Tighter sum uncertainty relations via metric-adjusted skew information

Uncertainty principle as a quintessential manifestation of quantum mechanics reveals the insights that distinguish quantum theory from classical theory. Heisenberg originally came up with the uncertainty principle 100 in 1927, which enunciates that the position and momentum of a particle cannot be determined simultaneously. Since then, the quantitative characterization of the uncertainty relation has received extensive attention, and many results have been obtained.

There are a host of methods to characterize the uncertainty principle, one of which is variance. This method, which was adopted by Robertson 101 and Schrödinger 102 , has been found that there exist lower bounds on the variances product for any two non-commuting observables. Subsequently, with regard to the sum of variance, the stronger uncertainty relations are provided 103 . And Wang $et~{}al.$ 104 verified the results in 103 through experiment. Later, some tighter uncertainty relations with respect to variance were given 105 ; 106 ; 107 ; 4 .

The other well-known method of characterizing uncertainty relation is entropy. Deutsch 110 first proposed a quantitative expression of the uncertainty principle by entropy for any two non-commuting observables, and then Maassen and Uffink 111 optimized the result in 1988. Furthermore, many scholars have put forward diverse uncertainty relations respecting distinct entropies 112 ; 121 ; 113 ; 114 . The uncertainty relations of entropy have numerous applications ranging from entanglement witnesses 16 ; 116 , quantum teleportation 117 , quantum steering 118 , quantum key distribution 15 ; 115 to quantum metrology 119 .

Recently, Luo 120 confirmed that skew information is an alternative approach to characterize the uncertainty relation. In 122 , Wigner and Yanase introduced the definition of skew information

Here $\rho$ and $M$ represent the quantum state and the observable, respectively. It can be considered as a measure and quantifies the information content included in the state $\rho$ regarding the conserved observables. Meanwhile, compared to the usual variance, it is better at times. The skew information, for pure states, is the same as the variance 125 , but they differ in mixed states. In the space of quantum states, skew information is convex, on the contrary for variance 125 , which is one of the remarkable differences between them. Later, Dyson put forth a quantitative way which is a generalization of skew information, its specific expression is

termed as Wigner-Yanase-Dyson skew information, and Lieb 123 resolved the convexity of this form on quantum states.

The sum of quantum uncertainty is crucial, because it is an effective tool for detecting quantum entanglement 126 ; 127 ; 128 ; 129 ; 130 . To this end, the sum of quantum Fisher information (QFI) which was defined by means of symmetric logarithmic derivative probably is superior to Wigner-Yanase skew information 128 , as in the quantum Cramér-Rao inequality. In 50 , Petz proposed the concept of monotone metric. After that, Hansen 51 further developed the notion of monotone metric which is metric-adjusted skew information, and QFI can be considered as a particular case of it. Consequently, we would like to further study tighter uncertainty relations regarding metric-adjusted skew information.

Quantum channel is essential in quantum theory. The uncertainty relation of channels has also been investigated extensively, and a large number of results have been yielded 2 ; 5 . Recently, some scholars generalized the uncertainty inequalities to metric-adjusted skew information for arbitrary finite quantum channels 1 ; 3 ; 27 .

The overall structure of this paper is as follows. In Sec. II, we recall the notion of metric-adjusted skew information. In Sec. III, we present some norm inequalities, and then new uncertainty relations of observables are given regarding metric-adjusted skew information. The distinct types of uncertainty relations of quantum channels as for metric-adjusted skew information are discussed in Sec. IV, and we proved that which of the two corresponding lower bounds obtained by the same norm inequality is better. At the same time, the above conclusions still hold for its special metrics. We also give several examples and compare the lower bounds obtained by us with the lower bounds in 3 ; 1 ; 27 . This more intuitively shows that our results are more accurate than the ones in 3 ; 1 ; 27 . The main conclusions are summarized in Sec. V.

Suppose that $M_{n}(\mathbb{C})$ is the set of all complex $n\times n$ matrices, $\mathscr{M}_{n}$ is the set of all positive definite $n\times n$ matrices with trace 1, where $n\in\mathbb{N}$. For any $A,B\in M_{n}(\mathbb{C})$, $\rho\in\mathscr{M}_{n}$, $K_{\rho}(\cdot,\cdot):M_{n}(\mathbb{C})\times M_{n}(\mathbb{C})\mapsto\mathbb{C}$ is termed as symmetric monotone metric 50 when it is content with the requirements

(i) $(A,B)\mapsto K_{\rho}(A,B)$ is sesquilinear, that is, the function $K_{\rho}(A,\cdot)$ is linear and $K_{\rho}(\cdot,B)$ conjugate linear.

(ii) $K_{\rho}(A,A)$ is nonnegative, $K_{\rho}(A,A)=0$ if and only if $A=0$.

(iii) $\rho\mapsto K_{\rho}(A,A)$ is continuous on $\mathscr{M}_{n}$.

(iv) $K_{T(\rho)}(T(A),T(A))\leq K_{\rho}(A,A)$ for any stochastic mapping $T$. A linear mapping $T:M_{n}(\mathbb{C})\rightarrow M_{m}(\mathbb{C})$ is called stochastic mapping if $T(\mathscr{M}_{n})\subset{\mathscr{M}_{n}}$ and $T$ is a completely positive.

(v) $K_{\rho}(A,B)=K_{\rho}(A^{\dagger},B^{\dagger})$.

The symmetric monotone metric $K_{\rho}(A,B)$ can be expressed as

where $L_{\rho}$ and $R_{\rho}$ are respectively left and right multiplication operators, $c$ is termed as Morozova-Chentsov function, and its form is

where the function $f$ is satisfied with the conditions: (a) $f:R_{+}\mapsto R_{+}$ is an operator monotone, where $R_{+}$ is the set of all positive real number, namely, if $A\geq B$, then $f(A)\geq f(B)$ for arbitrary $A,B>0$; (b) $tf(t^{-1})=f(t)$ for every $t>0$. Especially, it has been shown that if $K_{I/{n}}(I,I)=1$ holds, then the associated normalized function $f$ requires to admit $f(1)=1$. Here $I$ is $n$-dimensional identity operator.

In addition, in the space of quantum state, if the Morozova-Chentsov function associated with the symmetric monotone metric $K_{\rho}(\cdot,\cdot)$ satisfies

then $K_{\rho}(\cdot,\cdot)$ is known as regular 51 . $m(c)$ is called metric constant and $m(c)=f(0)$.

In 51 , Hansen introduced the metric-adjusted skew information $I_{\rho}^{c}(M)$ which is

where $c$ satisfies the constraint (5). Due to ${\rm i}[\rho,M]={\rm i}(L_{\rho}-R_{\rho})M$, then Eq. (6) can be rewritten as

where $\hat{c}(x,y)=(x-y)^{2}c(x,y)$, $x,y>0$.

When one chooses

and

the associated monotone metrics

and

are known as Wigner-Yanase metric and Wigner-Yanase-Dyson metric, respectively. Therefore, when $c=c^{\alpha}$, Eq. (6) turns into Eq. (2) which is Wigner-Yanase-Dyson skew information $I_{\rho}^{\alpha}(M)$. When $\alpha=\frac{1}{2}$, Eq. (2) reduces to Eq. (1) which is Wigner-Yanase skew information $I_{\rho}(M)$.

In this section, we first present some norm inequalities. By using these inequalities the new sum uncertainty relations of finite observables are given via metric-adjusted skew information, and the results also hold for its special metrics, such as those mentioned in Sec. II above. Then we provide two examples which show that our results are better than existing ones.

For finite $n$ observables $M_{1},M_{2},\cdot\cdot\cdot,M_{n}~{}(n>2)$, Cai 3 showed the uncertainty relation

Ren $et~{}al$. 1 gave an uncertainty inequality

Recently, Zhang $et~{}al$. 27 provided an uncertainty inequality

The inequalities (13) and (14) hold when $n\geq 2$. For simplicity, the lower bounds in (12), (13), and (14) are marked by $Lb_{1}$, $Lb_{2}$, and $Lb_{3}$, respectively.

Next we show various inequalities of the norm which are essential for the discussion of main results, so we take the inequalities as a Lemma.

Lemma 1. Suppose that ${\mathbf{x}}_{i}$ is a complex matrix, we can get

and

for arbitrary $m,l>0$, and

for $l>m>0$. Special we have

and

and

where $\|\cdot\|$ denotes the operator norm of a matrix.

Proof. By using the equations

and

we can derive that

for arbitrary $m,l\neq 0$ holds.

Then according to the inequality relations

we can get

and

for $m,l\geq 0$. Due to $\Bigg{\|}\sum\limits_{i=1}^{n}{\mathbf{x}}_{i}\Bigg{\|}^{2}\leq\frac{1}{(n-1)^{2}}\Bigg{(}\sum\limits_{1\leq i<j\leq n}\|{{\mathbf{x}}_{i}}+{{\mathbf{x}}_{j}}\|\Bigg{)}^{2}$, when $l>m>0$, we have

For special case $m=2,~{}l=1$, we obtain inequality (18). In the case $m=1,~{}l=2$, one gets inequalities (19) and (20). $\hfill\blacksquare$

When $n~{}(\geq 2)$ is determined, the larger $m$ and the smaller $l$, the bigger right side of inequalities (15) and (17), the larger $l$ and the smaller $m$, the bigger right side of inequality (16).

Note that for $m\geq l>0$ we have

and for $l\geq m>0$ one obtains

and for $l>m>0$ we get

Here the second inequality of (28) is strictly greater than when $m>l>0$, the second inequality of (29) and (30) is strictly greater than when $l>m>0$ and $n>2$. So the inequality (18) and the inequality in 5 have the following relation

and the inequality (19) and the inequalities in 2 ; 5 have the relation

and when $n>2$, the relation between the inequality (20) and the inequality in 28 is

These inequalities are helpful for us to explore the tighter uncertainty relations. Based on the inequalities (15)—(20), we give tighter sum uncertainty relations in the following Theorem.

Theorem 1. For finite $n$ observables $M_{1},M_{2},\cdot\cdot\cdot,M_{n}~{}(n\geq 2)$, the sum uncertainty relation with respect to metric-adjusted skew information is

Specially, we have

where the ineq37—ineq42 represent the lower bounds of inequality (37)—inequality (42), respectively.

Proof. Because the symmetric monotone metrics $K_{\rho}^{c}(\cdot,\cdot)$ are satisfied with the norm property, according to inequality (15), one has

for $m\geq l>0$. Multiply both sides of inequality (36) by a constant $\frac{f(0)}{2}$, we can derive

Using the similar procedure, for $l\geq m>0$ we can get

and for $l>m>0$ one reads

If we take $m=2$, $l=1$, the inequality (37) turns into

If we take $m=1$, $l=2$, the inequality (38) becomes

and the inequality (39) reduces to

For convenience, the lower bound of formula (34) is marked by $Lb$, that is, $Lb=\max\{\rm{ineq\ref{16},ineq\ref{15},ineq\ref{64}}\}$. $\hfill\blacksquare$

The following we illustrate that the lower bound obtained by us is stronger than the lower bounds in 3 ; 1 ; 27 . Because the uncertainty relations are derived by using the norm inequalities, according to the relationship between the norm inequalities presented in (28), (29), and (30), it is not difficult to show the lower bound $Lb$ obtained by us is more accurate than the lower bounds of inequalities (12), (13), and (14).

It is acknowledged that different results can be obtained by taking different Morozova-Chentsov functions for metric-adjusted skew information. Herein, we first consider the Morozova-Chentsov function with the form of Eq. (9). Meanwhile, the following results are obtained.

Corollary 1. For finite $n$ observables $M_{1},M_{2},\cdot\cdot\cdot,M_{n}~{}(n\geq 2)$, the sum uncertainty relations with respect to Wigner-Yanase-Dyson skew information are that for $m\geq l>0$ we can obtain

and for $l\geq m>0$ one derives

and for $l>m>0$ one reads