Multipartite uncertainty relation with quantum memory

We present a new quantum-memory-assisted entropic uncertainty relation for multipartite systems which shows the uncertainty principle of quantum mechanics. Notably, our results recover some well-known entropic uncertainty relations for two arbitrary incompatible observables that demonstrate the uncertainties about the results of two measurements. This uncertainty relation might play a critical role in the foundations of quantum theory.

where S(A|B) = S(ρ AB ) − S(ρ B ) is the conditional von-Neumann entropy of ρ AB with ρ B = tr A (ρ AB ) and S(ρ) = −tr(ρ log 2 ρ) is the von-Neumann entropy. Also, S(O|B) = S(ρ OB ) − S(ρ B ) with O ∈ {Q, R} is the conditional von-Neumann entropy of the post-measurement state after the quantum system A is measured, ρ QB = i ( q i A q i ⊗ I B )ρ AB ( q i A q i ⊗ I B ), likewise for ρ RB , and I B being an identity operator in Hilbert space of B.
In the literature, substantial efforts have been made to improve Berta et al. 's bound [9][10][11][12][13][14][15][16][17] . To be explicit, Pati et al. 9 improved Berta et al. 's bound by a term added to the right-hand side of inequality (2). Indeed, Pati et al. 's bound is tighter than Berta et al. 's bound if the quantum discord is larger than the classical correlation. Then, Pramanik et al. 10 obtained a new entropic uncertainty relation based on fine graining, which led to an ultimate limit on the accuracy achievable in measurements made on two incompatible observables in the presence of  15 improved the lower bounds for the entropic uncertainty relations via polynomial functions. Besides, Huang et al. 16 presented a Holevo bound for QMA-EUR, where the difference between the entropic uncertainties and the new bound is always a fixed value. More recently, Li and Qiao 17 proposed a method to decrease the local uncertainty. In this remarkable study a new kind of uncertainty relation based on conditional majorization [18][19][20] has been formulated, which can be calculated for any number of observables. According to this new class of uncertainty relation and in the presence of quantum memory, one can get lower bounds in comparison to the conditional entropic uncertainty relation. On the other hand, continuing progress has been done by some groups from an experimental viewpoint [21][22][23][24][25][26][27][28][29][30][31] . Furthermore, a promising effort was made by Adabi et al. 32 Interestingly, the bipartite QMA-EUR has been the topic of many works in recent years [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] (see Refs. 49 and 50 for detailed reviews on bipartite QMA-EUR). However, Renes and Boileau 51 showed that the bipartite QMA-EUR could be generalized to the tripartite case where two particles B and C are considered as the quantum memories. It is shown that the tripartite QMA-EUR can be written as In this scenario, Alice, Bob, and Charlie share a tripartite quantum state ρ ABC and then Alice carries out one of two observables (Q and R) on her system. Briefly, if Alice measures Q, then Bob's task is to minimize his uncertainty about Q and whenever she measures R, then Charlie's task is to minimize his uncertainty about R. Most recently, Ming et al. 52 improved the tripartite uncertainty bound of the inequality (4) as where Subsequently, Dolatkhah et al. 53 introduced a new lower bound for tripartite QMA-EUR that is tighter than Ming et al. 's bound. The new inequality can be derived by with Within the above, it is obvious that the studies only have been focused on the case of two measurements (Q and R). However, many attempts have been made to generalize the entropic uncertainty relations to more than two observables [54][55][56][57][58][59][60][61][62][63] . Here, one can refer to the result of Liu et al. 55 that they considered Maassen and Uffink bound for multi-observable ( > 2 ) with the state of the measured system A ≡ ρ A which is generally a mixed state, viz in which Furthermore, this inequality in the presence of quantum memory B is converted to 55 Until now, the inequalities revealed only for bipartite, tripartite, and multi-measurement cases, while the case of multipartite systems remains unstudied. In this paper, we will present a novel entropic uncertainty with quantum memory for multipartite systems where the memory is split into several parts. Herein, we highlight this relation    www.nature.com/scientificreports/ for its key role in quantum theory and potential wide applications, as well as expect that this uncertainty relation can be demonstrated in various physical systems.

Generalization of QMA-EUR
The following theorem reveals how to obtain the QMA-EUR for multipartite systems.   (14), one arrives at which can be rewritten as the desired outcome (12).

Corollary 1 If the prepared state is a bipartite state, our uncertainty relation will recover Adabi et al. 's result.
Proof For any bipartite state, we consider two observables ( O 1 = Q and O 2 = R ) and P 1 = P 2 = B . Therefore, from Eq. (12) we obtain the Eq. (3) with the complementarity b = c and κ = δ.

Corollary 2 If the prepared state is a tripartite state, our uncertainty relation recover Dolatkhah et al. 's result.
Proof For any tripartite state, we consider two observables ( O 1 = Q and O 2 = R ) and P 1 = B and P 2 = C . So, we restore the Eq. (7) with the complementarity b = c and κ = δ ′ .

Example: four-partite QMA-EUR
Let O 1 , O 2 , and O 3 be three incompatible observables for a four-partite system ρ ABCD which is generally a mixed state. The following four-partite uncertainty relation holds (with P 1 = B , P 2 = C , and P 3 = D)

Conclusion
In summary, we have presented a generalized uncertainty relation with quantum memory for multipartite systems and obtained a new QMA-EUR for four-partite quantum systems. This generalized entropic uncertainty depends on the conditional von-Neumann entropies, Holevo quantities, and the mutual information. We expect that the inequality will bring on more potential applications in quantum information and communication, e.g., entanglement detection 64 , multipartite entanglement-structure detection 65 , witnessing multipartite entanglement 66 , detection of genuine multipartite entanglement in multipartite systems 67 , exploring the efficient multipartite entanglement criteria [68][69][70] , analyzing the monogamy and polygamy relations of multipartite quantum states 71,72 , and so on. It means that our multipartite uncertainty relation will have significant applications in entanglement detection and precision measurements. In a forthcoming paper, one may motivate to extend the results to get an uncertainty relation for multipartite systems based on conditional majorization in comparison to recent study 17 .