Entropic No-Disturbance As a Physical Principle

Entropic No-Disturbance as a Physical Principle

Zhih-Ahn Jia∗,1, 2, † Rui Zhai*,3, ‡ Bai-Chu Yu,1, 2 Yu-Chun Wu,1, 2, § and Guang-Can Guo1, 2 1Key Laboratory of Quantum Information, Chinese Academy of Sciences, School of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China 2Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China 3Institute of Technical Physics, Department of Engineering Physics, Tsinghua University, Beijing 10084, P.R. China The celebrated Bell-Kochen-Specker no-go theorem asserts that quantum mechanics does not present the property of realism, the essence of the theorem is the lack of a joint probability distributions for some experiment settings. In this work, we exploit the information theoretic form of the theorem using information measure instead of probabilistic measure and indicate that quantum mechanics does not present such entropic realism neither. The entropic form of Gleason’s no-disturbance principle is developed and it turns out to be characterized by the intersection of several entropic cones. Entropic contextuality and entropic nonlocality are investigated in depth in this framework. We show how one can construct monogamy relations using entropic cone and basic Shannon-type inequalities. The general criterion for several entropic tests to be monogamous is also developed, using the criterion, we demonstrate that entropic nonlocal correlations are monogamous, entropic contextuality tests are monogamous and entropic nonlocality and entropic contextuality are also monogamous. Finally, we analyze the entropic monogamy relations for multiparty and many-test case, which plays a crucial role in quantum network communication.

Introduction.—It is common to assert that quantum than light. It has been shown that, in no-disturbance mechanics (QM) is beyond the classical conception of (nonsignalling) framework, contextuality (nonlocality) nature. There are many phenomena which indicate that tests are monogamous [14, 15, 17, 18], i.e., if one physi- classical description of our universe is not suit for quan- cist observes the contextuality (nonlocality) in an exper- tum world, for example, the uncertainty principle [1], iment, then the other one must not observe the phe- principle of complementarity [2,3], nonlocality [4–6], nomenon simultaneously. contextuality [7,8], negativity of quasi-probability [9] To understand these novel properties of quantum prob- and so on. Among which contextuality and its more abilities and find out the boundary between classical and restricted form, nonlocality, attract many interests for quantum theory, we need to explore the notions from as both their theoretical importance and broad applications. many aspects as possible. Sheaf theoretic [19, 20] and The works of Bell and of Kochen and Specker indicated graph theoretic [18, 21, 22] approaches have been well that quantum probabilities obtained from physical sys- exploited. The information theoretic method provides a tems with dimension greater than two are incompatible crucial and useful alternative, where the basic objects are with local hidden variable (LHV) model [10] and non- information entropy of the probabilities arising in a given contextual hidden variable (NCHV) models [7], which is theory. Entropy is a crucial concept of thermodynamics, now known as Bell-Kochen-Specker (BKS) theorem. The statistical mechanics and classical and quantum infor- essence of BKS theorem is that we can not preassign val- mation theory [23, 24]. It was Braunstein and Caves [25] ues to measurements in some consistent way, thus we can who first introduced the entropic form of Bell-Clauser- not always obtain a joint probability distributions for all Horne-Shimony-Holt (CHSH) nonlocality tests [4,5, 26], involved measurements [11]. then the entropic form [27] of Klyachko-Can-Binicio˘glu- One of the most important properties of quantum Shumovsky (KCBS) contextuality tests [28] were also ex- arXiv:1803.07925v1 [quant-ph] 21 Mar 2018 probabilities is Gleason’s no-disturbance principle [12– plored. Chaves, Fritz and Budroni [29–31] developed the 15], which asserts that the outcome of a measurements approach in a more systematic way using the entropic A does not depend on the (prior or simultaneous per- cone and entropic inequalities [32] , they also developed formed) compatible measurement B and vice versa, QM the entropic description of nonsignalling correlations. is in accordance with the no-disturbance principle [16]. Despite of all these progresses, many works still need When A, B are implemented by two spatially separated to be done. There is no existing information framework parties, the no-disturbance principle is also known as for Gleason’s no-disturbance principle and the entropic nonsignalling principle, which characterizes the prop- form of Bell-Kochen-Specker theorem is less explored. erty that information can not be transformed faster The relationship between entropic tests and usual tests of contextuality and nonlocality is not very clear neither. Moreover, the monogamy relation, which is a characteristic feature of nonlocality and contextuality remains ∗Two authors are of equal contributions. largely unexplored. In this work, we develop the informa- 2 tion framework of Gleason’s no-disturbance principle, the strong subadditivity ( or submodularity) I(A : B|C) = concepts of entropic nonlocality and entropic contextual- H(AC) + H(BC) − H(ABC) − H(C) ≥ 0, where we ity are introduced and their properties are extensively in- use notation A, B, C to represents some special measure- vestigated in this framework, and we present the entropic ments or collection of compatible measurements. It is form of Bell-Kochen-Specker theorem. Using the tools worth mentioning that Shannon-type inequalities are not of entropic vectors, entropic cone and computational ge- sufficient to completely characterize ΓE for |M| ≥ 4 [32], ometrical method like Fourier-Motzkin elimination, we they only give an outside approximation ΓSE , but they demonstrate the monogamy relations of entropic nonlo- are the most explored and most useful ones and have cality and entropic contextuality. Finally, we explore the been applied in many separated areas, including group method to construct new monogamy relation from exist- theory [32], Kolmogorov complexity [40]. ing ones, and we extend the entropic monogamy relations Entropic Bell-Kochen-Specker theorem.—Given a set to multiparty and many-test scenario, which will play an of measurements M = {A1, ··· ,An}, from QM we know crucial role in realization of quantum network communi- that only compatible measurements are jointly measur- cation. able. We introduce the notion of compatible graph Information measure of quantum probabilities.—In [15] G(M) for M, where each measurement is repre- generalized probability theory (which contains QM as a sented as a vertex, and if two measurements are com- special case), a measurement can be regarded as some patible, the corresponding vertices are connected by apparatus which accepts an input system and output an edge. The Bell-Kochen-Specker (BKS) scenario [4– random variables. If two measurements A and B are 7, 29] C(M) for M is defined as C(M) = {C1, ··· , Ck} compatible, then they admit a joint probability distribu- where Ci ⊂ M and all measurements Ai1 , ··· ,Aim ∈ tion p(a, b|A, B) where a, b are random outcomes of A, B Ci are pairwise compatible (they form a complete sub- respectively, and p(a|A), p(b|B) can be reproduced from graph of G(M)), joint probability distributions p(Ci) = p(a, b|A, B) by taking marginal distributions. To know p(ai1 , ··· , aim |Ai1 , ··· ,Aim ) are experimentally accessi- how much information we can obtain from the physical ble, and for any Ci ⊂ Cj, p(Ci) can be reproduced system when we are measuring quantitatively, we need to from p(Cj) by taking marginal distribution. Collect- introduce the concept of information measures which are ing these probabilities together, we will have a vector various types of entropies and its siblings [32]. Here we pobs = [p(Ck)]Ck∈C(M) which we refer to as the be- use the terminology information measures to refer to en- havior of M following the terminology introduced by tropy, conditional entropy, mutual information and con- Tsirelson [6, 41]. If there exists a joint probability dis- ditional mutual information. The importance of infor- tribution p(M) = p(a1, ··· , an|A1, ··· ,An) for all mea- mation measures not only reflects in the widespread use surements in M from which all experimentally accessi- in classical and quantum information theory [32–34], but ble probabilities can be reproduced as marginal distribu- also in statistical mechanics [35, 36], additive combinations, then we get a 2n-dimensional probabilistic vector torics [37], biodiversity studies [38] and so on. p = [p(∅), p(S1), ··· , p(M)] [42] where Sk are subsets of For a given set of pairwise compatible measurements M and p(Sk) are marginal distributions of p(M)[43–45]. M, there exists a joint entropy H(M), and for any sub- For a BKS scenario with observed probabilistic vector set S ⊆ M, there also exists a joint entropy H(S) which pobs, we know from Fine’s theorem [11] that the exis- can be reproduced from H(M). We now in a position tence of NCHV/LHV model for a pobs is equivalent to to introduce the concepts of entropic vectors and en- the existence of a joint probability distribution for all in- M tropic cones [29, 31, 32]. Let 2 be the power set of volved measurements. Thus observed data pobs is called M, the entropic vector is of the form h = [H(S]S∈2M non-contextual if it is consistent with a single well de- where we make the convention that H(∅) = 0. For a fined probability distribution p(A1, ··· ,An) for all mea- given set of measurements, since (i) 0 is entropic; (ii) surements. This further implies the existence of a prob- 0 0 h, h are entropic implies that ph + (1 − p)h and αh abilistic vector p which can project to pobs. This is the are entropic [32], all entropic vectors form a cone named well-known probabilistic vector formalism of contextual- entropic cone, denoted as ΓE. Since every closed con- ity and nonlocality. vex cone can be characterized by a system of linear in- Now, let us see how to understand contextuality and equalities, we may want to seek the precise linear in- nonlocality using information measures of quantum prob- equalities which bound it. Notwithstanding, to give an abilities. Suppose we are implementing measurements M explicit characterization of ΓE is still an open problem, with context set C(M), then the joint entropy H(Ck) for an outer approximation is known, which is Shannon en- each Ck ∈ C is experimentally accessible. Collecting all tropic cone ΓSE and is characterized by Shannon-type these data together we get an experimentally accessible inequalities [39], clearly speaking, the nonnegative lin- entropic vector hobs = [H(Ck)]Ck∈C(M). The existence ear combinations of the following inequalities: (i) mono- of joint probability distribution p(M) imposes a strict tonicity H(A|B) = H(AB) − H(B) ≥ 0; (ii) subaddi- constraints, contextual inequalities, on the experimen- tivity I(A : B) = H(A) + H(B) − H(AB) ≥ 0; (iii) tally accessible distributions pobs. We can define entropic 3

contextuality in a similar sense, hobs = [H(Ck)]Ck∈C(M) is called entropic non-contextual or entropic nonlocal if all entropies HCk are consistent with a single well deﬁned shannon entropy HM, i.e., there exist an entropic vec-

M tor h = {H(Sk)}Sk∈2 which can project to hobs. As have been proved for single system [27, 29] and for composite system [25], QM is inconsistent with the entropic NCHV/LNV models, we refer to such kind of contextuality and nonlocality as entropic ones. Therefore we obtain the following entropic Bell-Kochen-Specker theorem: FIG. 1: Pictorial illustration of the non-disturbance (ND) or nonsignaling (NS) cone, quantum mechanical (QM) cone Theorem 1 (Entropic Bell-Kochen-Specker). For phys- and non-contextual hidden variable (NCHV) or local hidden ical systems of dimension greater than two, there is no variable (LHV) cone. consistent way to assign an corresponding entropic vectors h(M) which can project to the observed entropic NCHV/LHV entropic cone, that is there exist some ob- vectors hobs(M) for all set of measurements M. served entropic vectors hobs which exhibit entropic con- We are now in a position to give some criteria to de- textuality but can not be detected in this way. Never- tect the entropic contextuality. Suppose that we have a theless, entropic cone obtained in this way is a good and set of n measurements M = {A1, ··· ,An} and the BKS useful approximation. scenario is C(M) = {{A1,A2}, {A2,A3}, ··· , {An,A1}}, Entropic no-disturbance principle.—No-disturbance i.e., each Ai and Ai+1 pair is compatible, thus there principle asserts that compatible measurements do not is a joint entropy H(Ai,Ai+1). If hobs is entropic disturb the experimentally accessible statistics of each non-contextual, then there exist a Shannon entropy other, it is a physically motivated principle proposed to H(A1 ··· ,An) which is consistent whit all H(Ai,Ai+1). explain the weirdness of QM [14, 15]. Mathematically, P P 0 0 Repeatedly using the chain rule H(Ai,Ai + 1) = it reads p(a|A) = b p(a, b|A, B) = b0 p(a, b |A, B ) 0 H(Ai|Ai+1) + H(Ai+1) and monotonicity conditions for any B,B compatible with A. Geometrically, the H(Ai|Ai+1) ≤ H(Ai) ≤ H(Ai,Ai + 1) which is noth- no-disturbance principle is characterized by the in- ing more than the Shannon-type inequalities, we obtain tersection of several probabilistic simplex polytopes. the inequality For instance, to characterize no-disturbance principle for KCBS test, the experimentally accessible data is n cycle X pobs = {p(a1, a2|A1,A2), ··· , p(a5, a1|A5,A1)} which is IM = H(A1|An) − H(A1|Ai) ≤ 0. (1) completely characterized by 5 × 22 probabilities (since i=2 A1, ··· ,A5 are all dichotomic measurements) and thus 5×22 It has been shown theoretically [27, 29] and then pobs ∈ R . There are two constraints imposed by 5×22 verified experimentally [46] in three level system properties of probabilities: (i) positivity pobs ∈ R≥0 ; and (ii) normalization P p(a a |A A ) = 1. with five measurements, the inequality can be vi- ai,ai+1 i i+1 i i+1 olated by QM. For composite system, distant Al- The no-disturbance constraints are then set of equalities: P p(a a |A A ) = P p(a a |A A ), thus ice and Bob choose to implement 2m measurements ai−1 i−1 i i−1 i ai+1 i i+1 i i+1 A , ··· ,A and B , ··· ,B with BKS scenario 5×22 0 m−1 0 m−1 no-disturbance vector form a affine subspace of R≥0 m C(M) = {{Ai,Bj}i,j=0}, then the entropic test inequal- which is the intersection of subspace Pi restricted by ities reads P p(a a |A A ) = P p(a a |A A ), i.e., ai−1 i−1 i i−1 i ai+1 i i+1 i i+1 ND = 1 ∩ · · · ∩ 5. m P P P B =H(A0|Bm−1) − [H(A0|B0) + H(B0|A1) + ··· In entropic formalism, to formalize the fact that the + H(Am−1|Bm−1)] ≤ 0. (2) outcomes of a measurement do not depend on the outcomes of the measurements compatible with it, we The compatible gragh of above entropic Bell inequality is need to introduce the notion of no-disturbance cone. a 2m bipartite cycle graph. It’s show that QM can violate For a given BKS scenario [M, C(M)] with C(M) = the inequalities for m = 2 case [25], which we also refer {C1, ··· , Cm}, there are m corresponding entropic cones CHSH to as entropic CHSH test, and denoted as B . En- ΓCk which are characterized by the basic inequalities for tropic vectors h which admits an entropic NCHV/LHV measurements contained in Ck. The no-disturbance cone model will form a cone named NCHV/LHV cone, de- is defined as the intersection of all entropic cones cor- noted as ΓNCHV or ΓLHV , see Fig.1 for a pictorial illus- responding to Ck ∈ C, i.e., ΓND = ΓC1 ∩ · · · ∩ ΓCm . tration. Note that since there exist inequalities which are This can be understand as follows: all ΓCk are character- not of Shannon-type, NCHV/LHV entropic cone based ized by the corresponding basic inequalities, when em- on Shannon-type inequalities are bigger than the true bedding the entropic vector in ΓCk into a bigger space, 4 the vectors not contained in it are not constrained. But that in QM framework the experiment can not reveal the if the vectors hobs contained in both ΓCk and ΓCl , it must contextuality or nonlocality. We then give the following satisfy the inequalities imposed by two cones simultane- monogamy criterion. ously. This is consistent with the probabilistic formalism. For a given BKS scenario, as depicted in Fig.1, Theorem 3 (Entropic monogamy criterion). Suppose there is a clear hierarchy among the NCHV/LHV cone two couple of physicists are simultaneously running two contextuality or nonlocality tests E1 and E2 with mea- ΓNCHV/LHV , QM cone ΓQM and no-disturbance cone surement sets M1 and M2 and test inequalities IM ≤ 0 ΓND:ΓNCHV/LHV ⊂ ΓQM ⊂ ΓND. 1 and I ≤ 0 respectively. If the compatible graph Monogamy relations in entropic no-disturbance frame- M2 work.—Quantum contextuality and nonlocality are cap- G(M1∪M2) corresponding to all involved measurements tured by the violations of the corresponding test inequali- can be decomposed into two chordal graphs, then two ties, one of the characteristic features of these inequalities tests are monogamous: is monogamy relations [14, 15, 17, 18] which play cru- Q cial roles in quantum key distribution [47, 48] and quan- IM1 + IM2 ≤ 0. (3) tum network [49]. The monogamy relation says that if one physicist observes the contextuality or nonlocality in See supplemental material [50] for the detailed proof. some experiment, then all other simultaneous test exper- To make things more concise, let us see some exam- iments (with measurement settings having some compat- ples. First we note that the monogamy of entropic ible part) can not observe the violated values. It is natu- CHSH inequalities has be proved by Chaves and Bu- ral to ask which part of QM is responsible for monogamy, droni [31] in the entropic nonsignalling framework (which no-disturbance (nonsignalling) principle is regarded as a is the special case of entropic no-disturbance principle), Q hopeful candidate [14, 15, 17]. One may ask if entropic CHSH CHSH BM + BM0 ≤ 0. With the similar strategy, we can contextuality and nonlocality also present monogamy re- also prove the monogamy of contextuality lations and if entropic no-disturbance can be used to ex- Q plain such kind of monogamy. Actually, this is the case, cycle cycle I + I 0 ≤ 0, (4) we are going to show that, in entropic no-disturbance M M framework entropic contextuality tests, entropic nonlo- 0 0 where M = {A1, ··· ,An}, M = {A1, ··· ,Am}, A2 = cality tests, and entropic contextuality and nonlocality 0 0 0 0 A2, An = Am, A1 is compatible with all A3, ··· ,Am−1 tests all present monogamy. 0 and A1 is compatible with all A3, ··· ,An−1; And Let us first illustrate how to construct a joint en- monogamy of entropic nonlocality and entropic contex- tropic vector of two given entropic vectors which tuality obey entropic no-disturbance principle. For vectors h(XY ) = [H(∅),H(X),H(Y ),H(XY )] and Q BCHSH + IKCBS ≤ 0. (5) h(XZ) = [H(∅),H(X),H(Z),H(XZ)], we can construct H(XYZ) = H(XY ) + H(XZ) − H(X) and thus a joint See supplementary material [50] for detailed proof of all entropic vector h(XYZ) which can project to h(XY ) above monogamy relations. and h(XZ). Using this strategy, we construct joint en- Constructing new monogamy relations from old tropic vector for any set of measurements with chordal ones.—Here we analyze how to construct new monogamy compatible graph. relations from the old ones, we will take entropic CHSH inequality as a prototypical example and the general- Lemma 2. For a given set of measurements M = ization to other cases are straightforward. Actually, {A , ··· ,A }, if the compatible graph G(M) is a chordal 1 n as what have been done patially in [15], if we have graph for which there exists no induced cycle of more some monogamy relations in hand, we can construct than 3 edges, then for each experimentally accessible CHSH more monogamy relations. For example, from Bij + entropic vectors hobs(M) which obey the entropic no- Q disturbance principle, there exists an entropic vector CHSH Sjk ≤ 0 where i, j, k label different parties, we can h(M) which can project to hobs(M). construct a new cycle monogamy relation,

m−1 The proof is similar as in [14], see supplemental ma- Q X CHSH terial [50] for details. Note that the construction is a Bij ≤ 0, (6) direct derivation of the entropic no-disturbance, we are i=0 now to investigate the monogamy relations of contextu- where we assume that all overlapped parties share their ality and nonlocality tests. First note that the existence measurements in each entropic CHSH experiment. To of a joint entropic vector h(M) for all h (M) which we obs see how this holds, we only need to copy each exper- obtain from a QM experiment will lead to the unviolated Pm CHSH Q iments, i.e., we have 2 i=1 Bij , then we can di- value of the test inequality IM ≤ 0, which further means vide them into m pairs of two-monogamy sums, e.g., 5

CHSH CHSH Bij + Bi+1j and sum in the subscript is module This work was supported by the National Key Re- m, then we arrive at the Eq. (6). With this method we search and Development Program of China (Grant No. obtain: 2016YFA0301700) and the Anhui Initiative in Quantum Information Technologies (Grants No. AHY080000) Corollary 4. For n parties labeled as P = {1, ··· , n} (drawn as vertices of the party graph G(P)) who im- CHSH plementing several entropic CHSH experiments Bij (represented as edges of the party graph G(P)) simultaneously and all measurements of the overlapped parties Supplemental Material are shared in each experiment, if the party graph G(P) is a connected graph which can be packed into some small Appendix A: Entropic vector formalism pieces of Euler graphs, complete graphs, 2k-edge lines, stars and cycles, then these experiments are monoga- 1. Probabilistic vectors and observed probabilistic mous, vectors

Q X CHSH Bij ≤ 0. (7) In a typical contextuality test experiment, we deal (ij)∈E[G(P)] with the observed probabilities involved in the experiment. For example, for KCBS test experiment See supplemental material [50] for detailed proof. [28], the experimentally accessible probabilities are Notice that for other type of entropic contextuality and pobs = [p(a1a2|A1A2), p(a2a3|A2A3), p(a3a4|A3A4), nonlocality tests, we have the similar result. Monogamy p(a4a5|A4A5), p(a5a1|A5A1)]. After we obtain the prob- relation plays an important role in quantum key distribu- abilities, we need to calculate a test parameter tion processes [47, 48]. But most results concentrated on 5 the two-test case, this corollary provides a generalization X hAiAi+1i ≥ −3, (A1) to multiparty and many-tests case, which will be useful i=1 for quantum network [49]. For theories of which entropic no-disturbance principle does not hold, it is possible to if the inequality is violated, then the observed probabili- observe violated values of two BKS tests simultaneously. ties are said to be contextual. For the general case, where However, the causal structure is broken in this case as we have a set of measurements M = {A1, ··· ,An}, some argued probabilistic case in [14]. More specifically, if two of which are compatible, thus the corresponding joint entropic nonsignalling Bell tests are not monogamous probabilities are experimentally accessible. A probabilis- in some theory, then there must be superluminal com- tic vector is p = [p(∅), p(S1), ··· , p(M)] where Si is a munications among three parties, this would broke the subset of M and all subsets are contained. We also make causal structure; if two entropic no-disturbance Kochen- the convention that p(∅) = 0. The observed probabilis- Specker test monogamy are violated, the entropic vectors tic vector is defined as the probabilistic vectors which obtained from the first experiment settings and from the only contain the experimentally accessible probabilities, compatible part of the other experiment are different, denoted as pobs. We can project a probabilistic vector then we can using the difference to propagating influence into an observed probabilistic vector. An observed prob- backward in time if two experiments are implemented se- abilistic vector pobs is said to be non-contextual if there quentially, which would also lead a violation of causality. exists a probabilistic vector p which can be projected to Conclusions and discussions.—Entropic formalism is pobs. an alternative approach to understand quantum contex- Let us take n-cycle contextuality test scenario [51] tuality and nonlocality [29–31], it has several advantages: as an example to illustrate all involved notions: a this formalism does not depend on the outcomes of the set of n measurements Mn = {A1, ··· ,An} is cho- measurement as strongly as usual probabilistic formal- sen to implemented, and each Ai is compatible with ism thus they can be applied to systems with arbitrary Ai+1 (we make the convention that n + 1 = 1), dimensions, and the detection inefficiencies is well dealt thus the corresponding compatible graph is an n- in this formalism. We exploited the entropic form of cycle. The corresponding BKS scenario is Cn = Bell-Kochen-Specker no-go theorem. Then we demon- {{A1,A2}, ··· , {An,A1}}, and the corresponding behav- strated that monogamy of Bell-Kochen-Specker tests seen ior is pobs = [p(a1, a2|A1,A2), ··· , p(an, a1|An,A1)]. To in probabilistic formalism can be extended to the entropic test if pobs admits an NCHV model, we need to check Pn formalism. Adopting the graph theoretic method, we de- a test parameter I = i=1 γihAiAi+1i where γi = ±1 velop the general criterion of monogamy of several Bell- and the number of minus ones is odd, and find out if Kochen-Specker tests. And we all analyze the multiparty I ≤ n − 2 or not. If we choose all measurements be and many-test case, which will be crucial for quantum dichotomic with outcomes ±1, then, the test parameter network communication. can also be written as I0 = P P p(ab|ii + a⊕b=0 i,γi=1 6

a entropic vector if it is the vectors of joint entropies of a set of n random variables M, i.e., hα = H(Xα) for all α ∈ 2Nn . Notice that H can be chosen as any information measure (any entropy function), not restricted to Shannon entropy. We now analyze the region (which we refer to as entropic quasi-cone)

∗ ΓE = {h|h is entropic}. (A2)

∗ We ﬁrst list some important properties of ΓE:

∗ 2n (1)Γ E contains the origin point of R ;

∗ 2n FIG. 2: Pictorial illustration of the non-disturbance (ND) (2)Γ E is a subset of the nonnegative orthant R≥0; or nonsignaling (NS) polytope, quantum mechanical (QM) ∗ ¯∗ polytope and non-contextual hidden variable (NCHV) or local (3) the closure ΓE, denoted as ΓE, is convex. hidden variable (LHV) polytope. From the above properties we know that

¯∗ 1) + P P p(ab|ii + 1) and the corresponding ΓE = ΓE, (A3) a⊕b6=0 i,γi=−1 NCHV inequality is I0 ≤ n − 1. It is clear that all above is a convex cone, which we will refer to as entropic cone. inequalities are just the inner product hs, pobsi for s are We also need to introduce the notion entropic ex- obtained from test parameter, thus they are all linear pression, which is a linear function of the entropies constraints on the behavior we have access to. The prob- f(H(X),H(X|Y ), ··· ). An entropic inequality is of the abilistic vectors allowed by NCHV model then form a form polytope known as NCHV polytope. No-disturbance principle asserts that compatible mea- f(H(X),H(X|Y ), ··· ) ≥ c, (A4) surements do not disturb the experimentally accessible statistics of each other, mathematically, it reads p(a|A) = or we can make it into f(H(X),H(X|Y ), ··· ) − c ≥ 0. P P 0 0 0 The Shannon entropic cone is characterized by Shannon- b p(a, b|A, B) = b0 p(a, b |A, B ) for any B,B compatible with A. Geometrically, the no-disturbance prin- type inequalities: (i) monotonicity H(A|B) = H(AB) − ciple is characterized by the intersection of several prob- H(B) ≥ 0; (ii) subadditivity I(A : B) = H(A) + H(B) − abilistic simplex polytopes. See Fig.2 for the pictorial H(AB) ≥ 0; (iii) strong subadditivity ( or submodular- illustration of the non-disturbance (ND) or nonsignaling ity) I(A : B|C) = H(AC)+H(BC)−H(ABC)−H(C) ≥ (NS) polytope, quantum mechanical (QM) polytope and 0, where we use notation A, B, C to represents some spe- non-contextual hidden variable (NCHV) or local hidden cial measurements or collection of compatible measure- variable (LHV) polytope. ments. Like in probabilistic formalism, for a given set of measurements M, we have access to the joint entropic H(S) 2. Entropic vector and Entropic cone compatible measurements Ai ∈ S, collecting these entropies together we will get the observed entropic vec- Let us take a close look at the entropic cone here, see tor hobs. If there exist a joint entropy H(M) for all involved measurements Ai ∈ M (thus an entropic vec- Ref. [32] for more details. we use the notation Nn to mean the set of ﬁrst n natural numbers, {1, 2, ··· , n}, tor h = [H(S)]S∈2M ), for which all experimentally ac- which we utilize as a label set to label n random variables cessible entropies can be reproduced from this entropy, then we say than hobs is entropic noncontextual. All entropic noncontextual entropic forms a cone named as M = {Xi|i ∈ Nn}. NCHV cone; entropic vectors arise in QM form a cone n The power set of Nn is denoted as 2N whose num- called QM cone; entropic vectors satisfy entropic no- ber of elements is 2n. Associated with M are 2n − disturbance principle form a cone we refer to as ND cone. 1 joint entropies. For instance, for N2, we have There is a hierarchy structure of these cones, see Fig.1. {H(X1),H(X2),H(X1,X2)}. n Now let α be a subset of Nn, i.e., α ∈ 2N , and let Cα = {Xi|i ∈ α} which we refer to as a context of M and Appendix B: Proofs of monogamy relations correspondingly HM(Cα) = H(Xα}. We will make the convention that H(∅) = 0. Consider the 2n dimensional In this section, we give the detailed proof the the the- 2n real vector space R , a vector h = (hα)α∈2Nn is called orem appeared in the main text. 7

1. General monogamy criterion

Lemma 2. For a given set of measurements M = {A1, ··· ,An}, if the compatible graph G(M) is a chordal K4 graph for which there exists no induced cycle of more than 3 edges, then for each experimentally accessible L4 entropic vectors hobs(M) which obey the entropic no- edge disturbance principle, there exists an entropic vector packing h(M) which can project to hobs(M). C6 PROOF. Inspired by the construction of Ramanathan et al. [14], we give a explicit entropic construction. Since G(M) is a chordal graph, thus G(M) contains no in- S5 duced cycle of length greater than 3. Let V [G(M)] i be the set of vertices of G(M), G3(M) = {K3} be E5 the set of all complete subgraphs (all vertices are pairwise connected) with more than or equal to 3 vertices, j G2(M) = {E2} be the set of edges which are not sub- i k graphs of any K3 ∈ G3(M), G1(M) = {V1 } be the set of FIG. 3: The depiction of the monogamy relation of entropic i vertices which are not subgraphs of any K3 ∈ G3(M) and CHSH experiments among many parties, each vertex repre- j E2 ∈ G2(M) and G = G3(M) ∪ G2(M) ∪ G1(M). From sents a party and each edge represents a entropic CHSH ex- the construction, it is clear that each edge of G(M) con- periment. On the left side, it is the whole party graph G(P) tained only one of G (M) or G (M) but they can appear of all involved experiments. The edge set of G(P) can be 3 2 packed into ﬁve small pieces as a 4-complete graph K , a 6- more than once in each set. Now we can construct a joint 4 cycle graph C , a 5-Euler graph E , a 4-edge line L and a entropy H(M) = H(A , ··· ,A ) for H(Ki ), H(Ej) and 6 5 4 1 n 3 2 5-star graph S5. Since all these small pieces are monogamous k H(V1 ): then using the triangle inequality of absolute value function, we can get the monogamy relation corresponding to G(P). X i X j H(M) = H(K3) + H(E2) i j K ∈G3(M) E ∈G (M) 3 3 2 Q X k and IN ≤ 0. Since we have IM + IM = IN + IN , + H(V1 ) 2 1 2 1 2 k we get the required result. V1 ∈G1(M) 1 X Corollary 4. For n parties which are labeled as P = − H(Kl ∩ Ks). (B1) 2 {0, ··· , n} (represented as vertices of the party graph l s l s K 6=K ,K ,K ∈G G(P)) who implementing several entropic CHSH exper- CHSH Thus we can construct the entropic vector h(M) which iments Bij (represented as edges of the party graph can project to hobs = [H(K)] where K is taken over all G(P)) simultaneously and all measurements of the over- complete subgraphs of G(M), therefore we completes the lapped parties are shared in each experiment, if the party proof. graph G(P) is a connected graph which can be divided into some small pieces of Euler graphs, complete graphs, Theorem 3 (Entropic monogamy criterion). Suppose two 2k-edge lines, stars and cycles, then these experiments couple of physicists are simultaneously running two con- are monogamous, i.e. textuality or nonlocality tests E1 and E2 with measurement sets M and M and test inequalities I ≤ 0 Q 1 2 M1 X CHSH Bij ≤ 0. (B3) and IM2 ≤ 0 respectively. If the compatible graph (ij)∈E[G(P)] G(M1∪M2) corresponding to all involved measurements can be decomposed into two chordal graphs, then two PROOF. The process is actually the edge packing, since tests are monogamous: we can using the copy method to make the sum of all in- Q dividual edges into the sum of the groups of edges of IM1 + IM2 ≤ 0. (B2) each packed small pieces, we only need to show that there is a monogamy relation for each small piece, i.e., we PROOF. Suppose G(M ∪M ) can be decomposed into 1 2 need to show that there is monogamy relations for Euler two chordal graph G(N ) and G(N ) where N ∪ N = 1 2 1 2 graphs, complete graphs, 2k-edge lines, stars and cycles, M ∪ M , then there always exist two entropic vectors 1 2 this can be done in the same spirit as we have done for h(N ) and h(N ) which can project to h (N ) and 1 2 obs 1 star graph case for Eq. (6). For example, as depicted in hobs(N2), thus the correspond tests of N1 and N2 can Q Fig.3, we have the whole party graph G(P), since its not get violated value in QM framework, i.e., IN1 ≤ 0 edge set E[G(P)] can be packed into ﬁve small pieces, a 8

4-complete graph K4, 6-cycle graph C6, a 5-Euler graph Adding all these element inequalities together and us- E5, a 4-edge line L4 and a 5-star graph S5 then we have ing the chain rule of conditional entropy, we obtain the X CHSH inequality Bij (ij)∈E[G(P)] 0 0 H(A1|An)−[H(A1|A2)+···+H(An−1|An)]+H(A1|An) X CHSH X CHSH X CHSH = Bij + Bij + Bij Q − [H(A0 |A0 ) + ··· + H(A0 |A0 )] ≤ 0, (ij)∈K4 (ij)∈C6 (ij)∈S5 1 2 n−1 n X CHSH X CHSH + Bij + Bij which is nothing more than the monogamy relation of

(ij)∈E5 (ij)∈L4 contextuality:

Then using the monogamy relations of each small piece, Q cycle cycle I + I 0 ≤ 0. (B5) we arrive at the required result. M M

2. Monogamy of entropic nonlocality 4. Monogamy of entropic nonlocality and entropic contextuality

Suppose Alice is running entropic mi-measurement entropic Bell experiment with k Bob simultaneously, and Suppose that Alice and Bob are running a entropic Alice shares at least two measurements for all experi- nonlocality test experiment with M = {A0,A2,B0,B1} ments, then we have the monogamy relations: and C(M) = {{A0,B0}, {A0,B1}, {A2,B0}, {A2,B1}} where A0,A2 are implemented by Alice and B0,B1 are Q Bm1 + ··· + Bmk ≤ 0, (B4) implemented by Bob. If Alice runs an entropic contextu- 0 0 ality text with M = {A0,A1, ··· ,A4} and C(M ) = where Bm1 = H(Ai |Bi )−[H(Ai |Bi )+H(Bi |Ai )+···+ 0 mi 0 0 0 1 {{A0,A1}, ··· , {A4,A0}} simultaneously, then two ex- H(Ai |Bi )], Ai and Bi are measurements implemented mi mi j l periments are monogamous. by Alice and i-th Bob. For measurement set {B1,A2, ··· ,A0} we have Actually, this is a direct corollary of theorem3 and corollary4, since the party graph G(P) is a star graph H(B1A0) ≤ H(B1A2 ··· An) with k edges. H(B1|A2 ··· A0) ≤ H(B1|A2)

H(A2|A3 ··· A0) ≤ H(A2|A3)

3. Monogamy of entropic contextuality H(A3|A4A0) ≤ H(A3|A4)

For measurement set {A0,A1,A2,B0} we have For BKS scenario (M, C) with M = {A1, ··· ,An}, C = C(M) = {{A1,A2}, ··· , {An,A1}} and BKS sce- 0 0 0 0 0 H(A1A0) ≤ H(A1B0A2A0) nario (M , C) with M = {A1, ··· ,Am}, C = C(M) = 0 0 0 0 0 0 H(A1|A2B0A0) ≤ H(A1|A2) {{A1,A2}, ··· , {Am,A1}}, where A2 = A2, An = Am, 0 0 0 A1 is compatible with all A3, ··· ,Am−1 and A1 is com- H(A2|B0A0) ≤ H(A2|B0) patible with all A3, ··· ,An−1, we must to give the needed basic entropic inequalities. For measurement set 0 {A1,A2, ··· ,An} we have Adding all these element inequalities together and us- 0 0 ing the chain rule of conditional entropy, we obtain the H(A1An) ≤ H(A1A2 ··· An) inequality 0 0 H(A1|A2 ··· An) ≤ H(A1|A2)

H(A2|A3 ··· An) ≤ H(A2|A3) H(A1|A0) − [H(A1|A2) + ··· + H(A4|A0)] + H(B1|A0) Q . . − [H(B1|A2) + H(A2|B0) + H(B0|A0)] ≤ 0, H(An−2|An−1An) ≤ H(An−2|An−1) which is nothing more than the monogamy relation of 0 0 entropic contextuality and entropic nonlocality: For measurement set {A1,A2, ··· ,An} we have 0 0 0 Q H(A1Am) ≤ H(A1A2 ··· Am) CHSH KCBS 0 0 0 BM + IM0 ≤ 0. (B6) H(A1|A2 ··· Am) ≤ H(A1|A2) 0 0 0 0 0 H(A2|A3 ··· Am) ≤ H(A2|A3) . . 0 0 0 0 0 H(An−2|Am−1Am) ≤ H(Am−2|Am−1) † Electronic address: [email protected] 9

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