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Experimental Investigation of Entropic Uncertainty Relations and Coherence Uncertainty Relations

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Experimental Investigation of Entropic Uncertainty Relations and Coherence Uncertainty Relations Experimental investigation of entropic uncertainty relations and coherence uncertainty relations Zhi-Yong Ding,1, 2, 3 Huan Yang,1, 4, 5 Dong Wang,1, 6, ∗ Hao Yuan,1, 6, 7 Jie Yang,1 and Liu Ye1, y 1School of Physics and Material Science, Anhui University, Hefei 230601, China 2School of Physics and Electronic Engineering, Fuyang Normal University, Fuyang 236037, China 3Key Laboratory of Functional Materials and Devices for Informatics of Anhui Educational Institutions, Fuyang Normal University, Fuyang 236037, China 4Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China 5Department of Experiment and Practical Training Management, West Anhui University, Lu’an 237012, China 6CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China 7Key Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education, Anhui University, Hefei 230601, China (Dated: December 19, 2019) Uncertainty relation usually is one of the most important features in quantum mechanics, and is the back- bone of quantum theory, which distinguishes from the rule in classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states with high fidelity, i.e., Bell-like states and Bell-like diagonal states, we carry on local projective measurements over a complete set of mutually unbiased bases on the measured subsystem. In terms of quantum tomography, the density matrices of the initial states and the post-measurement states are reconstructed. It shows that our experimental results coincide with the theoretical predictions very well. Additionally, we also verify that the lower bounds of both the entropy-based and coherence-based uncer- tainty can be tightened by imposing the Holevo quantity and mutual information, and the entropic uncertainty is inversely correlated with the coherence. Our demonstrations might offer an insight into their uncertainty re- lations and their connection to quantum coherence in quantum information science, which might be applicable to the security analysis of quantum key distributions. I. INTRODUCTION Notably, the entropy-based uncertainty relation mentioned previously is suitable for observing the law of incompatible Heisenberg’s uncertainty principle [1] is deemed as one of measurements in a single-particle system. As to a composite the most fundamental features of quantum world, which es- system ρ^AB with A to be probed and B as a quantum memory, sentially is different from classical world. The uncertainty Berta et al.[8] had put forward the quantum-memory-assisted principle manifests that it is impossible to accurately predict entropic uncertainty relation, which is given by both the position and the momentum of a microscopic particle. 1 S(P jB) + S(QjB) ≥ log + S(AjB); (2) In other words, the more accurate the position measurement is, 2 c the more uncertain the momentum measurement will be, and vice versa [2,3]. Before long, Robertson [4] generalized the where S(AjB) = S(^ρAB)−S(^ρB) represents the conditional uncertainty principle to an arbitrary pair of non-commuting von Neumann entropy of ρ^AB and ρ^B = TrAρ^AB is the re- observables. Notably, there is a conceptual shortcoming in duced state, S(P jB) = S(^ρPB) − S(^ρB) denotes the con- Robertson’s inequality, leading to a trivial result when the sys- ditional von Neumann entropy of the post-measurement state temic state is in the state of one of eigenstates of the measured ρ^PB, which can be obtained by performing the local projec- observables. Afterwards, Deutsch had established the cele- tive measurements fjpii hpijg on the subsystem A. Subse- brated entropic uncertainty relation (EUR) [5,6], in terms of quently, Li et al. [9] and Prevedel et al. [10] experimentally Shannon entropy, which was improved by Maassen and Uffink demonstrated the quantum-memory-assisted entropic uncer- arXiv:1912.08457v1 [quant-ph] 18 Dec 2019 into the form of [7] tainty relation through two back-to-back all-optical configu- rations. 1 Technically, EURs had yielded many potential applications H(P ) + H(Q) ≥ log ; (1) 2 c in the domain of quantum information science, e.g., quantum key distribution [8,9], entanglement witness [8–11], quantum where P and Q are two arbitrary observables, H(·) denotes teleportation [12], quantum steering [13, 14], and quantum 2 the Shannon entropy, and c = maxi;jjhpijqjij stands for the metrology [15]. Accordingly, EURs has received much atten- maximum overlap between any two eigenvectors (fjpiig and tion and several tighter lower bounds of the uncertainty were fjqjig) of P and Q. proposed by means of various methods [11, 16–19]. Later on, the entropic uncertainty relations are generalized to multi- observable versions [20, 21]. Soon afterwards, Xing et al. [22] reported an experimental investigation of entropic uncer- ∗ [email protected] (D. Wang) tainty relations with regard to three measurements in a pure y [email protected] (L. Ye) diamond system. In addition, there exhibit some promising 2 experiments to verify different types of uncertainty relations In general, the unilateral coherence of the bipartite system [23–26]. can be written as [32, 33] On the other hand, quantum coherence, the embodiment M of the superposition principle of quantum states, is also one Cre (^ρAB) = S(^ρMB) − S(^ρAB): (5) of the most fundamental features that marks the violation of quantum mechanics from the classical world [27–30]. It has Furthermore, for a bipartite state ρ^AB and N projective been widely concerned and studied since Baumgratz et al. M = f um um g (m = 1; ··· ;N) measurements m im im per- [31] established a rigorous framework for the quantification formed on subsystem A, the entropic uncertainty relations of coherence. Generally, quantum coherence is related to the with regard to multiple measurements can be described by characteristics of the whole system. In order to reveal the re- [20] lationship between quantum coherence and quantum correla- N tions, the unilateral coherence [32, 33] is introduced for a bi- X 1 S(M jB) ≥ log + (N − 1)S(AjB); (6) partite system. Intrinsically, quantum coherence of a quantum m 2 b state depends on the choice of the reference basis. If two or m=1 more incompatible reference bases are selected, coherence- based uncertainty relations (CURs) can be established [34– here b = max PiN−1 max[c(u1 ; u2 )] QN−1 c(um ; um+1 ) i2 i1 i2 m=2 im im+1 37]. Soon after, Lv et al. [38] demonstrated an all-optical iN i1 c = max c(u1 ; u2 ) = max j u1 u2 j2 experiment of CURs in two different reference bases. with i1 i2 i1 i2 denoting the i1;i2 i1;i2 In this paper, we focus on achieving experimental investiga- largest overlap between any two eigenvectors of the given tions with respect to EURs and CURs in an all-optics frame- observables. According to Eqs. (5) and (6), we can obtain the work. By means of preparing two kinds of bipartite initial uncertainty relations for unilateral coherence under multiple states, i.e., Bell-like states (pure) and Bell-like diagonal states measurements [36], viz. (mixed), we then perform local projective measurements over a complete set of mutually unbiased bases (MUBs) on one N X 1 of the subsystem [39, 40]. By utilizing quantum tomogra- CMm (^ρ ) ≥ log − S(AjB); (7) re AB 2 b phy [41], we obtain the density matrices of the initial states m=1 and the post-measurement states, as well as the corresponding measurement probability. It shows our experimental results Specifically, for a d × d-dimensional bipartite state ρ^AB, if coincide with the theoretical predictions very well. Moreover, we select a complete set of d + 1 MUBs applied on subsystem we also demonstrate that the lower bounds of both the en- A, the EURs and CURs in MUBs can be derived into tropic and coherence-based uncertainty can be strengthened d+1 based on the Holevo quantity and mutual information. X S(MmjB) ≥ log2d + d · S(AjB); (8a) m=1 II. THEORETICAL FRAMEWORK d+1 X Mm Cre (^ρAB) ≥ log2d − S(AjB); (8b) m=1 Briefly, we herein review the EURs for multiple measure- ments, and then render a theoretical framework of the CURs. respectively. Note that, the lower bound of the EURs can be For a quantum state ρ^ in a d-dimensional Hilbert space H, strengthened by adding a term relating to the Holevo quantity once resorting to an arbitrary set of basis fjiig, the relative and mutual information [18, 36] entropy of coherence can be written as [31] d+1 X (i) (i) S(M jB) ≥ log d + d · S(AjB) + Maxf0; δg; (9) Cre (^ρ) = S(^ρdiag) − S(^ρ); (3) m 2 m=1 where S(^ρ) = −Tr(^ρ logρ ^) is the von Neumann entropy, and Pd+1 (i) P where δ = (d + 1)I(A : B) − m=1 I(Mm : B), I(A : ρ^diag = i hij ρ^jii jii hij is the diagonal part of ρ^ in the basis B) = S(^ρA) + S(^ρB) − S(^ρAB) is the mutual information fjiig. For a bipartite state ρ^AB in a composite Hilbert space of the bipartite system ρ^AB, and I(Mm : B) = S(^ρB) − HA ⊗ HB, after performing a local projective measurement P pi S(^ρB ) is the Holevo quantity [42], which is an up- M = fju i hu jg on subsystem A, the probability of obtain- im m m i i per bound on the accessible information by the local projec- ing result i is quantified by p = Tr[(ju i hu j⊗ )^ρ ] and i i A i IB AB tive measurement on the subsystem A.
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