Generalized uncertainty relations for multiple measurements

The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.


Introduction
It is widely known that the uncertainty principle proposed by Heisenberg in 1927 [1] is one of the most elemental and important characteristics in the matter of quantum mechanics, which distinguishes from the classical counterpart. Canonically, the uncertainty principle suggests that one cannot accurately predict the momentum p and position x of a particle simultaneously. Later, Kennard [2] had proven and optimized the position-momentum uncertainty relation, and Robertson [3] further formulated the uncertainty principle via the standard deviation for two arbitrary incompatible observables R and S in the given system , expressed as [4] R S 1 2 R, S .
( 1 ) Apparently, the lower bound of Robertson's relation depends on state as shown in Eq. (1). As a matter of fact, a trivial result, that the bound becomes zero, will take place if the systemic state is prepared in one of eigenstates of the two observables R and S. With the advent of quantum information theory, Deutsch imposed Shannon entropy as an alternative measure of uncertainty and proposed the so-called entropic uncertainty relations (EURs) [5]. Technically, EUR is also regarded as the achievement of the combination of quantum mechanics and classical information theory.
where H R i p i log 2 p i represents the Shannon entropy and p i i i . The maximal overlap © The Author(s). 2022 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. c R, S max j,k j k 2 with j and k respectively referring to the eigenvectors of R and S. Due to the fact that c R, S is correlated to the two observables themselves, it clearly demonstrates that the lower bound of EUR is state-independent.
Recently, quantum-memory-assisted entropic uncertainty relation (QMA-EUR) has been proposed by Renes et al. [8] and Berta et al. [9] for a bipartite system AB. Where, S R B S RB S B is denoted as the conditional von Neumann entropy [10,11] of post-measurement states (ii) when the measured particle A and the memory particle B are entangled, the bound can be reduced because that the conditional von Neumann entropy S A B can be negative. Specifically, if A and particle B are maximally entangled, we have S R B S S B 0 on account of S A B log 2 d (d is the dimension of the measured particle), which reflects that Bob is able to predict perfectly Alice's measured outcomes; (iii) if the quantum memory is absent, Eq. (3) becomes H R H S S A log 2 c R, S , which yields a tighter bound due to S A 0. As far as the QMA-EUR is concerned, it can be applied to a number of quantum tasks [12][13][14][15][16][17][18] including quantum teleportation [12], quantum key distribution [13], entanglement witness [9], quantum metrology [14,15], quantum steering [16] and so forth. Additionally, some promising improvements had been made on QMA-EUR [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. To be specific, by considering the second largest value of the overlap c R, S , Coles and Piani [26] presented the uncertainty relation with a tighter bound. In 2015, Liu et al. [31] presented the uncertainty relation of multi-observable scenario. In 2016, Adabi et al. [32] optimized the lower bound by adding mutual information and Holevo quantity. Huang et al. [33] proposed a Holevo bound of QMA-EUR. More recently, Xie et al. [34] improved the lower bound of the entropic uncertainty relation for multiple measurements in bipartite systems. Until now, several groups have dedicated to make sustained progress in terms of experiments [36][37][38][39][40][41][42][43]. Note that, the tripartite EUR had been proposed by Renes and Boileau [44] This inequality can be interpreted by the so-called monogamy game. Very recently, Ming et al. [45] put forward a tighter tripartite EUR compared with Renes-Boileau's result. Within the above, with mutual information A : B S A S B S AB and Holevo quantity R : B S B i p i S B i . One performs measurement R on particle A and obtains the i-th measurement outcome with probability p i Tr AB A i AB A i . The above relations are suitable for issuing the cases in two or three-particle systems. While, the correlated many-body systems are usual and are frequently required to achieve the realistic quantum information processing, where the mentioned relations are ineffective. Especially, EUR takes an irreplaceable role in the security analysis of quantum key distribution within the multipartite setting. In this sense, we here would like to raise an open question: how to characterize the measurement uncertainty with respect to multipartite systems? Inspired by this, we will focus on addressing this issue in this article.

Generalized EUR
Theorem 1 In the case of multipartite systems, the generalized entropic uncertainty relation for multi-observable measurements can be described as where O i represents the i-th operator measured on subsystem A, and B i denotes the i-th quantum memory other than particle A in multipartite system.
Proof Based on the tripartite EUR expressed in Eq. (6), one can write m m 1 2 inequalities for m-observable measurements in m 1 -party system Next, all these m m 1 2 inequalities are summed and both sides of the total inequality is divided by the quantity m 1 . As a consequence, we have Likewise, by making use of Renes-Boileau's result in Eq. and achieve the optimization of the two inequalities, i.e., taking the maximum between m and zero. We thereby obtain the expected generalized EUR for multi-observable in multipartite systems expressed as Eq. (8).

Examples
We have already obtained the generalized EUR for multi-observable in multipartite systems, and for the sake of illustrations, we here present some meaningful observables in practical quantum information processing. Assuming two groups of orthogonal bases where by virtue of Pauli measurement , , , where B MU1 log 2 c , , B MU2 log 2 c , , B MU3 log 2 c , . It is apparently that all is equal to 1, and 3 2 is obtained. In the following, one can apply our result to the case of three measurements, and obtain a tighter bound from the following relation Now, we turn to compare our proposed bound with the previous derived from Renes-Boileau's result in the case of different four-qubit state scenarios.

Mixed GHZ-type states
First of all, we take into account a type of mixed GHZtype states. Generally, a four-qubit GHZ-type state can be written as 1234 GHZ GHZ 1 16 1 16 16 , (17) with the purity of the state [ 0, 1] and GHZ cos 0000 sin 1111 with [ 0, 2 , and 1 16 16 denotes an identity 16 16 matrix. In this scenario, we plot the uncertainty, our bound and Renes-Boileau's bound in Eq. (15) as Fig. 1a and b, with 4 and 0.5, respectively. It is straightforward to display that our derived bound is higher than Renes-Boileau's one, i.e., U RB U O U, which is reflecting that our bound outperforms the previous.

W-type states
In addition, we proceed by considering a type of W-type four-qubit states, expressed as in the Hilbert space spanned by 0 , 1 , with [ 0, 2 . In this scenario, we have computed the uncertainty, our bound and Renes-Boileau's result, which have been numerically indicated in Fig. 2. Following the figure, it is easy to realize that our bound is stronger than the previous one.

Random four-qubit states
To verify our obtained results, we consider more general states, i.e., arbitrary sets of random four-qubit states, which in principle contain both pure and mixed states.
To begin with, let us introduce an effective approach to generate arbitrary random four-qubit states. Actually, arbitrary random four-qubit states can be expressed as the form of 16 n 1 n n n , where n and n are regarded as the n-th eigenvalues and the eigenstates of . Incidentally, the eigenvalue n quantifies the probability that the systematic state is in the pure state n . Next, one can set up an arbitrary unitary operation by using the normalized eigenvector n . An arbitrary fourqubit state also can be composed by arbitrary probabilities n and arbitrary . As a result, one can construct arbitrary four-qubit states by making use of the arbitrary set of probabilities and arbitrary unitary operation. In what follows, we utilize the random function x 1 , x 2 to randomly generate a real number in a given interval [ x 1 , x 2 ]. Generally, one can generate random probabilities P n by the following method where i 1, 2, 3, , 15 . Then a set of random probabilities Based on Eqs. (19) and (20), one can obtain a set of probabilities in descending order. By way of constructing random unitary operation, we randomly generate one 16-order real matrix by the Fig. 3 Uncertainty and our derived bound in regard to 1.5 10 5 four-qubit random states. X-axis represents the lower bound proposed by us and Y-axis represents the uncertainty, respectively. Green line stands for the proportion function with the unitary slope random function 1, 1 with the given interval [ 1,1]. Through using the real matrix , a random Hermitian matrix is given by where , and represent diagonal, strictly lower-and strictly upper-triangular part of the real matrix , respectively, and T represents the transposition of the matrix .
Following this calculation, one can get sixteen normalized eigenvectors n of the Hermitian matrix , which forms the random unitary operation . With the above procedures, one can perfectly construct random four-qubit quantum states by the expression 16 n 1 n n n . In order to testify our result, we take 1. 5 10 5 random states to show the uncertainty and our bound as shown in Fig. 3. It is straightforward to demonstrate that the uncertainty U is great than or equal to the lower bound proposed by us, which also supports that our result is available for the arbitrary random states. In this sense, we claim that our result in Eq. (8) is universal. Moreover, we also compare our bound with Renes-Boileau's bound with regard to randomly generated four-qubit states in Fig. 4. It is obvious that our bound is higher than Renes-Boileau's bound, illustrating that our derived inequality is optimal.

Conclusions
To summarize, we have derived general entropic uncertainty relations for multiple-observable measurements in multipartite correlated systems, which is essentially deemed as a universal architecture in quantum information theories. Herein, we employ three two-dimensional Pauli measurement , , and as the incompatibility in the framework of arbitrary four-qubit systems, including GHZ-type states, W-type states and random four-qubit states. It turns out that our derived bound is tighter than Renes-Boileau's one, which shows that our generalized EUR is optimal. In this regard, it is believed that our result will be instrumental in various measurement-based quantum information processing, especially quantum precision measurements [34] and improving quantum secret key rate [46] in multipartite systems. Hence, we argue that our explorations would pave the avenue on quantum measurement estimation for multiple observables within many-body systems in the area of quantum information science.