyson ((α, γ) WWYD) skew information in [34].Motivated by the equation (5 ) above and equation (3 ) in [19], we introduced the (α, β, γ) weighted Wigner–Yanase–Dyson ((α, β, γ) WWYD) skew information [35], 7 ) reduces to equation (7 ) in [35] and equation (6 ) reduces to equation (5 ) when β = 1 − α, respectively.
$\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }(A) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}\left({\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },A\right]}^{2}{\rho }^{1-\alpha -\beta }\right)\\ & = & \displaystyle \frac{1}{2}{\parallel \left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },A\right]{\rho }^{\displaystyle \frac{1-\alpha -\beta }{2}}\parallel }^{2},\\ & & \alpha ,\beta \geqslant 0,\,\alpha +\beta \leqslant 1,0\leqslant \gamma \leqslant 1,\end{array}\end{eqnarray}$
and its non-Hermitian extension, the (α, β, γ) modified weighted Wigner–Yanase–Dyson ((α, β, γ) MWWYD) skew information [35], $\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }(E) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}\left([(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{E}^{\dagger }]\right.\\ & & \left.\times \,[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },E]{\rho }^{1-\alpha -\beta }\right)\\ & = & \displaystyle \frac{1}{2}{\parallel \left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },E\right]{\rho }^{\displaystyle \frac{1-\alpha -\beta }{2}}\parallel }^{2},\\ & & \alpha ,\beta \geqslant 0,\,\alpha +\beta \leqslant 1,0\leqslant \gamma \leqslant 1.\end{array}\end{eqnarray}$
Here equation (As the broadest form of measurement [36, 37], quantum channels play a crucial role in quantum theory. Chen et al [38] investigated the summation form of the uncertainty relations based on WY skew information for observables. Fu et al [39] explored the uncertainty relations for quantum channels in terms of WY skew information. Recently, the summation form of the uncertainty relations associated with skew information for arbitrary finite quantum observables and quantum channels has also been derived [40–45].
The paper is structured as follows. In section 2 , by using operator norm inequalities, new uncertainty relations of observables are given in terms of the (α, β, γ) WWYD skew information. We present two distinct types of uncertainty relations for quantum channels with respect to the (α, β, γ) MWWYD skew information and establish an optimal lower bound in section 3 . The tighter uncertainty relations of unitary channels are presented in section 4 . We conclude with a summary in section 5 .
2. Sum uncertainty relations for arbitrary finite N observables
For arbitrary finite N observables A1, A2, ⋯ ,AN, Xu et al [34] provided the following sum uncertainty relations, 8 ) and N ≥ 2 for the inequalities (9 –10 ). For convenience we denote by Lb1, Lb2 and Lb3 the right-hand sides of (8 ), (9 ) and (10 ), respectively.
$\begin{eqnarray}\begin{array}{rcl} & & \displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i})\geqslant \displaystyle \frac{1}{N-2}\left[\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})\right.\\ & & \left.-\displaystyle \frac{1}{{\left(N-1\right)}^{2}}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})}\right)}^{2}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl} & & \displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i})\geqslant \displaystyle \frac{1}{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{i=1}^{N}{A}_{i}\right)\\ & & +\displaystyle \frac{2}{\left.{N}^{2}(N-1\right)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}-{A}_{j})}\right)}^{2}\,,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}) & \geqslant & \mathop{\max }\limits_{x\in \{0,1\}}\displaystyle \frac{1}{2(N-1)}\\ & & \times \,\left[\displaystyle \frac{2}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left({A}_{i}+{\left(-1\right)}^{x}{A}_{j}\right)}\right)}^{2}\right.\\ & & \left.+\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{\left(-1\right)}^{x+1}{A}_{j})\right],\end{array}\end{eqnarray}$
where α, β ≥ 0, α + β ≤ 1, 0 ≤ γ ≤ 1, N > 2 for the inequality (The following relations are given in the appendix B in [45],
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}\parallel {u}_{i}{\parallel }^{2} & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}+{u}_{j}\parallel \right)}^{2}\right.\\ & & +\,\left.M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}-{u}_{j}{\parallel }^{2}+(M-L){\parallel \displaystyle \sum _{i=1}^{N}{u}_{i}\parallel }^{2}\right\},\end{array}\end{eqnarray}$
for M ≥ L > 0, $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}\parallel {u}_{i}{\parallel }^{2} & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}-{u}_{j}\parallel \right)}^{2}\right.\\ & & +\,\left.L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}+{u}_{j}{\parallel }^{2}+(M-L){\parallel \displaystyle \sum _{i=1}^{N}{u}_{i}\parallel }^{2}\right\},\end{array}\end{eqnarray}$
for L ≥ M > 0, $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}\parallel {u}_{i}{\parallel }^{2} & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \,\times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}+{u}_{j}\parallel \right)}^{2}\right.\\ & & +\,\left.M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}-{u}_{j}{\parallel }^{2}+L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\parallel {u}_{i}+{u}_{j}{\parallel }^{2}\right\}\end{array}\end{eqnarray}$
for arbitrary L > M > 0. By replacing ui and uj with $\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{A}_{i}\right]{\rho }^{\tfrac{1-\alpha -\beta }{2}}$ and $\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{A}_{j}\right]{\rho }^{\tfrac{1-\alpha -\beta }{2}}$, respectively, in the above inequalities, we obtain the following inequalities.For arbitrary finite N observables ${A}_{1},{A}_{2},\cdots ,{A}_{N}$ ($N\geqslant 2$), we have the following sum uncertainty relation via $(\alpha ,\beta ,\gamma )$ WWYD skew information,
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i})\geqslant \max \{{\overline{{Lb}}}_{1},{\overline{{Lb}}}_{2},{\overline{{Lb}}}_{3}\},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\overline{{Lb}}}_{1} & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})}\right)}^{2}\right.\\ & & +\,\left.M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}-{A}_{j})+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{i=1}^{N}{A}_{i}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\overline{{Lb}}}_{2} & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}-{A}_{j})}\right)}^{2}\right.\\ & & +\,\left.L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{i=1}^{N}{A}_{i}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\overline{{Lb}}}_{3} & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})}\right)}^{2}\right.\\ & & +M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}-{A}_{j})\\ & & \left.+L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({A}_{i}+{A}_{j})\right\},\end{array}\end{eqnarray}$
$\alpha ,\beta \geqslant 0$, $\alpha +\beta \leqslant 1$, $0\leqslant \gamma \leqslant 1$, the parameters L, M in the expressions of ${\overline{{Lb}}}_{1}$, ${\overline{{Lb}}}_{2}$ and ${\overline{{Lb}}}_{3}$ satisfy $M\geqslant L\gt 0$, $L\geqslant M\gt 0$ and $L\gt M\gt 0$, respectively.For convenience we denote $\overline{{Lb}}=\max \{{\overline{{Lb}}}_{1},{\overline{{Lb}}}_{2},{\overline{{Lb}}}_{3}\}$ the right-hand side of (14 ). In [45] Li et al proved that (11 ), (12 ) and (13 ) are strictly tighter than those of norm inequalities related to (8 ), (9 ) and (10 ) for appropriate M and L. If we take M = L, ${\overline{{Lb}}}_{1}$ and ${\overline{{Lb}}}_{2}$ reduces to the cases of Lb3. For fixed N(≥ 2), larger M and smaller L give rise to larger right-hand sides of the inequalities (11 ) and (13 ). Conversely, smaller M and larger L result in larger right-hand side of the inequality (12 ).
For finite N observables ${A}_{1},{A}_{2},\cdots ,{A}_{N}$ ($N\geqslant 2$), the sum uncertainty relations with respect to WWYD skew information are given by
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}) & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}+{A}_{j})}\right)}^{2}\right.\\ & & +\,M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}-{A}_{j})\\ & & \left.+(M-L){{\rm{K}}}_{\rho }^{\alpha }\left(\displaystyle \sum _{i=1}^{N}{A}_{i}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}) & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}-{A}_{j})}\right)}^{2}\right.\\ & & +\,L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}+{A}_{j})\\ & & \left.+(M-L){{\rm{K}}}_{\rho }^{\alpha }\left(\displaystyle \sum _{i=1}^{N}{A}_{i}\right)\right\},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}) & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}{\left(\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}+{A}_{j})}\right)}^{2}\right.\\ & & \left.+M\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}-{A}_{j})+L\displaystyle \sum _{1\leqslant i\lt j\leqslant N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i}+{A}_{j})\right\},\end{array}\end{eqnarray}$
where $0\leqslant \alpha \leqslant 1$, the parameters L, M in (Denote rhs18, rhs19 and rhs20 the right-hand sides of the inequalities (18 ), (19 ) and (20 ), respectively. Corollary 1 implies that ${\sum }_{i=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({A}_{i})\geqslant \max \{{rhs}19,{rhs}20,{rhs}21\}$. In particular, (33 ), (34 ) and (35 ) in [45] are just the special cases of the inequalities (18 ), (19 ) and (20 ) for $\alpha =\tfrac{1}{2}$, respectively. Next we prove that our new lower bound $\overline{{Lb}}$ is tighter than the existing ones by a detailed example. We consider the WWYD skew information as a special case, and take M = 2, L = 1 for ${\overline{{Lb}}}_{1}$, and M = 1, L = 2 for ${\overline{{Lb}}}_{2}$ and ${\overline{{Lb}}}_{3}$.
Consider the pure state $\rho =\tfrac{1}{2}({{\rm{I}}}_{2}+{\boldsymbol{r}}\cdot {\boldsymbol{\sigma }})$, where ${\boldsymbol{r}}=\left(\tfrac{\sqrt{2}}{2}\cos \theta ,\tfrac{\sqrt{2}}{2}\sin \theta ,\tfrac{\sqrt{2}}{2}\right)$, ${{\rm{I}}}_{2}$ is the 2 × 2 identity matrix and ${\boldsymbol{\sigma }}=({\sigma }_{1},{\sigma }_{2},{\sigma }_{3})$ is composed of Pauli matrices. We compare $\overline{{Lb}}$ with Lb1, Lb2 and Lb3 for any α and $\alpha =\tfrac{1}{3}$, respectively. It can be seen that $\overline{{Lb}}$ is tighter than Lb1, Lb2 and Lb3 for arbitrary α, see figure 1.
Figure 1. Comparison of the lower bound (Lb) $\overline{{Lb}}$ with Lb1, Lb2 and Lb3. (a) The red surface and the magenta surface represent $\overline{{Lb}}$ and Lb1, respectively. (b) The blue surface represents the difference value (DV) between $\overline{{Lb}}$ and Lb2. (c) The green surface represents the difference value between $\overline{{Lb}}$ and Lb3. (d) For fixed $\alpha =\tfrac{1}{3}$, the black, red dashed, magenta, blue and green curves represent the summation ${{\rm{K}}}_{\rho }^{\tfrac{1}{3}}({\sigma }_{1})+{{\rm{K}}}_{\rho }^{\tfrac{1}{3}}({\sigma }_{2})+{{\rm{K}}}_{\rho }^{\tfrac{1}{3}}({\sigma }_{3})$, $\overline{{Lb}}$, Lb1, Lb2 and Lb3, respectively. |
3. Sum uncertainty relations for finite quantum channels
In this section, we give two different types of uncertainty relations associated with arbitrary finite number of quantum channels based on (α, β, γ) MWWYD skew information. We derive an optimal lower bound and show that our bounds are tighter than the existing ones by a detailed example.
Let Φ be a quantum channel with Kraus representation, ${\rm{\Phi }}(\rho )={\sum }_{i=1}^{n}{E}_{i}\rho {E}_{i}^{\dagger }$. In [34] the authors have presented the uncertainty quantification with respect to a channel Φ via (α, β, γ) MWWYD skew information, 22 ) and N ≥ 2 for the inequalities (23 –24 )). Xu et al. [34] gave the following sum uncertainty quantifications associated with the channels, 22 ), (23 ) and (24 ), respectively.
$\begin{eqnarray}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({\rm{\Phi }})=\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{i}),\end{eqnarray}$
where α, β ≥ 0, α + β ≤ 1, 0 ≤ γ ≤ 1. For arbitrary N quantum channels ,Φ1, ⋯ ,ΦN with Kraus representations ${{\rm{\Phi }}}_{t}{(\rho )={\sum }_{i=1}^{n}{E}_{i}^{t}\rho ({E}_{i}^{t})}^{\dagger },\,t\,=\,1,2,\cdots ,N$ (N > 2 for the inequality ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{N-2}\\ & & \times \,\left\{\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\right.\\ & & \left.-\displaystyle \frac{1}{{\left(N-1\right)}^{2}}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\\ & & \times \,\left\{\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right.\\ & & \left.+\displaystyle \frac{2}{{N}^{2}(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{2(N-1)}\\ & & \times \,\left\{\displaystyle \frac{2}{N(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}\pm {E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & \left.+\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}\mp {E}_{{\pi }_{s}(i)}^{s})\right\},\end{array}\end{eqnarray}$
where α, β ≥ 0, α + β ≤ 1, 0 ≤ γ ≤ 1, Sn is the n-element permutation group and πt, πs ∈ Sn are arbitrary n-element permutations. We denote by LB1, LB2 and LB3 the right-hand sides of (Let ${{\rm{\Phi }}}_{1},\cdots ,{{\rm{\Phi }}}_{N}$ be N quantum channels with Kraus representations ${{\rm{\Phi }}}_{t}(\rho )={\sum }_{i=1}^{n}{E}_{i}^{t}\rho {\left({E}_{i}^{t}\right)}^{\dagger },\,t\,=\,1,2,\cdots ,N$ ($N\geqslant 2$). We have
$\begin{eqnarray}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t})\geqslant \max \{{\overline{{LB}}}_{1},{\overline{{LB}}}_{2},{\overline{{LB}}}_{3}\},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\overline{{LB}}}_{1} & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & \left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})+(M-L)\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\overline{{LB}}}_{2} & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & \left.+L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})+(M-L)\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\overline{{LB}}}_{3} & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+\,M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\right\},\end{array}\end{eqnarray}$
$\alpha ,\beta \geqslant 0$, $\alpha +\beta \leqslant 1$, $0\leqslant \gamma \leqslant 1$, ${\pi }_{t},{\pi }_{s}\in {S}_{n}$ are arbitrary n-element permutations, the parameters L, M in ${\overline{{LB}}}_{1}$, ${\overline{{LB}}}_{2}$ and ${\overline{{LB}}}_{3}$ satisfy $M\geqslant L\gt 0$, $L\geqslant M\gt 0$ and $L\gt M\gt 0$, respectively.According to the inequalities (
$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}) & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}\left[{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & \left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray*}$
for $M\geqslant L\gt 0$, $\begin{eqnarray*}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}) & \geqslant & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}\left[{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray*}$
for $L\geqslant M\gt 0$, $\begin{eqnarray*}\begin{array}{l}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t})\geqslant \displaystyle \frac{1}{{MN}+(N-2)L}\\ \,\times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}\left[{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ \,\left.+L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\right.\\ \,\left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\right\},\end{array}\end{eqnarray*}$
for $L\gt M\gt 0$. Summing over the index i, which implies theorem 2.Let ${{\rm{\Phi }}}_{1},\cdots ,{{\rm{\Phi }}}_{N}$ be N quantum channels with Kraus representations ${{\rm{\Phi }}}_{t}{(\rho )={\sum }_{i=1}^{n}{E}_{i}^{t}\rho ({E}_{i}^{t})}^{\dagger },\,t\,=\,1,2,\cdots ,N$ ($N\geqslant 2$), we have the sum uncertainty relations with respect to WWYD skew information,
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & +M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+(M-L)\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+(M-L)\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{t}) & \geqslant & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}\left[\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right)}^{2}\right]\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho }^{\alpha }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\right\},\end{array}\end{eqnarray}$
where $0\leqslant \alpha \leqslant 1$, the parameters L, M in (Thus we have ${\sum }_{i=1}^{N}{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{t})\geqslant \max \{{rhs}29,{rhs}30,{rhs}31\}$, where rhs29, rhs30 and rhs31 represent the right-hand sides of inequalities (29 ), (30 ) and (31 ), respectively.
The uncertainty quantification of quantum channel Φ based on (α, β, γ) MWWYD skew information can also be written as [35], 11 –13 ), we have the following theorem.
$\begin{eqnarray}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({\rm{\Phi }})=\displaystyle \frac{1}{2}\mathrm{Tr}({u}^{\dagger }u)=\displaystyle \frac{1}{2}\parallel u{\parallel }^{2},\end{eqnarray}$
where α, β ≥ 0, α + β ≤ 1, 0 ≤ γ ≤ 1, $u=(\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{E}_{1}\right]{\rho }^{\tfrac{1-\alpha -\beta }{2}}$, $\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{E}_{2}\right] \times {\rho }^{\tfrac{1-\alpha -\beta }{2}}$, $\cdots$ ,$\left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{E}_{n}\right]{\rho }^{\tfrac{1-\alpha -\beta }{2}})$. Therefore, on the basis of $\parallel {u}_{t}\pm {u}_{s}{\parallel }^{2}=2{\sum }_{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{i}^{t}\pm {E}_{i}^{s})$, $\parallel {u}_{t}{\parallel }^{2}\,=2{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t})$ and the inequalities (Let ${{\rm{\Phi }}}_{1},\cdots ,{{\rm{\Phi }}}_{N}$ be N quantum channels with Kraus representations ${{\rm{\Phi }}}_{t}(\rho )={\sum }_{i=1}^{n}{E}_{i}^{t}\rho {\left({E}_{i}^{t}\right)}^{\dagger },\,t=1,2,\cdots ,N$ ($N\geqslant 2$). We have
$\begin{eqnarray}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t})\geqslant \max \{\overline{{LB}}1,\overline{{LB}}2,\overline{{LB}}3\},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\overline{{LB}}1 & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right]}^{2}\right.\\ & & +M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+(M-L)\displaystyle \sum _{i\,=\,1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\overline{{LB}}2 & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})}\right]}^{2}\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+(M-L)\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{E}_{{\pi }_{t}(i)}^{t}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\overline{{LB}}3 & = & \mathop{\max }\limits_{{\pi }_{t},{\pi }_{s}\in {S}_{n}}\displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})}\right]}^{2}\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}+{E}_{{\pi }_{s}(i)}^{s})\\ & & \left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\displaystyle \sum _{i=1}^{n}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({E}_{{\pi }_{t}(i)}^{t}-{E}_{{\pi }_{s}(i)}^{s})\right\},\end{array}\end{eqnarray}$
$\alpha ,\beta \geqslant 0$, $\alpha +\beta \leqslant 1$, $0\leqslant \gamma \leqslant 1$, ${\pi }_{t},{\pi }_{s}\in {S}_{n}$ are arbitrary n-element permutations, and the parameters L, M in $\overline{{LB}}1$, $\overline{{LB}}2$ and $\overline{{LB}}3$ satisfy $M\geqslant L\gt 0$, $L\geqslant M\gt 0$ and $L\gt M\,\gt 0$, respectively.Motivated by the results given in appendix D of [45], we have an optimal lower bound, 37 ), $\overline{{LB}}=\max \{\overline{{LB}}1,\overline{{LB}}2,{\overline{{LB}}}_{3}\}$.
$\begin{eqnarray}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({{\rm{\Phi }}}_{t})\geqslant \max \{\overline{{LB}}1,\overline{{LB}}2,{\overline{{LB}}}_{3}\},\end{eqnarray}$
where ${\overline{{LB}}}_{3}$ is given in theorem 2. For convenience we denote by $\overline{{LB}}$ on the right-hand side of (To illustrate our results, we consider the MWWYD skew information as a special case, and take M = 2, L = 1 for $\overline{{LB}}1$, and M = 1, L = 2 for $\overline{{LB}}2$ and ${\overline{{LB}}}_{3}$.
Consider the mixed state given by the Bloch vector ${\boldsymbol{r}}=(\tfrac{\sqrt{2}}{2}\cos \theta ,\tfrac{\sqrt{2}}{2}\sin \theta ,0)$, $\rho =\tfrac{1}{2}({{\rm{I}}}_{2}+{\boldsymbol{r}}\cdot {\boldsymbol{\sigma }})$, where $0\leqslant \theta \leqslant \pi $, ${{\rm{I}}}_{2}$ is the 2 × 2 identity matrix and ${\boldsymbol{\sigma }}=({\sigma }_{1},{\sigma }_{2},{\sigma }_{3})$ is given by the Pauli matrices. We consider the following three quantum channels:
| i | (i)the phase damping channel ${{\rm{\Phi }}}_{{\rm{PD}}}$, $\begin{eqnarray*}\begin{array}{l}{{\rm{\Phi }}}_{{\rm{PD}}}(\rho )=\displaystyle \sum _{i=1}^{2}{A}_{i}\rho {A}_{i}^{\dagger },\quad {A}_{1}=| 0\rangle \langle 0| +\sqrt{1-q}| 1\rangle \langle 1| ,\\ {A}_{2}=\sqrt{q}| 1\rangle \langle 1| ;\end{array}\end{eqnarray*}$ |
| ii | (ii)the amplitude damping channel ${{\rm{\Phi }}}_{{\rm{AD}}}$, $\begin{eqnarray*}\begin{array}{l}{{\rm{\Phi }}}_{{\rm{AD}}}(\rho )=\displaystyle \sum _{i=1}^{2}{B}_{i}\rho {B}_{i}^{\dagger },\quad {B}_{1}=| 0\rangle \langle 0| +\sqrt{1-q}| 1\rangle \langle 1| ,\\ {B}_{2}=\sqrt{q}| 0\rangle \langle 1| ;\end{array}\end{eqnarray*}$ |
| iii | (iii)the bit flip channel ${{\rm{\Phi }}}_{{\rm{BF}}}$, $\begin{eqnarray*}\begin{array}{l}{{\rm{\Phi }}}_{{\rm{BF}}}(\rho )=\displaystyle \sum _{i=1}^{2}{C}_{i}\rho {C}_{i}^{\dagger },\quad {C}_{1}=\sqrt{q}(| 0\rangle \langle 0| +| 1\rangle \langle 1| ),\\ {C}_{2}=\sqrt{1-q}(| 0\rangle \langle 1| +| 1\rangle \langle 0| ),\end{array}\end{eqnarray*}$ with $0\leqslant q\lt 1$. We compare the lower bounds $\overline{{LB}}$, LB, LB3, LB2, LB1 with the sum $={{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{\rm{AD}}})+{{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{\rm{PD}}})\,+{{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{\rm{BF}}})$ for $\alpha =\tfrac{1}{4}$, q = 0.3. It is shown that our new lower bound $\overline{{LB}}$ is tighter than LB, LB3, LB2 and LB1, see figure 2. |
Figure 2. (a) The solid black, solid red and solid blue curves represent the sum $=\,{{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{AD}})+{{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{PD}})+{{\rm{K}}}_{\rho }^{\tfrac{1}{4}}({{\rm{\Phi }}}_{{BF}})$ , the lower bounds (Lb) $\overline{{LB}}$ and LB, respectively. The dashed green, dashed blue and dashed magenta curves represent the lower bounds LB3, LB2 and LB1, respectively. (b-d) The solid green curve denotes the difference value (DV) between $\overline{{LB}}$ and LB, LB3 and LB2, respectively. |
4. Sum uncertainty relations for finite unitary channels
In [35] we introduced the (α, β, γ) MWWYD skew information of a unitary operator U,
$\begin{eqnarray}\begin{array}{rcl}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }(U) & = & -\displaystyle \frac{1}{2}\mathrm{Tr}([(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },{U}^{\dagger }][(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },U]{\rho }^{1-\alpha -\beta })\\ & = & \displaystyle \frac{1}{2}{\parallel \left[(1-\gamma ){\rho }^{\alpha }+\gamma {\rho }^{\beta },U\right]{\rho }^{\displaystyle \frac{1-\alpha -\beta }{2}}\parallel }^{2},\\ & & \alpha ,\beta \geqslant 0,\,\alpha +\beta \leqslant 1,\,0\leqslant \gamma \leqslant 1.\end{array}\end{eqnarray}$
Associated with a unitary channel ΦU(ρ) = UρU†, we denote ${{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }(U)$ the related quantity of the unitary channel ΦU. By employing $\parallel \left({\sum }_{t=1}^{N}{u}_{t}\right){\parallel }^{2}\,=\,2{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left({\sum }_{t=1}^{N}{U}_{t}\right)$, $\parallel {u}_{t}{\parallel }^{2}=2{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t})$, $\parallel {u}_{t}\pm {u}_{s}{\parallel }^{2}=2{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}\pm {U}_{s})$ and the inequalities (11 –13 ) , we have the following theorem.
Let ${U}_{1},\cdots ,{U}_{N}$ be N unitary operators, we have
$\begin{eqnarray}\displaystyle \sum _{t=1}^{N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t})\geqslant \max \{\overline{{Lb}}1,\overline{{Lb}}2,\overline{{Lb}}3\},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\overline{{Lb}}1 & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2L}{N(N-1)}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}+{U}_{s})}\right]}^{2}\right.\\ & & +M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}-{U}_{s})+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\\ & & \left.\times \,\left(\displaystyle \sum _{t=1}^{N}{U}_{t}\right)\right\},\end{array}\end{eqnarray}$
with $M\geqslant L\gt 0$, $\begin{eqnarray}\begin{array}{rcl}\overline{{Lb}}2 & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{2M}{N(N-1)}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}-{U}_{s})}\right]}^{2}\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}+{U}_{s})\\ & & \left.+(M-L){{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }\left(\displaystyle \sum _{t=1}^{N}{U}_{t}\right)\right\},\end{array}\end{eqnarray}$
with $L\geqslant M\gt 0$, $\begin{eqnarray}\begin{array}{rcl}\overline{{Lb}}3 & = & \displaystyle \frac{1}{{MN}+(N-2)L}\\ & & \times \,\left\{\displaystyle \frac{M-L}{{\left(N-1\right)}^{2}}{\left[\displaystyle \sum _{1\leqslant t\lt s\leqslant N}\sqrt{{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}+{U}_{s})}\right]}^{2}\right.\\ & & +L\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}+{U}_{s})\\ & & \left.+M\displaystyle \sum _{1\leqslant t\lt s\leqslant N}{{\rm{K}}}_{\rho ,\gamma }^{\alpha ,\beta }({U}_{t}-{U}_{s})\right\},\end{array}\end{eqnarray}$
with $L\gt M\gt 0$, $\alpha ,\beta \geqslant 0$, $\alpha +\beta \leqslant 1$ and $0\leqslant \gamma \leqslant 1$.For convenience we denote by $\widetilde{{Lb}}$ on the right-hand side of (39 ), $\widetilde{{Lb}}=\max \{\overline{{Lb}}1,\overline{{Lb}}2,\overline{{Lb}}3\}$ and compare the lower bound with ${Lb}=\max \{{Lb}1,{Lb}2,{Lb}3\}$ in [35]. In the example below we consider the MWWYD skew information with M = 2, L = 1 for $\overline{{Lb}}1$, and M = 1, L = 2 for $\overline{{Lb}}2$ and $\overline{{Lb}}3$.
Suppose that $\rho =\tfrac{1}{2}({{\rm{I}}}_{2}+{\boldsymbol{r}}\cdot {\boldsymbol{\sigma }})$ with ${\boldsymbol{r}}\,=(\tfrac{\sqrt{3}}{3}\cos \theta ,\tfrac{\sqrt{3}}{3}\sin \theta ,0)$. Consider the following three unitary operators generated by Pauli matrices,
$\begin{eqnarray*}\begin{array}{l}{U}_{1}={{\rm{e}}}^{\tfrac{{\rm{i}}\pi {\sigma }_{1}}{8}}=\left(\begin{array}{c}\cos \displaystyle \frac{\pi }{8}\ \mathrm{isin}\displaystyle \frac{\pi }{8}\\ \mathrm{isin}\displaystyle \frac{\pi }{8}\ \cos \displaystyle \frac{\pi }{8}\end{array}\right),\\ {U}_{2}={{\rm{e}}}^{\tfrac{{\rm{i}}\pi {\sigma }_{2}}{8}}=\left(\begin{array}{c}\cos \displaystyle \frac{\pi }{8}\ \sin \displaystyle \frac{\pi }{8}\\ -\sin \displaystyle \frac{\pi }{8}\ \cos \displaystyle \frac{\pi }{8}\end{array}\right),\\ {U}_{3}={{\rm{e}}}^{\tfrac{{\rm{i}}\pi {\sigma }_{3}}{8}}=\left(\begin{array}{c}{e}^{{\rm{i}}\displaystyle \frac{\pi }{8}}\quad 0\\ \ 0\ {e}^{-{\rm{i}}\displaystyle \frac{\pi }{8}}\end{array}\right),\end{array}\end{eqnarray*}$
which correspond to the Bloch sphere rotations of $-\tfrac{\pi }{4}$ around the x axis, the y axis and z axis, respectively. We compare the lower bounds $\widetilde{{Lb}}$, Lb with the sum $=\,{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{{\rm{AD}}})+{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{{\rm{PD}}})+{{\rm{K}}}_{\rho }^{\alpha }({{\rm{\Phi }}}_{{\rm{BF}}})$ via MWWYD skew information for $\alpha =\tfrac{1}{3}$ and $\alpha =\tfrac{1}{5}$, respectively. Our new lower bound $\widetilde{{Lb}}$ is tighter than Lb, see figure 3.
Figure 3. The solid black, red and green dashed curves represent the sum $=\,{{\rm{K}}}_{\rho }^{\alpha }({U}_{1})+{{\rm{K}}}_{\rho }^{\alpha }({U}_{2})+{{\rm{K}}}_{\rho }^{\alpha }({U}_{3})$, $\widetilde{{Lb}}$ and Lb, respectively. (a) $\alpha =\tfrac{1}{3};$ (b) $\alpha =\tfrac{1}{5}$. |
5. Conclusions
We have presented tighter uncertainty relations via (α, β, γ) WWYD skew information for multiple observables, quantum channels and unitary channels. By explicit examples, we have shown that our uncertainty inequalities are tighter than the existing results given in [34, 35]. Our results are also valid for the WY, WWYD and (α, γ) WWYD skew information as the special cases. Uncertainty relations give rise to fundamental limitations on quantum physical quantities, and our results may shed new light on the understanding of uncertainty relations and their applications in quantum information processing such as communication security.