Limitations on sharing Bell nonlocality between sequential pairs of observers
Shuming Cheng, Lijun Liu, Travis J. Baker, Michael J. W. Hall
LLimitations on sharing Bell nonlocality between sequential pairs of observers
Shuming Cheng,
1, 2, 3
Lijun Liu, and Michael J. W. Hall The Department of Control Science and Engineering, Tongji University, Shanghai 201804, China Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai 201804, China Institute for Advanced Study, Tongji University, Shanghai, 200092, China College of Mathematics and Computer Science, Shanxi Normal University, Linfen 041000, China Department of Theoretical Physics, Research School of Physics,Australian National University, Canberra ACT 0200, Australia (Dated: March 2, 2021)We give strong analytic and numerical evidence that, under mild measurement assumptions, twoqubits cannot both be recycled to generate Bell nonlocality between multiple independent observerson each side. This is surprising, as under the same assumptions it is possible to recycle just one ofthe qubits an arbitrarily large number of times [P. J. Brown and R. Colbeck, Phys. Rev. Lett. ,090401 (2020)]. We derive corresponding ‘one-sided monogamy relations’ that rule out two-sidedrecycling for a wide range of parameters, based on a general tradeoff relation between the strengthsand maximum reversibilities of qubit measurements. We also show if the assumptions are relaxedto allow sufficiently biased measurement selections, then there is a narrow range of measurementstrengths that allows two-sided recycling for two observers on each side, and propose an experimentaltest. Our methods may be readily applied to other types of quantum correlations, such as steeringand entanglement, and hence to general information protocols involving sequential measurements.
Introduction—
Sharing entanglement between a pairof observers, for information tasks such as quantum tele-portation [1] and quantum key distribution [2], typicallyrequires generating entangled states which are measuredand then discarded by the observers. A more efficient useof entanglement resources would be to pass the measuredsystems on, to allow a second pair of observers to exploitany residual entanglement, then onto a third pair, etc.(see Fig. 1). This possibility of recycling entanglementresources has become of great interest recently, both the-oretically [3–16] and experimentally [17–22].As a notable example, it is possible to use two entan-gled qubits to generate Bell nonlocality between a firstobserver holding the first qubit and each one of an arbi-trarily long sequence of independent observers that holdthe second qubit in turn [13]. This recycling of the sec-ond qubit allows the first observer to implement device-independent information protocols, such as quantum keydistribution [2, 23] and randomness generation [22, 24],with each one of the other observers.In contrast, we show here that there are surprisinglystrong limitations on recycling both qubits in this way,so as to generate Bell nonlocality for multiple observerson each side. This is so even for just two observers oneach side (see Fig. 1). For example, we give strong nu-merical evidence for the conjecture that if two observersindependently make one of two equally-likely two-valuedmeasurements on their qubits, as in [13], then a secondpair of observers cannot observe Bell nonlocality if thefirst pair does. This limitation, to one-sided recycling,may be regarded as a type of sharing monogamy, whichwe demonstrate analytically for a wide range of parame-ters via corresponding ‘one-sided monogamy relations’.The physical intuition behind such limitations is that ifthe measurements made by the first pair of observers are sufficiently strong to demonstrate Bell nonlocality, thenthey are also sufficiently irreversible to leave the qubits ina Bell-local state. Correspondingly, the analytic resultsrely on a tradeoff between the strength and maximumreversibility of general two-valued qubit measurements,as shown below, of some interest in its own right.We show that the validity of the conjecture onlyrequires explicit consideration of the 16-parameter setof observables measured by the first pair of observers,on a 1-parameter class of pure initial states, makinga numerical test feasible. Further, we obtain ana-lytic monogamy relations for two 14-parameter subsets,corresponding to the observables having either equalstrengths or orthogonal measurement directions for eachside. These monogamy relations hold for all initial stateswith maximally-mixed marginals, and for arbitrary ini-tial states if the observables also have zero biases.Finally, by allowing the first pair of observers to se-lect one of their measurements with high probability( > One-sided monogamy conjecture—
Before proceedingto details, we formally state the main conjecture and pre-view the numerical evidence. First, if an observer A ( B ) measures either of two observables X or X (cid:48) ( Y or Y (cid:48) ), with outcomes labelled by ±
1, then Bell nonlocal- a r X i v : . [ qu a n t - ph ] F e b S Bob 1
Alice 2
Alice 1
Bob 2 S Bob 1
Alice 2
Alice 1
Bob 2
FIG. 1. Sequential Bell nonlocality with multiple observers on each side. A source S generates two qubits on each run, whichare received by observers Alice 1 and Bob 1 ( A and B in the main text). Each makes one of two local measurements ontheir qubit with equal probabilities; records their result; and passes their qubit onto independent observers Alice 2 and Bob 2,respectively ( A and B in the main text). It is known that Alice 1 can demonstrate Bell nonlocality with each of an arbitrarynumber of Bobs in this way, via recycling of the second qubit [13]. However, strong analytic and numerical evidence supportsthe conjecture that, surprisingly, Alice 1 and Bob 1 can demonstrate Bell nonlocality via their measurements only if Alice 2and Bob 2 cannot, and vice versa. A similar conjecture and evidence applies to the pairs (Alice 1, Bob 2) and (Alice 2, Bob 1).Thus, the qubits cannot both be recycled to sequentially generate Bell nonlocality in this way. However, by significantly relaxingthe equal-probability assumption, this limitation can be overcome to allow all four pairs to demonstrate a small degree of Bellnonlocality and thus to implement device-independent protocols such as randomness generation. ity is characterised by the value of the Clauser-Horne-Shimony-Holt (CHSH) parameter [27, 28] S ( A, B ) := (cid:104) XY (cid:105) + (cid:104) XY (cid:48) (cid:105) + (cid:104) X (cid:48) Y (cid:105) − (cid:104) X (cid:48) Y (cid:48) (cid:105) . (1)In particular, a violation of the CHSH inequality S ( A, B ) ≤ Conjecture:
If observers A and B independentlymake one of two equally-likely measurements on a firstand second qubit, respectively, and the qubits are passedon to observers A and B , respectively, then the pairs ( A j , B k ) and ( A j (cid:48) , B k (cid:48) ) can each violate the CHSH in-equality only if they share a common observer, i.e., S ( A j , B k ) , S ( A j (cid:48) , B k (cid:48) ) > j = j (cid:48) or k = k (cid:48) . (2)Thus, the conjecture asserts that sequential Bell nonlo-cality is only possible via a fixed observer on one side inthis scenario, as in [13], but not for multiple observerson each side. In particular, it implies that at most oneof the pairs ( A , B ) and ( A , B ) can violate the CHSHinequality, and similarly at most one of the pairs ( A , B )and ( A , B ). Here A corresponds to Alice 1 in Fig. 1,etc. Numerical evidence for the conjecture is summarisedin Fig. 2. We note the conjecture does not generally ex-tend to, e.g., Bell inequalities with more measurementsper observer on higher-dimensional systems [29]. Qubit measurements—
To proceed, consider mea-surement of a (generalised) two-valued observable de-scribed by a positive operator valued measure (POVM) { X + , X − } . Thus, X ± ≥ X + + X − = , andthe observable is equivalently represented by the opera-tor X := X + − X − , with − ≤ X ≤ . For qubits, X can be decomposed as X = B + S σ · x , (3)with respect to the Pauli spin operator basis σ ≡ ( σ , σ , σ ), with S ≥ | x | = 1. Here B and S definethe outcome bias and strength of the observable, and x isa direction associated with the observable. For projectiveobservables one has B = 0 and S = 1, while for the trivialobservable with POVM { , } one has B = 1 and S = 0.More generally, the bias B is the difference of the +1 and − ρ = , and |B| + S ≤ X ≡ { X + , X − } can be implemented by the square-rootmeasurement that takes the state ρ to φ ( ρ ) := X / ρX / + X / − ρX / − . (4)More generally, any measurement of X takes ρ to φ G ( ρ ) = φ + ( X / ρX / ) + φ − ( X / − ρX / − ), for twoquantum channels φ + and φ − [13], equivalent to firstcarrying out the square-root measurement and then ap-plying a quantum channel that may depend on the out-come. Now, a quantum channel is reversible if and onlyif it is unitary, implying the square-root measurementmap φ can be recovered from φ G only if φ ± are unitarytransformations, and indeed the same unitary transfor-mation if the outcome is unknown (as is the case for in-dependent sequential measurements). Hence, the square-root measurement φ is optimal, in the sense of being themaximally reversible measurement of X (up to a unitarytransformation), and we follow [13] in confining attentionto such measurements.To quantify the degree of maximum reversibility, we FIG. 2. One-sided monogamy for CHSH Bell nonlocality. Numerical values of | S ( A , B ) | in Eq. (1) and the proxy quantities S ∗ ( A , B ) , S ∗ ( A , B ) , S ∗ ( A , B ) in Eqs. (10)–(12) are plotted for various pairs of observers from Fig. 1, where A denotesAlice 1, etc. The values are generated by a uniform search over possible two-valued observables for A and B , for several qubitstates [25] (pink dots denote the maximally entangled state α = π/ α = 0 . , . , . S X = S X (cid:48) and S Y = S Y (cid:48) the one-sided monogamy relations in Eqs. (13)and (14) are analytically satisfied, corresponding to values below the red lines in the left and right hand panels, respectively.Similar relations hold for orthogonal spin directions and/or zero Bloch vectors (see main text). The dashed lines indicate themaximum quantum value of 2 √ note that explicit calculation gives [25] φ ( ρ ) = P x ρP x + P − x ρP − x + R ( P x ρP − x + P x ρP − x ) , (5)where P x = ( + σ · x ) denotes the projection onto unitspin direction x , and R := (cid:112) (1 + B ) − S + (cid:112) (1 − B ) − S . (6)Thus, the off-diagonal elements of ρ in the σ · x basisare scaled by R , with R = 0 for projective observables( B = 0 , S = 1), and R = 1 for trivial observables ( S = 0).Hence, R is a natural measure of the maximum reversibil-ity associated with the measurement of a given observ-able. For convenience we will often simply refer to R as the reversibility in what follows (we remark that R generalises the ‘quality factor’ F defined for a class ofunbiased weak qubit measurements [3]). Tradeoff between strength and reversibility—
Equa-tion (6) is a general relation connecting outcome bias,strength, and maximum reversibility, and implies the fun-damental tradeoff relation [25] R + S ≤ , (7)between reversibility and strength. Equality holds onlyfor unbiased observables, i.e., for B = 0. This tradeoffis very useful for studying the shareability of Bell non-locality via sequential measurements, and may also be used to reinterpret the information-disturbance relationgiven in [30] (in the case of qubit measurements) [25].Equations (5) and (7) also suggest a natural definition of“minimal decoherence”, D = √ − R ≥ S . Simplifying the conjecture—
Fortunately, the validityof the conjecture does not require explicit considerationof all possible initial states, nor of all possible observablesmeasured by the four observers. For example, since theCHSH parameters S ( A j , B k ) in Eq. (1) are convex-linearin the initial state ρ shared by A and B , only pure ini-tial states ρ = | ψ (cid:105)(cid:104) ψ | need be considered to test the jointranges of the parameters for any given measurements.We can further restrict to the 1-parameter class | ψ (cid:105) = cos α | (cid:105) ⊗ | (cid:105) + sin α | (cid:105) ⊗ | (cid:105) , α ∈ [0 , π/
2] (8)in the σ -basis {| (cid:105) , | (cid:105)} , as Bell nonlocality is invariantunder local unitaries [28].Moreover, once A ’s and B ’s observables X, X (cid:48) , Y, Y (cid:48) have been specified, the optimal choices for A and B areuniquely determined, on each run, by which observer onthe other side they are trying to generate Bell nonlocalitywith. For example, it follows from Eq. (5) that if A and B choose between their measurements with equalprobabilities, as per the conjecture, and T denotes theinitial spin correlation matrix (with coefficients T jk :=Tr [ ρ ( σ j ⊗ σ k ]), then the correlation matrix shared by A and B is KT L , with [25] K := ( R X + R X (cid:48) ) I + (1 −R X ) xx (cid:62) + (1 −R X (cid:48) ) x (cid:48) x (cid:48)(cid:62) (9) L := ( R Y + R Y (cid:48) ) I + (1 − R Y ) yy (cid:62) + (1 − R Y (cid:48) ) y (cid:48) y (cid:48)(cid:62) (here R X denotes the reversibility associated with ob-servable X , etc.). It follows immediately from theHorodecki criterion that A and B can violate the CHSHinequality only if [31] S ∗ ( A , B ) := 2 (cid:112) s ( KT L ) + s ( KT L ) > , (10)where s ( M ) and s ( M ) denote the two largest singularvalues of matrix M .Similarly, it can be shown that pairs ( A , B ) and( A , B ) can violate the CHSH inequality only if [25] S ∗ ( A , B ) := (cid:12)(cid:12) ( B X + B X (cid:48) ) L b + LT (cid:62) ( ˜ x + ˜ x (cid:48) ) (cid:12)(cid:12) + (cid:12)(cid:12) ( B X − B X (cid:48) ) L b + LT (cid:62) ( ˜ x − ˜ x (cid:48) ) (cid:12)(cid:12) (11) S ∗ ( A , B ) := | ( B Y + B Y (cid:48) ) K a + KT ( ˜ y + ˜ y (cid:48) ) | + | ( B Y − B Y (cid:48) ) K a + KT ( ˜ y − ˜ y (cid:48) ) | (12)are greater than 2, respectively, where ˜ x := S X x , ˜ x (cid:48) := S X (cid:48) x (cid:48) , etc, and a and b are the initial Bloch vectors ofthe first and second qubits.Hence, to verify the conjecture one need only con-sider the values of S ( A , B ) and the proxy quan-tities S ∗ ( A , B ) , S ∗ ( A , B ) , S ∗ ( A , B ), over the 16-parameter set of observables X, X (cid:48) , Y, Y (cid:48) and a 1-parameter set of initial states. Moreover, if the observ-ables are unbiased (i.e., B X = B X (cid:48) = B Y = B Y (cid:48) = 0),then it may be shown that the conjecture need only beverified for a 9-parameter set of observables on a sin-gle maximally entangled state [25]. Note that the priorwork [3–22] is restricted to unbiased observables. Evidence for conjecture—
Our numerical resultsbased on a brute-force search support the conjecture,as shown in Fig. 2 (see also [25]). Moreover, we candirectly prove the conjecture in a number of scenarios,using tradeoff relation (7). For example, for the caseof equal strengths for each side, i.e., S X = S X (cid:48) and S Y = S Y (cid:48) , the one-sided monogamy relation S ∗ ( A , B ) + S ∗ ( A , B ) ≤ ρ with maximally mixed marginals,i.e., a = b = 0, and for arbitrary states if the observablesare unbiased [25], implying Eq. (2) holds for these pairs.The upper bound corresponds to the red curve on theright hand panel of Fig. 2. This relation also holds for thealternative case of orthogonal measurement directions oneach side, i.e., x · x (cid:48) = y · y (cid:48) = 0 [25].One-sided monogamy relations for the pairs ( A , B )and ( A , B ) can also be derived, such as | S ( A , B ) | + S ∗ ( A , B ) ≤ / (3 √
2) for the case of orthogonal directions. However,since these relations require considerably more work (in-cluding a significant generalisation of the Horodecki cri-terion), they are only derived under joint strength andorthogonality assumptions in [25], with the general casesleft to a companion paper [26].
Sequential Bell nonlocality via biased measurementselections—
The conjecture and above results requirethat observers A and B each select their measurementswith equal probabilities. However, if they instead eachchoose between making a relatively weak measurement,with sufficiently high probability, and a relatively strongmeasurement, with correspondingly low probability, thenthe average disturbance to their shared state can be smallenough to allow all four pairs to generate Bell nonlocal-ity.To show this, suppose that A measures X and X (cid:48) with probabilities 1 − (cid:15) and (cid:15) , and B similarly mea-sures Y and Y (cid:48) with probabilities 1 − (cid:15) and (cid:15) , on a sin-glet state (i.e., T = − I ). Suppose further that X and Y are measured with strengths S X = S Y = S and re-versibilities R X = R Y = R ; that X (cid:48) and Y (cid:48) are pro-jective, with strengths S X (cid:48) = S Y (cid:48) = 1 and reversibilities R X (cid:48) = R Y (cid:48) = 0; with the measurement directions beingthe optimal CHSH directions [27] (i.e., x and x (cid:48) are or-thogonal with y = ( x + x (cid:48) ) / √ y (cid:48) = ( x − x (cid:48) ) / √ | S ( A , B ) | = ( S + 1) / √ A and B can violate the CHSH inequality only if thisis larger than 2, i.e., only if S > / − ∼ . . (15)Further, the state shared by A and B has spin corre-lation matrix T = − K (cid:15) L (cid:15) , with K (cid:15) = (1 − (cid:15) )[ R I + (1 −R ) aa (cid:62) ] + (cid:15) a (cid:48) a (cid:48)(cid:62) , and a similar expression for L (cid:15) with x , x (cid:48) replaced by y , y (cid:48) , implying via Eq. (10) that A and B can violate the CHSH inequality only if [25] R > [ √ − (1 − (cid:15) ) ] / − (cid:15) − (cid:15) > ( √ − / ∼ . . (16)Equations (7), (15) and (16) yield a narrow range ofstrengths, 0 . < S < . A , B )and ( A , B ) can generate Bell nonlocality by recyclingboth qubits. Further, this range constrains the probabil-ity of making the projective measurement to [25] (cid:15) < (cid:15) max ∼ . . (17)The pairs ( A , B ) and ( A , B ) can also generate Bellnonlocality under these conditions [25]. Discussion—
Strong numerical and analytic evidencehas been given to support an unexpected one-sidedmonogamy conjecture, that limits sequential violationof the CHSH inequality to one-sided qubit recycling, asin [13], if observers make unbiased measurement selec-tions. Conversely, allowing sufficiently biased selectionspermits a narrow range of measurement strengths withinwhich two-sided qubit recycling is possible. We pro-pose testing the latter experimentally, as it in principlepermits four independent pairs of observers to generateBell nonlocality, and hence to carry out device indepen-dent quantum information protocols such as randomnessgeneration, via the recycling of a two-qubit state. Fur-ther details and generalisations of our methods are givenin [26], and we expect these methods can also be read-ily applied to the sequential sharing of quantum proper-ties such as entanglement [32], Einstein-Podolsky-Rosensteering [33], and random access codes [34–36].We thank Yong Wang, Xinhui Li, Jie Zhu, MengjunHu and Ad´an Cabello for helpful discussions. S.C. issupported by the Fundamental Research Funds for theCentral Universities (No. 22120200415) and the NationalNatural Science Foundation of China (No. 62088101).L.L. is supported by National Natural Science Founda-tion of China (No. 61703254).
Note:
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SUPPLEMENTAL MATERIAL
I. BIAS, STRENGTH AND MAXIMUMREVERSIBILITY OF TWO-VALUED QUBITMEASUREMENTSA. Measurement strength and outcome bias
Recall from the main text that a general two-valuedqubit observable, with POVM { X + , X − } , is equivalentlyrepresented by the operator X = X + − X − = B + S σ · x , (S.1)where S ≥ B denote the strength and outcome bias of the observable, respectively, and x is the associatedmeasurement direction, with | x | = 1. The POVM ele-ments are determined uniquely by X via X ± = ( ± X ),and the positivity requirement X ± ≥ − ≤ X ≤
1, i.e., to the condition |B| + S ≤ . (S.2)on the strength and bias.It is convenient for later purposes to also write X inthe form X := x + P x + x − P − x , (S.3)where P x := 1 + σ · x x . Hence, x ± = B ± S , (S.5)and X ± = 1 ± x + P x + 1 ± x − P − x . (S.6) B. Maximum reversibility
To obtain Eqs. (5) and (6) of the main text, note fromEq. (S.6) that X / ± = (cid:114) ± x + P x + (cid:114) ± x − P − x . (S.7) Hence, the square-root measurement operation corre-sponding to X takes state ρ to the state φ ( ρ ) := X / ρX / + X / − ρX / − = (cid:18) x + − x + (cid:19) P x ρP x + (cid:18) x − − x − (cid:19) P − x ρP − x + (cid:112) (1 + x + )(1 + x − ) ( P x ρP − x + P x ρP − x )+ (cid:112) (1 − x + )(1 − x − ) ( P x ρP − x + P x ρP − x )= P x ρP x + P − x ρP − x + R ( P x ρP − x + P x ρP − x ) , (S.8)as per Eq. (5), with the maximum reversibility R givenby R := (cid:112) (1 + x + )(1 + x − ) + (cid:112) (1 − x + )(1 − x − ) . (S.9)Equation (S.5) then yields R = (cid:112) (1 + B ) − S + (cid:112) (1 − B ) − S (S.10)as per Eq. (6) of the main text. C. Fundamental tradeoff relation
Squaring each side of Eq. (S.10), then rearranging toput the square-root term and squaring again, leads to thetradeoff relation R + S = 1 − B ( 1 R − ≤ d -dimensional system, Banaszek defines a corresponding‘mean operation fidelity’ F , related to the disturbancecaused by the measurement, and a ‘mean estimation fi-delity’ G , related to the average information gain or qual-ity of the measurement, and shows that these satisfy thegeneral information-disturbance relation [29] (cid:114) F − d + 1 ≤ (cid:114) G − d + 1 + (cid:114) ( d − d + 1 − G ) . For a two-valued qubit measurement with Kraus opera-tors M ± , corresponding to the POVM { X ± = M †± M ± } ,these quantities may be calculated explicitly, yielding6 F = 2 + | Tr [ M + ] | + | Tr [ M − ] | = 2 + (cid:12)(cid:12)(cid:12) Tr (cid:104) U + X / (cid:105)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Tr (cid:104) U − X / − (cid:105)(cid:12)(cid:12)(cid:12) ≤ (cid:104) X / (cid:105) + Tr (cid:104) X / − (cid:105) = 2 R + 4 (S.12)upplemental Material – 2/9(where the second line uses the polar decomposition M ± = U ± M ± for unitary operators U ± and the last linefollows via Eqs. (S.7) and (S.9)), and6 G = 2 + λ max ( X + ) + λ max ( X − ) = S + 3 , (S.13)where λ max ( M ) denotes the maximum eigenvalue of M .Thus, F and G can be reinterpreted in terms of the max-imum reversibility and strength of the measurement forthis case (note also the maximum reversibility property of R is emphasised by the inequality in Eq. (S.12), which issaturated for the square-root measurement M ± = X / ± ).Further, our tradeoff relation (S.11) for R and S can berewritten as2 + 2 R ≤ (cid:112) − S = (cid:16) √ S + √ − S (cid:17) , which, on taking the square root and substituting theabove expressions for F and G , yields √ F − ≤ √ G − √ − G. (S.14)Thus, the fundamental tradeoff relation implies Ba-naszek’s information-disturbance relation for qubit mea-surements.Generalisations and further applications of tradeoff re-lation (S.11) will be discussed elsewhere [26]. II. SPIN CORRELATION MATRIX, BLOCHVECTORS, AND THE FORMS OF K AND L The spin correlation matrix T and Bloch vectors a , b of a two-qubit state ρ are defined by T jk := Tr [ ρ ( σ j ⊗ σ k )] , (S.15) a j := Tr [ ρ ( σ j ⊗ )] , b j := Tr [ ρ ( ⊗ σ j )] , (S.16)for j, k = 1 , ,
3. To calculate the effect of a maximallyreversible measurement of X by A on these quantities,note first that substituting P x = ( + σ · x ) in Eq. (S.8)gives φ ( ρ ) = ( ρ + σ · x ρ σ · x ) + R ( ρ − σ · x ρ σ · x ) . (S.17)Thus, φ ( ) = (i.e., the map is unital). Further, forPauli operators defined on the first qubit we have theidentity( σ · x ) σ j ( σ · x ) = x k x l σ k σ j σ l = x k x l σ k ( δ jl + i(cid:15) jlm σ m )= x j ( σ · x ) + ix k x l (cid:15) jlm ( δ km + i(cid:15) kmn σ n )= x j ( σ · x ) + x k x l (cid:15) jlm (cid:15) knm σ n = x j ( σ · x ) + x k x l ( δ jk δ ln − δ jn δ kl ) σ n = 2 x j ( σ · x ) − σ j , (S.18) with summation over repeated indices. Hence, φ ( σ j ) = x j ( σ · x ) + R ( σ j − x j σ · x ) . (S.19)Thus, noting from Eq. (S.8) that the map is self-dual, i.e.,Tr [ φ ( M ) N ] = Tr [ M φ ( N )], the spin correlation matrix T changes to T X where T Xjk := Tr [( φ ⊗ I )( ρ ) ( σ j ⊗ σ k )] = Tr [ ρ φ ( σ j ) ⊗ σ k (cid:105) ]= (cid:104) [ x j ( σ · x ) + R ( σ j − x j σ · x )] ⊗ σ k (cid:105) = R(cid:104) σ j ⊗ σ k (cid:105) + (1 − R ) x j x l (cid:104) σ l ⊗ σ k (cid:105) = R T jk + (1 − R ) x j x l T lj . (S.20)Hence, T X = K X T, K X := R X I + (1 − R X ) xx (cid:62) , (S.21)where I is the 3 × R by R X to indicate we are referring to the measurement X . One similarly finds that the Bloch vector a changesto T X a , while the Bloch vector b of the second qubit isof course unchanged by a measurement on the first.Similarly, if instead observer B measures the POVM Y ≡ { Y + , Y − } , then the spin correlation matrix changesto T Y = T L Y , L Y := R Y I + (1 − R Y ) yy (cid:62) , (S.22)and the Bloch vector of the second qubit to L Y b . It fur-ther follows that if both observers make a measurement,then the spin matrix and Bloch vectors a , b change to T XY = K X T L Y , K X a , L Y b , (S.23)respectively.Finally, if A and B instead each measure one of twoPOVMs, X or X (cid:48) and Y or Y (cid:48) , with equal probabilities,it follows that K X and L Y above are replaced by the‘average’ matrices K and L with with K := ( K X + K X (cid:48) ) , L := ( L Y + L Y (cid:48) ) , (S.24)as per Eq. (9) of the main text. III. SIMPLIFYING THE CONJECTUREA. Derivation of S ∗ ( A , B ) and S ∗ ( A , B ) We first demonstrate that, for given observables
X, X (cid:48) , Y, Y (cid:48) measured by A and B , that A and B canviolate the CHSH inequality only if S ∗ ( A , B ) > A and B can violate the CHSH inequality only if S ∗ ( A , B ) > A , B ), suppose that A measures observables corresponding to W = B W +upplemental Material – 3/9 S W σ · w and W (cid:48) = B W (cid:48) + S W (cid:48) σ · w (cid:48) . Hence, the CHSHparameter for ( A , B ) follows from Eqs. (1) and (3) as S ( A , B ) = (cid:104) W ⊗ ( Y + Y (cid:48) ) (cid:105) + (cid:104) W (cid:48) ⊗ ( Y − Y (cid:48) ) (cid:105) = B W (cid:104) Y + Y (cid:48) (cid:105) + S W (cid:104) σ · w ⊗ ( Y + Y (cid:48) ) (cid:105) + B W (cid:48) (cid:104) Y − Y (cid:48) (cid:105) + S W (cid:48) (cid:104) σ · w (cid:48) ⊗ ( Y − Y (cid:48) ) (cid:105) . (S.25)This is linear in the bias B W and B W (cid:48) , and hence, re-calling that S + |B| ≤
1, it achieves its extremal valuesfor fixed strengths S W , S W at B W = α (1 − S W ) and B W (cid:48) = β (1 − S W (cid:48) ), for α, β = ±
1, yielding S ( A , B ) ≤ max α,β f αβ (S.26)with f αβ := α (1 − S W ) (cid:104) Y + Y (cid:48) (cid:105) + S W (cid:104) σ · w ⊗ ( Y + Y (cid:48) ) (cid:105) + β (1 − S W (cid:48) ) (cid:104) Y − Y (cid:48) (cid:105) + S W (cid:48) (cid:104) σ · w (cid:48) ⊗ ( Y − Y (cid:48) ) (cid:105) . Again, since f αβ is linear in the measurement strengths S W , S W (cid:48) , its extreme values must be achieved at S W , S W (cid:48) = 0 ,
1. Hence, we only need to analyse its val-ues at these points. First, for S W = S W (cid:48) = 0 we have f αβ = ( α + β ) (cid:104) Y (cid:105) + ( α − β ) (cid:104) Y (cid:48) (cid:105) ≤ | α + β | + | α − β | = 2,and so the CHSH inequality cannot be violated for thischoice. Second, for S W = 0 and S W (cid:48) = 1 we have f αβ = α ( (cid:104) Y + Y (cid:48) (cid:105) + (cid:104) σ · w (cid:48) ⊗ ( Y − Y (cid:48) ) (cid:105) = 2 α (cid:0) (cid:104) P α w (cid:48) ⊗ Y (cid:105) + (cid:104) P − α w (cid:48) ⊗ Y (cid:48) (cid:105) (cid:1) ≤ , where P x = ( + σ · x ) denotes the projection ontounit spin direction x , and so the CHSH inequality againcannot be violated for this choice, nor, by symmetry, forthe choice S W = 1 and S W (cid:48) = 0. Thus, it is only possiblefor ( A , B ) to violate the inequality for the remainingchoice S W = S W (cid:48) = 1, for which we have, via Eq. (S.26), S ( A , B ) ≤ (cid:104) w · σ ⊗ ( Y + Y (cid:48) ) (cid:105) + (cid:104) w (cid:48) · σ ⊗ ( Y − Y (cid:48) ) (cid:105) , (S.27)Note that equality holds for W = σ · w and W (cid:48) = σ · w (cid:48) . Hence, ( A , B ) can violate the CHSH inequality ifand only if they can violate it via A making projectivemeasurements. A similar result holds for ( A , B ) bysymmetry.Moreover, we can find the optimal projective measure-ments for A to make (and similarly for B ), as fol-lows. First, for projective measurements W = σ · w and W (cid:48) = σ · w (cid:48) , we have S ( A , B ) = (cid:104) W ⊗ ( Y + Y (cid:48) ) (cid:105) + (cid:104) W (cid:48) ⊗ ( Y − Y (cid:48) ) (cid:105) = ( B Y + B Y (cid:48) ) (cid:104) W (cid:105) + (cid:104) W ⊗ σ · ( ˜ y + ˜ y (cid:48) ) (cid:105) + ( B Y − B Y (cid:48) ) (cid:104) W (cid:48) (cid:105) + (cid:104) W (cid:48) ⊗ σ · ( ˜ y − ˜ y (cid:48) ) (cid:105) = w · [( B Y + B Y (cid:48) ) K a + KT ( ˜ y + ˜ y (cid:48) )]+ w (cid:48) · [( B Y − B Y (cid:48) ) K a + KT ( ˜ y − ˜ y (cid:48) )] ≤ | ( B Y + B Y (cid:48) ) K a + KT ( ˜ y + ˜ y (cid:48) ) | + | ( B Y − B Y (cid:48) ) K a + KT ( ˜ y − ˜ y (cid:48) ) | = S ∗ ( A , B ) , (S.28)where ˜ y := S Y y , ˜ y (cid:48) := S Y (cid:48) y (cid:48) , a is A ’s Bloch vectorfor the initial shared state, and T is the spin correlationmatrix for the initial shared state. We have used the factthat A ’s Bloch vector is K a , and the spin correlationmatrix for the state shared by A and B is KT (seeSec. II above). Equality holds in the last line by choosing w to be the unit vector in the ( B Y + B Y (cid:48) ) K a + KT ( ˜ y +˜ y (cid:48) ) direction and w (cid:48) to be the unit vector in the ( B Y −B Y (cid:48) ) K a + KT ( ˜ y − ˜ y (cid:48) ) direction. Hence, A and B canviolate the CHSH inequality only if S ∗ ( A , B ) >
2, asclaimed in the main text. It may similarly be shownthat A and B can violate the CHSH inequality only if S ∗ ( A , B ) > B. Optimality of the singlet state forunbiased observables
Consider now the case that the observables are unbi-ased (i.e., B X = B X (cid:48) = B Y = B Y (cid:48) = 0). Prior work[3–22] has in fact been restricted to this case. We showthat validity of the conjecture can then be reduced totesting it on the singlet state for a 9-parameter subset ofobservables.First, using Eq. (1) of the main text and Eqs. (S.21)–(S.23) above, it follows for unbiased observables thatthe CHSH parameters for each pair ( A j , B k ) are convex-linear in the spin correlation matrix T of the initial state(and are independent of the Bloch vectors). Further, anyphysical spin correlation matrix T can be written as aconvex combination of the spin correlation matrices ofmaximally entangled states, as follows from the proof ofProposition 1 of [36] (in particular, T can be expressedas a mixture of the four Bell states corresponding to abasis in which T is diagonal).Now, any maximally entangled spin correlation matrixcan be written as T me = R (cid:48) T R (cid:48)(cid:48)(cid:62) , where T = − I is thespin correlation matrix of the singlet state and R (cid:48) , R (cid:48)(cid:48) arelocal rotations of the first and second qubits. Hence, sincethe set of possible measurements is invariant under suchrotations, it follows that searching the CHSH parametersover all measurements for a given T me is equivalent tosearching over all measurements for T , i.e, for the singletupplemental Material – 4/9state (corresponding to taking α = − π/ R (cid:48) = R (cid:48)(cid:48) ,the measurement direction x for X can be fixed withoutloss of generality, as can the plane spanned by measure-ment directions x and x (cid:48) . Hence, the directions corre-sponding to X, X (cid:48) , Y, Y (cid:48) that need to be considered, forthe purposes of the conjecture, form a 5-parameter set(the angle between x and x (cid:48) in the given plane, and theangles specifying y and y (cid:48) ).Finally, for unbiased observables the only remainingfree parameters are the four measurement strengths, S X , S X (cid:48) , S Y , S Y (cid:48) . Hence, the conjecture need only betested for this case, whether numerically or analyti-cally, for a 9-parameter subset of observables on a fixedmaximally-entangled state, as claimed in the main text. IV. NUMERICAL EVIDENCE FORTHE CONJECTURE
As per the main text, verification of our conjectureonly requires consideration of the values of S ( A , B ) andthe proxy quantities S ∗ ( A , B ) , S ∗ ( A , B ) , S ∗ ( A , B )in Eqs. (10)–(12) of the main text, for the set of pureinitial states in Eq. (8) of the main text. We have numer-ically sampled these values over the 16-parameter spaceof possible observables X, X (cid:48) , Y, Y (cid:48) for A and B , totest the conjecture, for several values of α in Eq. (8), in-cluding the case α = π/ choices of ob-servables, for the initial states corresponding to sin 2 α =0 . , . , . , . , . , . , . α = π/ ∼ data points are depicted in each panel of Fig. 2 ofthe main text.It can be seen that the data points in the left handpanel of Fig. 2 are asymmetrically distributed, with arelatively small number of points corresponding to values | S ( A , B ) | >
2, in comparison to a relatively large num-ber of points corresponding to values of S ∗ ( A , B ) > S ∗ ( A , B ) rather than of S ( A , B ). In par-ticular, most choices of the observables measured by thepair ( A , B ) are typically non-optimal, and so will typ-ically yield values of | S ( A , B ) | less than 2 even whenit is possible for the pair to violate the CHSH inequal-ity. However, these choices directly determine the valueof S ∗ ( A , B ) via Eq. (10) of the main text, where thisvalue is always greater than 2 whenever ( A , B ) can vi-olate the CHSH inequality (and corresponds in this caseto the optimal measurements specified by the Horodeckicriterion). Thus, the value of S ∗ ( A , B ) involves an inherent degree of optimisation, explaining the relativeasymmetry. Indeed, if instead values of | S ( A , B ) | wereplotted, based on same choices of observables for the pair( A , B ), then these values would typically be less than2 just as for | S ( A , B ) | .Finally, it is seen from Fig. 2 that the one-sidedmonogamy relations in Eqs. (13) and (14) of the maintext do not hold for all possible choices of observables by A and B , since some points lie above the red curves.However, it is of interest to ask whether these rela-tions might hold for the special case of unbiased ob-servables, i.e., B X = B X (cid:48) = B Y = B Y = 0. Forthis case only the singlet state need be considered (seeSec. III.B above), and hence we sampled over 10 pointsfor this state, to investigate this question. The resultssupport a conjecture that the one-sided monogamy rela-tion | S ( A , B ) | + S ∗ ( A , B ) ≤ all unbiased observables, i.e., even without making equalstrength and/or orthogonality assumptions. In contrast,the monogamy relation S ∗ ( A , B ) + S ∗ ( A , B ) ≤ V. ONE-SIDED MONOGAMY RELATIONS
Equation (2) in the conjecture given in the main text isequivalent to the requirement that the CHSH parameterssatisfy the general one-sided monogamy relations (cid:12)(cid:12) | S ( A , B ) | + | S ( A , B ) | − (cid:12)(cid:12) + (cid:12)(cid:12) | S ( A , B ) | − | S ( A , B ) | (cid:12)(cid:12) | ≥ , (S.29) | S ( A , B ) | + | S ( A , B ) | − | ++ || S ( A , B ) | − | S ( A , B ) || ≥ . (S.30)In particular, the first relation rules out values of S ( A , B ) and S ( A , B ) that are both greater than 2,and the second relation similarly rules out values of S ( A , B ) and S ( A , B ) that are both greater than 2.Note that these relations are saturated (up to the maxi-mum values of 2 √
2. For example, if A and B measurethe trivial observables X = X (cid:48) = Y = Y (cid:48) = B , then S ( A , B ) = 2 B ranges over [0 ,
2] while the lack of dis-turbance allows | S ( A , B ) | to range over the full quan-tum range [0 , √ A and B measure these trivial observables, and asimilar saturation is obtained via ( A , B ) and ( A , B )making trivial measurements.It follows from the main text that the con-jecture is also equivalent to the above relationswith S ( A , B ) , S ( A , B ) , S ( A , B ) replaced by theproxy quantities S ∗ ( A , B ) , S ∗ ( A , B ) , S ∗ ( A , B ), cor-responding to requiring the points in Fig. 2 of the maintext to lie outside the shaded regions. Here we deriveupplemental Material – 5/9the (less general but stronger) one-sided monogamy rela-tions for the proxy quantities discussed in the main text.These hold for the cases of (i) unbiased observables, i.e.,with B X = B X (cid:48) = B Y = B Y (cid:48) = 0 , (S.31)(note that prior work [3-22] is confined to this case),and/or (ii) states with zero Bloch vectors, i.e., with a = b = . (S.32)(which includes all maximally entangled states), in com-bination with any of several mild measurement assump-tions. A. One-sided monogamy relation for ( A , B ) and ( A , B ) Here we prove the monogamy relation in Eq. (13) ofthe main text, i.e., S ∗ ( A , B ) + S ∗ ( A , B ) ≤ , (S.33)for each of the cases in Eqs. (S.31) and (S.32), under theadditional assumption of equal measurement strengthsfor each side. We also prove this relation holds under thealternative additional assumption of orthogonal measure-ment directions for each side. It follows that the proxyquantities S ∗ ( A , B ) and S ∗ ( A , B ) cannot both vio-late the CHSH inequality under such restrictions. Hence,as per Sec. III.A above, neither can both S ( A , B ) and S ( A , B ), thus confirming the conjecture under theserestrictions.
1. Convexity considerations
To derive the monogamy relations, some convexityproperties are needed to simplify the dependence of thequantities on the spin correlation matrices.First, note for either of the above cases in Eqs. (S.31)and (S.32), that Eqs. (11) and (12) of the main text sim-plify to S ∗ ( A , B ) = | LT (cid:62) ( ˜ x + ˜ x (cid:48) ) | + | LT (cid:62) ( ˜ x − ˜ x (cid:48) ) | , (S.34) S ∗ ( A , B ) = | KT ( ˜ y + ˜ y (cid:48) ) | + | KT ( ˜ y − ˜ y (cid:48) ) | , (S.35)where ˜ x = S X x , etc. Importantly, these quantitiesare convex-linear with respect to the initial spin corre-lation matrix T . Further, the latter can always be writ-ten as a convex-linear combination T = (cid:80) j w j T j of atmost four spin correlation matrices T j , corresponding tomaximally-entangled states [36] (specifically, to the fourBell states defined by the local basis sets in which T isdiagonal). Moreover, any maximally entangled state isrelated to the singlet state by local rotations, implying that T j = R (cid:48) j T R (cid:48)(cid:48) j = − R (cid:48) j R (cid:48)(cid:48) j =: − R j , where R (cid:48) j , R (cid:48)(cid:48) j and R j = R (cid:48) j R (cid:48)(cid:48) j are rotation matrices, and T = − I is thespin correlation matrix of the singlet state.Hence, since | z | is a convex function, S ∗ ( A , B ) ≤ (cid:88) j w j | LR (cid:62) j ( ˜ x + ˜ x (cid:48) ) | + | LR (cid:62) j ( ˜ x − ˜ x (cid:48) ) |≤ max R {| LR (cid:62) ( ˜ x + ˜ x (cid:48) ) | + | LR (cid:62) ( ˜ x − ˜ x (cid:48) ) |} , (S.36)where the maximum is over all rotations R , and similarly S ∗ ( A , B ) ≤ max R {| KR ( ˜ y + ˜ y (cid:48) ) | + | KR ( ˜ y − ˜ y (cid:48) ) |} . (S.37)These results will be used in obtaining Eq. (S.33) forequal measurement strengths.Further, since | z | is a convex function it also followsthat S ∗ ( A , B ) + S ∗ ( A , B ) ≤ max R (cid:110) [ | KR ( ˜ y + ˜ y (cid:48) ) | + | KR ( ˜ y − ˜ y (cid:48) ) | ] + (cid:2) | LR (cid:62) ( ˜ x + ˜ x (cid:48) ) | + | LR (cid:62) ( ˜ x − ˜ x (cid:48) ) | (cid:3) (cid:111) . (S.38)This result will be used in obtaining Eq. (S.33) for or-thogonal measurement directions.
2. Equal strengths for each side
Under the additional assumption that the measure-ments X and X (cid:48) by A have equal strengths, and simi-larly for the measurements Y and Y (cid:48) by B , i.e., S X = S X (cid:48) , S Y = S Y (cid:48) , (S.39)Choosing R to be the rotation saturating Eq. (S.36) thenyields, S ∗ ( A , B ) ≤S X (cid:0) | LR (cid:62) ( x + x (cid:48) ) | + | LR (cid:62) ( x − x (cid:48) ) | (cid:1) =4 S X (cid:0) | LR (cid:62) x | cos θ + | LR (cid:62) x | sin θ (cid:1) ≤ S X (cid:0) x (cid:62) RL (cid:62) LR (cid:62) x + x (cid:62) RL (cid:62) LR (cid:62) x (cid:1) ≤ S X (cid:2) s ( L (cid:62) L ) + s ( L (cid:62) L ) (cid:3) = S X (cid:2) (1 + R Y ) + (1 + R Y (cid:48) ) +2(1 − R Y )(1 − R Y (cid:48) )( y · y (cid:48) ) (cid:3) ≤S X (cid:2) (1 + R Y ) + (1 + R Y (cid:48) ) +2(1 − R Y )(1 − R Y (cid:48) )] ≤S X (cid:2) (1 + R Y ) + (1 + R Y (cid:48) ) +(1 − R Y ) + (1 − R Y (cid:48) ) (cid:3) =2 S X (cid:2) R Y + R Y (cid:48) ) (cid:3) . (S.40)Here, x := 2 cos θ ( x + x (cid:48) ) and x := 2 sin θ ( x − x (cid:48) ) are or-thogonal unit vectors defined via the half-angle θ betweenupplemental Material – 6/9 x and x (cid:48) (implying that R (cid:62) x and R (cid:62) x are similarly or-thogonal), and the singular values of L (cid:62) L (equivalent tothe eigenvalues thereof) have been calculated via Eq. (9)of the main text. We similarly find, via Eq. (S.37), that S ∗ ( A , B ) ≤ S Y (cid:2) R X + R X (cid:48) (cid:3) . (S.41)Finally, noting from the fundamental tradeoff rela-tion (S.11) that S X ≤ min { (cid:113) − R X , (cid:113) − R X (cid:48) } , (S.42) S Y ≤ min { (cid:113) − R Y , (cid:113) − R Y (cid:48) } , (S.43)Eqs. (S.40) and (S.41) yield S ∗ ( A , B ) + S ∗ ( A , B ) ≤ S X (cid:2) (1 + R Y ) + (1 + R Y (cid:48) ) (cid:3) + 2 S Y (cid:2) (1 + R X ) + (1 + R X (cid:48) ) (cid:3) . ≤ − R X )(1 + R Y ) + 2(1 − R Y )(1 + R X )+ 2(1 − R X (cid:48) )(1 + R Y (cid:48) ) + 2(1 − R Y (cid:48) )(1 + R X (cid:48) )= 4(1 − R X R Y ) + 4(1 − R X (cid:48) R Y (cid:48) ) ≤ , (S.44)as claimed in Eq. (13) of main text and Eq. (S.33) above.
2. Orthogonal measurement directions for each side
We now drop the equal strength assumption (S.39),and instead assume that observables X and X (cid:48) have or-thogonal measurement directions, as do observables Y and Y (cid:48) , i.e., that x · x (cid:48) = 0 , y · y (cid:48) = 0 . (S.45)We first show that we only need to consider the casewhere x , x (cid:48) , R y , R y (cid:48) lie in the same plane, for any ro-tation R in Eq. (S.38). In particular, defining x (cid:48)(cid:48) := x × x (cid:48) , y (cid:48)(cid:48) := y × y (cid:48) , note it follows from Eq. (9) of themain text and the orthogonality condition (S.45) that K = 1 + R X (cid:48) xx (cid:62) + 1 + R X x (cid:48) x (cid:48)(cid:62) + R X + R X (cid:48) x (cid:48)(cid:48) x (cid:48)(cid:48)(cid:62) (S.46) L = 1 + R Y (cid:48) yy (cid:62) + 1 + R Y y (cid:48) y (cid:48)(cid:62) + R Y + R Y (cid:48) y (cid:48)(cid:48) y (cid:48)(cid:48)(cid:62) . Hence, for a given rotation matrix R , one finds againusing the orthogonality condition that | KR ( ˜ y ± ˜ y (cid:48) ) | = 14 (cid:8) (1 + R X (cid:48) ) [ x · R ( ˜ y ± ˜ y (cid:48) )] + (1 + R X ) [ x (cid:48) · R ( ˜ y ± ˜ y (cid:48) )] + ( R X + R X (cid:48) ) [ x (cid:48)(cid:48) · R ( ˜ y ± ˜ y (cid:48) )] (cid:9) . (S.47) Since R X , R X (cid:48) ≤ | KR ( ˜ y ± ˜ y (cid:48) ) | ismaximised for any R by choosing directions such that x (cid:48)(cid:48) is orthogonal to R ( ˜ y ± ˜ y (cid:48) ), i.e., such that x , x (cid:48) liein the same plane as R y , R y (cid:48) . One similarly finds that | LR (cid:62) ( ˜ x ± ˜ x (cid:48) ) | is maximised by choosing directions suchthat y , y (cid:48) lie in the same plane as R (cid:62) x , R (cid:62) x (cid:48) , i.e., againsuch that x , x (cid:48) lie in the same plane as R y , R y (cid:48) . Notethat the latter two vectors are also orthogonal to eachother.Thus, choosing R to be the rotation saturatingEq. (S.38), and introducing the parameter β to charac-terise the relative angles between (coplanar) x , x (cid:48) and R y , R y (cid:48) , i.e., R y = x cos β + x (cid:48) sin β, R y (cid:48) = x sin β − x (cid:48) cos β. (S.48)and x = R y cos β + R y (cid:48) sin β, x (cid:48) = R y sin β − R y (cid:48) cos β, (S.49)we find via Eq. (S.47) that | KR ( ˜ y ± ˜ y (cid:48) ) | = 14 (1 + R X (cid:48) ) [ x · R ( S Y y ± S Y (cid:48) y (cid:48) )] + 14 (1 + R X ) [ x (cid:48) · R ( S Y y ± S Y (cid:48) y (cid:48) )] = 14 (1 + R X (cid:48) ) ( S Y cos β ± S Y (cid:48) sin β ) + 14 (1 + R X ) ( S Y sin β ∓ S Y (cid:48) cos β ) . (S.50)Hence, using (( a + b ) ≤ ( a + b ) + ( a − b ) = 2( a + b )and the fundamental tradeoff relation (S.11) yields( | KR ( ˜ y + ˜ y (cid:48) ) | + | KR ( ˜ y − ˜ y (cid:48) ) | ) ≤ | KR ( ˜ y + ˜ y (cid:48) ) | + | KR ( ˜ y − ˜ y (cid:48) ) | )= (1 + R X (cid:48) ) ( S Y cos β + S Y (cid:48) sin β )+ (1 + R X ) ( S Y sin β + S Y (cid:48) cos β ) ≤ (1 + R X (cid:48) ) (1 − R Y ) cos β + (1 + R X (cid:48) ) (1 − R Y (cid:48) ) sin β + (1 + R X ) (1 − R Y ) sin β + (1 + R X ) (1 − R Y (cid:48) ) cos β = (1 + R X ) (1 − R Y ) + (1 + R X (cid:48) ) (1 − R Y (cid:48) )+ (cid:2) (1 + R X (cid:48) ) − (1 + R X ) (cid:3) ( R Y (cid:48) − R Y ) cos β. (S.51)Similarly, we obtain( | LR (cid:62) ( ˜ x + ˜ x (cid:48) ) | + | LR (cid:62) ( ˜ x − ˜ x (cid:48) ) | ) ≤ (1 − R X )(1 + R Y ) + (1 − R X (cid:48) )(1 + R Y (cid:48) ) + ( R X (cid:48) − R X ) (cid:2) (1 + R Y (cid:48) ) − (1 + R Y ) (cid:3) cos β. (S.52)Substituting Eqs. (S.51) and (S.52) into Eq. (S.38) thenupplemental Material – 7/9gives S ∗ ( A , B ) + S ∗ ( A , B ) ≤ (1 − R Y )(1 + R X ) + (1 − R Y (cid:48) )(1 + R X (cid:48) ) + (1 − R X )(1 + R Y ) + (1 − R X (cid:48) )(1 + R Y (cid:48) ) + ( R Y (cid:48) − R Y ) (cid:2) (1 + R X (cid:48) ) − (1 + R X ) (cid:3) cos β + ( R X (cid:48) − R X ) (cid:2) (1 + R Y (cid:48) ) − (1 + R Y ) (cid:3) cos β = P + ( R X − R X (cid:48) )( R Y − R Y (cid:48) ) Q cos β, (S.53)where P :=(1 − R Y )(1 + R X ) + (1 − R Y (cid:48) )(1 + R X (cid:48) ) + (1 − R X )(1 + R Y ) + (1 − R X (cid:48) )(1 + R Y ) (S.54)and Q :=( R X + R X (cid:48) + 2)( R Y + R Y (cid:48) )+ ( R X + R X (cid:48) )( R Y + R Y (cid:48) + 2) . (S.55)Finally, noting that P, Q ≥
0, if ( R X − R X (cid:48) )( R Y −R Y (cid:48) ) ≤
0, then Eq. (S.53) is maximised for cos β = 0,implying S ∗ ( A , B ) + S ∗ ( A , B ) ≤ (1 − R Y )(1 + R X ) + (1 − R Y (cid:48) )(1 + R X (cid:48) ) + (1 − R X )(1 + R Y ) + (1 − R X (cid:48) )(1 + R Y (cid:48) ) ≤ R X , R Y (cid:2) (1 − R Y )(1 + R X ) + (1 − R X )(1 + R Y ) (cid:3) = 8 , (S.56)while if ( R X − R X (cid:48) )( R Y − R Y (cid:48) ) ≥
0, then Eq. (S.53) ismaximised for cos β = 1, implying S ∗ ( A , B ) + S ∗ ( A , B ) ≤ (1 − R Y (cid:48) )(1 + R X ) + (1 − R X )(1 + R Y (cid:48) ) + (1 − R Y (cid:48) )(1 + R X ) + (1 − R X )(1 + R Y (cid:48) ) ≤ R X , R Y (cid:48) (cid:2) (1 − R Y (cid:48) )(1 + R X ) + (1 − R X )(1 + R Y (cid:48) ) (cid:3) = 8 . (S.57)Thus, in either case we again obtain the one-sidedmonogamy relation in Eq. (S.33), as claimed in the maintext. B. One-sided monogamy relation for ( A , B ) and ( A , B ) We now prove the one-sided monogamy relation inEq. (14) of the main text, under the combined assump-tions of unbiased observables, and equal strengths andorthogonal measurement directions for each side. In fact, for this combination the upper bound can be improvedto | S ( A , B ) | + S ∗ ( A , B ) ≤ √ ∼ . < , (S.58)as noted in the main text. Hence, S ( A , B ) and theproxy quantity S ∗ ( A , B ) cannot both violate the CHSHinequality, implying that neither can both S ( A , B ) and S ( A , B ) (see main text), thus confirming the conjecturefor this case. One-sided monogamy relations for moregeneral cases, in which either of the equal strength ororthogonality assumptions is dropped, will be derived ina forthcoming paper [26], via a significant generalisationof the Horodecki criterion.First, under the above assumptions it follows fromSec. III.B that the result needs only to be proved forthe singlet state, i.e, for T = − I and a = b = 0. But forthis state the CHSH parameter for the pair ( A , B ) canbe calculated via Eqs. (1) and (3) of the main text andEq. (S.31), (S.39) and (S.45) above to give | S ( A , B ) | = S X S Y | x · y + x · y (cid:48) + x (cid:48) · y − x (cid:48) · y (cid:48) |≤ S X S Y [ | x · ( y + y (cid:48) ) | + | x (cid:48) · ( y − y (cid:48) ) | ] ≤ S X S Y [ | y + y (cid:48) | + | y − y (cid:48) | ]= 2 √ S X S Y = 2 √ (cid:113) − R X (cid:113) − R Y , (S.59)where the last line follows by noting that the fundamentaltradeoff relation (S.11) is saturated for unbiased observ-ables.Moreover, from the Horodecki criterion in Eq. (9) ofthe main text, it follows for the singlet state that S ∗ ( A , B ) ≤ s ( KL ) + s ( KL ) ] ≤ s ( K ) s ( L ) + s ( K ) s ( L ) ] , (S.60)using Theorem IV.2.5 of Ref. [38]. Further, the matrices K and L follow from Eq. (9) of the main text under theequal strengths assumption as (again noting the satura-tion of tradeoff relation (S.11)) K = R X I + (1 − R X )( xx (cid:62) + x (cid:48) x (cid:48)(cid:62) ) , (S.61) L = R Y I + (1 − R Y )( yy (cid:62) + y (cid:48) y (cid:48)(cid:62) ) . (S.62)The orthogonality assumption then yields s ( K ) = s ( K ) = (1 + R X ) and s ( L ) = s ( L ) = (1 + R Y )by inspection, and so S ∗ ( A , B ) ≤ (1 + R X ) (1 + R Y ) . (S.63)Finally, defining f ( x ) = 2 √ − x and g ( x ) := 1 + x ,upplemental Material – 8/9Eqs. (S.59) and (S.63) give | S ( A , B ) | + | S ∗ ( A , B ) |≤ f ( R X ) f ( R Y ) + g ( R X ) g ( R Y ) √ ≤ (cid:112) f ( R X ) + g ( R X ) (cid:112) f ( R Y ) + g ( R Y ) √ ≤ max R f ( R ) + g ( R ) √
2= max R − (1 − R ) √
2= 163 √ , (S.64)as claimed, where the second inequality follows via m · n ≤ | m || n | for m = ( f ( R X ) , g ( R X )) and n =( f ( R Y ) , g ( R Y )). Note that the maximum is achievedfor S X = S Y = 2 √ ∼ . , R X = R Y = 13 , (S.65)and the optimal CHSH directions. VI. BIASED MEASUREMENT SELECTIONS
Recall that, in the example of the main text, X and Y are each selected with probability 1 − (cid:15) and mea-sured with strengths S X = S Y = S and reversibilities R X = R Y = R ; X (cid:48) and Y (cid:48) are selected with probability (cid:15) and are projective, with strengths S X (cid:48) = S Y (cid:48) = 1 andreversibilities R X (cid:48) = R Y (cid:48) = 0; and the measurement di-rections correspond to the optimal CHSH directions [27],i.e., x and x (cid:48) are orthogonal with y = ( x + x (cid:48) ) / √ y (cid:48) = ( x − x (cid:48) ) / √ K and L in Eq. (S.24) to K (cid:15) := (1 − (cid:15) ) K X + (cid:15)K X (cid:48) = (1 − (cid:15) )[ R I + (1 − R ) xx (cid:62) ] + (cid:15) x (cid:48) x (cid:48)(cid:62) , (S.66) L (cid:15) := (1 − (cid:15) ) L Y + (cid:15)L Y (cid:48) = (1 − (cid:15) )[ R I + (1 − R ) yy (cid:62) ] + (cid:15) y (cid:48) y (cid:48)(cid:62) , (S.67)as per the main text.For the optimal CHSH directions this gives, in the { x , x (cid:48) , x × x (cid:48) } basis, K (cid:15) = − (cid:15) R + (cid:15) (1 − R ) 00 0 (1 − (cid:15) ) R , (S.68) L (cid:15) = − (cid:15) ) R − (1 − (cid:15) ) R − (cid:15) − (1 − (cid:15) ) R − (cid:15) − (cid:15) ) R
00 0 (1 − (cid:15) ) R . (S.69) Only the principal 2 × K (cid:15) and L (cid:15) contribute to the two largest singular values of K (cid:15) L (cid:15) (the smallest value is (1 − (cid:15) ) R ), implying s ( K (cid:15) L (cid:15) ) + s ( K (cid:15) L (cid:15) ) is given by the trace of the 2 × K (cid:15) L (cid:15) )( K (cid:15) L (cid:15) ) (cid:62) = K (cid:15) L (cid:15) K (cid:15) , which may beevaluated to give s ( K (cid:15) L (cid:15) ) + s ( K (cid:15) L (cid:15) ) = (cid:2) R − (cid:15) (1 − R + R )+ (cid:15) (2 − R + R ) (cid:3) . (S.70)It immediately follows via Eq. (10) of the main text that A and B can violate the CHSH inequality only if S ∗ ( A , B ) = √ (cid:2) R − (cid:15) (1 − R + R )+ (cid:15) (2 − R + R ) (cid:3) > , (S.71)which is equivalent to R > R − ( (cid:15) ) := [ √ − (1 − (cid:15) ) ] / − (cid:15) − (cid:15) , (S.72)as per Eq. (16) of the main text. This is monotonic in-creasing in (cid:15) , implying in particular that one must have R > R − (0) = ( √ − / ∼ . . (S.73)This corresponds, via tradeoff relation (S.11), to the up-per bound S < S max := (cid:112) − R − (0) = (2 − √ / ∼ . . (S.74)on the strength S , as noted in the main text.Note that the requirement R − ( (cid:15) ) ≤ (cid:15) of selecting a projective measurement X (cid:48) or Y (cid:48) cannot be arbitrarily large. In particular, Eq. (S.72)can be inverted to give (cid:15) − ( R ) = 1 − R + R − (cid:113) √ − R + R ) − − R + R , (S.75)which is a monotonic increasing function of R , and henceviolation of the CHSH inequality by A and B is onlypossible at all if (cid:15) < (cid:15) − (1) = 1 − ( √ − / ∼ . . (S.76)Moreover, for both pairs ( A , B ) and ( A , B ) to vio-late the CHSH inequality one further requires that thestrength satisfy S > S min := 8 / − ∼ . , (S.77)as per Eq. (16) of the main text, which together withtradeoff relation (S.11) and Eq. (S.72) implies that R and (cid:15) must satisfy R − ( (cid:15) ) < R < R + , (S.78)upplemental Material – 9/9with R + := (cid:113) − S = 2 / (cid:112) / − ∼ . . (S.79)Eq. (S.78) can clearly be satisfied for sufficiently smallvalues of (cid:15) , i.e., for sufficiently large selection biases (since R − (0) < R + ). Hence, it is possible to have two-qubit re-cycling of Bell nonlocality for sufficiently biased measure-ment selections. The maximum possible value of (cid:15) thatallows such a violation of one-sided monogamy followsfrom Eqs. (S.75) and (S.78) as (cid:15) max = (cid:15) − ( R + ) ∼ . ∼ . , (S.80)as per Eq. (17) of the main text.Finally, as noted in the main text, the pairs ( A , B )and ( A , B ) can also violate a CHSH inequality underthe above conditions. This expected, on the grounds thatthese share qubits of which only one qubit has been re-cycled following measurement, so that it should be eveneasier for them to violate a Bell inequality than it is for( A , B ), as both qubits of the latter pair have been recy-cled. We demonstrate this explicitly below by consideringthe optimal case that (cid:15) approaches zero.In particular, in this limit K and L in Eqs. (10)and (11) of the main text are replaced by K and L inEqs. (S.66) and (S.67). Further, for the optimal CHSHdirections one finds K ˜ y = S √ (cid:18) R (cid:19) , K ˜ y (cid:48) = 1 √ (cid:18) −R (cid:19) , (S.81)and hence, noting that the Bloch vectors a and b vanishfor the singlet state, Eq. (10) of the main text is replacedby S ∗ ( A , B ) = 1 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) S−R (1 − S ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 1 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − S−R (1 + S ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 1 √ (cid:2) (1 + S ) + R (1 − S ) (cid:3) / + 1 √ (cid:2) (1 − S ) + R (1 + S ) (cid:3) / (S.82) in the limit (cid:15) →
0. Using the tradeoff relation (S.11) thengives S ∗ ( A , B ) ≤ √ (cid:2) (1 + S ) + (1 − S )(1 − S ) (cid:3) / + 1 √ (cid:2) (1 − S ) + (1 − S )(1 + S ) (cid:3) / (S.83)in this limit. One similarly finds L ˜ x = S (cid:18) R − R (cid:19) , L ˜ x (cid:48) = 12 (cid:18) − R R (cid:19) , (S.84)which yields the same upper bound for S ∗ ( A , B ) viavia Eq. (11) of the main text.Hence, both pairs can violate a CHSH inequality if thisupper bound is greater than 2, corresponding to S < S := (cid:113) √ − ∼ . , (S.85)or equivalently to R > R := (cid:113) − √ ∼ . . (S.86)Noting that S > S max in Eq. (S.74, it follows that bothpairs ( A , B ) and ( A , B ) can violate the CHSH in-equality if ( A , B2