Some quantum measurements with three outcomes can reveal nonclassicality where all two-outcome measurements fail to do so
SSome quantum measurements with three outcomes can reveal nonclassicalitywhere all two-outcome measurements fail to do so
H. Chau Nguyen ∗ and Otfried G¨uhne † Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany (Dated: December 14, 2020)Measurements serve as the intermediate communication layer between the quantum world andour classical perception. So, the question which measurements efficiently extract information fromquantum systems is of central interest. Using quantum steering as a nonclassical phenomenon, weshow that there are instances, where the results of all two-outcome measurements can be explainedin a classical manner, while the results of some three-outcome measurements cannot. This points atthe important role of the number of outcomes in revealing the nonclassicality hidden in a quantumsystem. Moreover, our methods allow to improve the understanding of quantum correlations bydelivering novel criteria for quantum steering and improved ways to construct local hidden variablemodels.
Introduction.—
It is widely believed that, at the fun-damental level, our world behaves according to the lawsof quantum mechanics, although we can only perceive itclassically [1]. In fact, realizing the hidden potential ofquantum mechanical systems in information processinghas ignited the burst of quantum information and quan-tum computation during the last years [2]. To transferthe quantum mechanical concepts to that of our familiarclassicality, quantum measurements are required [3]. Thequestion how to use quantum measurements to interactefficiently with quantum mechanical systems is thus ofcentral interest in quantum information theory [3].In 1964, Bell found that measurements performed lo-cally on a bipartite quantum system can yield resultswhich cannot be explained with a classical intuition basedon the assumptions of locality and realism [4, 5]. Thisphenomenon manifests itself as the violation of Bell in-equalities, and a famous example of such an inequalityis the Clauser-Horne-Shimony-Holt (CHSH) inequality,designed for two parties with two measurements, havingtwo outcomes each. Not all entangled states violate theCHSH inequality [6], and one may wonder whether theusage of measurements with more outcomes helps in ob-serving nonclassical behaviors. Is there a quantum statefor which the infinite set of all possible two-outcome mea-surements does not lead to nonclassical effects, but somethree-outcome measurements lead to a Bell inequality vi-olation? This question has not been answered despitedecades of research, arguably due to the complex struc-ture of Bell correlations.There are, however, other nonclassical correlations inquantum mechanics besides the violation of Bell inequal-ities. An important one is captured by the notion ofquantum steering [7, 8]. This phenomenon goes back toSchr¨odinger’s observation that in the Einstein-Podolsky-Rosen argument, one party (typically called Alice) cansteer the state of the other party (called Bob) by makingsuitable measurements [9]. The modern formulation ofthis effect has been given by Wiseman and coworkers [10]and since then it was found to be connected to many sub-
FIG. 1. The set of all bipartite quantum states (here for thecase of dimenion d = 3) can be divided into subsets, depend-ing on how many outcomes measurements must have in orderto show steering, and some states (dark brown, inner most)are not steerable at all. For a given state ρ , the distance tothese sets defines the so-called critical radii R k or R PVM forprojective measurements. We prove that in general R > R ,demonstrating that some states require three-outcome mea-surements for steering. jects in quantum information processing. For instance,it has been shown that the measurements made by Alicehave to be incompatible, implying that commuting mea-surements as in classical physics are not suitable [11, 12].Furthermore, the theory of quantum steering has turnedout to be useful to solve long standing open problems con-cerning Bell inequalities, e.g., the construction of stateshaving a positive partial transpose, but violating a Bellinequality [13, 14].The goal of this paper is twofold. First, we will showthat for some quantum states a finite number of measure-ments of three outcomes can reveal quantum steering,while the infinite set of all measurements with two out-comes cannot. This proves that the number of measure-ment outcomes can be important to the question whethernonclassical effects can be observed or not. We note thatin recent works it has been demonstrated that the cor-relations of certain multi-outcome measurements cannotbe explained by assuming that these measurements them-selves have only two effective outcomes [15–17]. Thus thisdoes not concern the fundamental limitation of the whole a r X i v : . [ qu a n t - ph ] D ec infinite set of two-outcome measurements as comparisonto those with more outcomes in revealing quantum cor-relations.Second, the methods developed in this paper allow oneto advance the theory of quantum steering in several di-rections. In particular, we derive novel criteria for steer-ability and unsteerability, and present significantly im-proved local hidden variable models for so-called Wernerstates, which show that they do not violate any Bell in-equality, even if the most general measurements are con-sidered [18]. Quantum steering.—
Consider the situation where Al-ice and Bob share a bipartite quantum state ρ andAlice performs a measurement (denoted by x ) with n outcomes. This is generally described by a collectionof n positive operators, { E ( x ) a } na =1 , E ( x ) a ≥
0, normal-ized by (cid:80) na =1 E ( x ) a =
1, which form a so-called posi-tive operator valued measure (POVM). Bob’s system isthen found in the ensemble of conditional states { ρ a | x =Tr A [ ρ ( E ( x ) a ⊗ } . It has been noted early that by choos-ing different measurements, Alice can steer Bob’s sys-tem to ensembles that are intuitively ‘incompatible’ witheach other, such as pure eigenstates of noncommutativeobservables, conflicting with our intuition of classical lo-cality [4, 9]. However, it was not until 2007 that thisnaive notion of ‘incompatibility’ gained a precise defini-tion. Wiseman et al. [10] pointed out that incompatibleensembles in general mean that they cannot be derivedfrom a single collection of states, called a local hiddenstate (LHS) ensemble. a LHS ensemble is simply a dis-tribution µ on Bob’s pure states | λ (cid:105) . The different ensem-bles { Tr A [ ρ ( E ( x ) a ⊗ } corresponding to different mea-surement choices x can be derived from the single LHSensemble µ if one can reach any conditional state a LHSTr A [ ρ ( E ( x ) a ⊗ | λ (cid:105) via classical postpro-cessing. That means that there are probabilities G ( x ) a ( λ )such thatTr A [ ρ ( E ( x ) a ⊗ (cid:90) d µ ( λ ) G ( x ) a ( λ ) | λ (cid:105)(cid:104) λ | , (1)where the integration is taken over Bob’s pure states. Ifthis is the case, one says that ρ admits a LHS model, orin short, ρ is unsteerable . The postprocessing functions G ( x ) a ( λ ) are called Alice’s response functions . Being prob-abilities, the response functions G ( x ) a ( λ ) are constrainedby 0 ≤ G ( x ) a ( λ ) ≤ (cid:80) na =1 G ( x ) a ( λ ) = 1. If such a LHSmodel does not exist, one says that ρ is steerable [10]. The role of measurements.—
Crucially for our purpose,Alice’s steering abilities depend on the set of measure-ments M she can potentially make. This allows one toquantify how much steering the measurements of a class M reveal for a state ρ . Specifically, we define the steer-ing critical radius of ρ with respect to measurements in M , denoted R M ( ρ ), to be the maximum of the mixing parameter η such that ρ η = ηρ + (1 − η )( A ⊗ ρ B ) /d A isunsteerable with measurements in M , R M ( ρ ) = max { η ≥ ρ η is unsteerable w.r.t. M } . (2)Here A denotes the identity operator acting on system A and ρ B denotes the reduced state of system B , ρ B =Tr A ( ρ ). Geometrically 1 − R M ( ρ ) measures the distancefrom ρ to the surface separating steerable/unsteerablestates (with measurements in M ) relatively to the noisyand unsteerable state ( A ⊗ ρ B ) /d A , see also Fig. 1.This special choice of the separable state is explainedin Ref.[19, Section A]. There we also show how this defi-nition stems from the critical radius defined in Ref. [20],originally measures certain geometrical object associatedto a two-qubit system. In a similar fashion, we define S ( ρ ) to be the maximum mixing parameter η such that ρ η is separable, i.e., it can be written as a convex combi-nation of product states [6]. The structure of measurements.—
The set of POVMshas a nested structure: measurements with n outcomesare naturally a subset of that of measurements with n +1outcomes. Measurements with two outcomes, so-calleddichotomic measurements, are the most elementary, andalso among the measurements that are most often per-formed in experiments. Measurements whose effects E a are rank-1 projections will be referred to as projectivemeasurements which are the standard measurements oc-curring in textbooks.For M being the set of POVMs of n outcomes, or pro-jective measurements, we simply denote the critical radiiby R n , and R PVM , respectively. Since any POVM canbe written as a mixture of POVMs with at most d out-comes, measurements with n > d A outcomes do not bringany more steerability to Alice [18, 21]. So we can alsodenote R POVM = R d A . Because measurements with n outcomes form a subset of that with n + 1 outcomes, andprojective measurements form a subset of measurementswith d A outcomes, the critical radii organize in the fol-lowing sequence R PVM ≥ R ≥ · · · ≥ R d A ≥ · · · ≥ R d A = R POVM , (3)which is valid for any state. Fig. 1 illustrates this se-quence geometrically.Although difficult to compute, already in their earlypaper, Wiseman et al. [10] remarked that R PVM canbe computed for the Werner states and the isotropicstates. More recently, it has been shown that R canalso be computed for arbitrary two-qubit states [20, 22–24]. Further, numerical evidences suggested that for two-qubit states, the chain in fact collapses to a single value R = R PVM = R = R [20].Here we report a practically closed formula for R forthe high-dimensional isotropic states and Werner states d . η d . . . . FIG. 2. Summary of the results on the steering critical radiifor Werner states (left) and isotropic states (right). Fromtop to bottom, we show the steering critical radii R for di-chotomic measurements from Eqs. (6, 7) (violet), R PVM forprojective measurements from Ref. [10] (green), a lower boundon R POVM for Werner states from Eq. (9) (red), lower boundson R POVM from Ref. [18, 25] (grey), and the separability limit S (orange). and show that R > R PVM ≥ R POVM for systems otherthan qubits. This is in particular true for dimension d = 3: R > R for the three-dimensional isotropic andWerner states. Since by replacing the infinite set of 3-POVMs by a finite subset of measurements, one can ap-proach R (from above) as close as possible. So, there ex-ists a finite set of measurements of three outcomes whichgives a smaller critical radius than R . These three-outcome measurements can then reveal nonclassicality,where all two-outcome measurements cannot. Werner states and the isotropic states.—
Recall thatthe fully antisymmetric state of dimension d × d is de-fined by W d = 2 π − / ( d − d ), where π − is the projectiononto the antisymmetric subspace of C d ⊗ C d , spanned byvectors of the type | ij (cid:105) − | ji (cid:105) [6]. The Werner state atmixing probability η is then defined by mixing this pro-jection with the white noise, W dη = ηW d + (1 − η )( /d ) ⊗ ( /d ). This is in line with the notation introduced beforeEq. (2), as we have Tr A ( W d ) = /d. By construction, the Werner states are symmetric un-der application of the same local unitary operation U ∈ U( d ) on both parties, namely, W dη = ( U ⊗ U ) W dη ( U † ⊗ U † ) [6]. It has been shown that Werner states are separa-ble if and only if η ≤ / ( d +1) [6], which, can be written inthe above notation as S ( W d ) = 1 / ( d + 1). Werner statesare unsteerable with projective measurements if and onlyif η ≤ − /d [6, 10], thus R PVM ( W d ) = 1 − /d .To define the isotropic states, one first considers themaximally entangled state on C d ⊗ C d , defined by S d = | φ + (cid:105)(cid:104) φ + | , where | φ + (cid:105) = 1 / √ d (cid:80) dk =1 | k (cid:105) ⊗ | k (cid:105) . Theisotropic state at mixing probability η is then S dη = ηS d + (1 − η )( /d ) ⊗ ( /d ). The isotropic state alsohas a symmetry under local unitaries U ∈ U( d ), as S dη = ( U ⊗ U ∗ ) S dη ( U † ⊗ ( U ∗ ) † ), where U ∗ stands forthe complex conjugate of U [26]. It is well-known that S ( S d ) = 1 / ( d +1) [26], and R PVM ( S d ) = ( H d − / ( d − H d = 1 + 1 / · · · + 1 /d [10]. The uniform distribution as LHS ensemble.—
When writing down a LHS model as in Eq. (2) for Werner statesor isotropic states, it is known [10, 27] that one can re-strict the attention to a probability distribution which isthe uniform distribution according to the Haar measure,denoted by ω , over Bob’s Bloch sphere. It is easily seefrom the argument given in Ref. [27] that this remainstrue also if the measurements are limited to generalisedones of any fixed number of outcomes.To proceed, we consider the set of conditional statesAlice can simulate using this distribution ω , which isgiven by the convex set K ( ω ) = (cid:26) K = (cid:90) d ω ( λ ) g ( λ ) | λ (cid:105)(cid:104) λ | : 0 ≤ g ( λ ) ≤ (cid:27) . (4)The set K ( ω ) is known as the capacity of ω [20, 27]. Inhigher-dimensional spaces, K ( ω ) has complicated struc-ture and no complete characterization of its geometry isknown. However, we will see that even a partial informa-tion of K ( ω ) will be sufficient to characterize quantumsteering of Werner states and isotropic states. Dichotomic measurements.
Each dichotomic measure-ment is completely characterized by one of its two effects,say M , since the other is − M . It follows directly fromthe definition of quantum steering that Werner states andisotropic states are unsteerable if and only if the corre-sponding conditional state Tr A [ ρ ( M ⊗ K ( ω )(and so is Tr A [ ρ (( − M ) ⊗ M on Alice’s side.Let us have a closer look at the set of measurement ef-fects on Alice’s side, { M : 0 ≤ M ≤ } . This is a convexset, of which the extreme points are precisely the projec-tion operators. These can be organized in hyperplanescorresponding to different ranks of the projections. Itis then natural to introduce finer subsets of 2-POVMswhose two effects are projections and the lower rank is r .Accordingly, we use R r ( ρ ) to denote the steering criticalradius corresponding to this subset of measurements. Wethen have R = min r =1 ,..., (cid:98) d/ (cid:99) R r , (5)where (cid:98) d/ (cid:99) is the maximal integer not greater than d/ Reducing the dimension and main result.—
The fol-lowing observation is crucial to computing R r : ForWerner states and isotropic states, a conditional stateof Bob’s system corresponding to a projection P onAlice’s side belongs to a special two-dimensional planespanned by the projection itself and the identity oper-ator, span { P, } . This is easily verified by direct com-putation of the conditional states in these cases. Con-sequently, instead of considering the general capacity K ( ω ), we can consider its cross-section with these two-dimensional subspaces and the original high-dimensionalproblem is now reduced to a two-dimensional one. Fortu-nately, in these two-dimensional spaces, the cross-sectionwith K ( ω ) can be computed exactly. The formulae aresomewhat cumbersome, but can be explicitly given [19,Section B and C]. To find the critical radii R r of the fullyantisymmetric state and the maximally entangled state,we simply identify the critical mixing probability thresh-old at which Bob’s conditional states corresponding to aprojection of rank r is at the border of this cross-section;for the details, see [19, Section C and D].The remaining step is the discrete minimization of R r with respect to the rank r of the projection in Eq. (5).We find that for both Werner states and isotropic states, R r is always minimal at r = 1 for all dimensions d ≤ and conjecture that this holds in general. In other words,among dichotomic measurements, those with a rank-1effects are conjectured to be most useful for quantumsteering. This eventually leads to the steering criticalradius R ( W d ) = ( d − [1 − (1 − /d ) / ( d − ] (6)for Werner states, and R ( S d ) = 1 − d − / ( d − (7)for isotropic states. These critical radii are presented inFig. 2 together with other known thresholds for these twofamilies of states.As an example, for the system of two qutrits, d = 3, wefind for the Werner state R ( W ) = 4(1 − (cid:112) / ≈ . R PVM ( W ) = 2 / ≈ . R ( S ) = 1 − / √ ≈ . R PVM ( S ) = 5 / ≈ . Steering with arbitrary POVMs.—
As long as quan-tum steerability is concerned, it follows from Ref. [18]that without loss of generality, one can assume that Al-ice’s measurements consist of d rank-1 effects, E =( E , E , · · · , E d ) with E a = α a P a , where P a are rank-1projections, 0 ≤ α a ≤
1, and (cid:80) d a =1 α a = d . Let us con-sider the Werner state W dη . For outcome a of Alice’s mea-surement, Bob’s system is steered to Tr A ( W dη E a ⊗
1) = α a Tr A [ W dη P a ⊗ α a , the conditional states are essentiallythat of n = d dichotomic measurements ( P a , − P a ).But even if the state is unsteerable with dichotomic mea-surements and the explicit response functions are given,it is not possible to directly combine them to form a re-sponse function for the general POVM E , which requiresthe normalization for the response function as probabili-ties, (cid:80) d a =1 G a ( λ ) = 1. To achieve the normalization, onehas to soften the response functions for the dichotomicmeasurements in a suitable way. Barrett was the firstwho used this idea to construct a local hidden variablemodel with POVMs for certain entangled Werner states,which later turns out to be a LHS model [18, 28]. As itturns out, his construction is in fact most suitable when the two parties are correlated, such as when they sharean isotropic state. For the Werner states, the two par-ties are however anticorrelated. We therefore propose thefollowing response function for the Werner state, G a ( λ ) = α a (cid:104) λ | − P a d − | λ (cid:105) Θ(1 /d − (cid:104) λ | P a | λ (cid:105) ) (8)+ α a d (cid:34) − n (cid:88) b =1 α b (cid:104) λ | − P b d − | λ (cid:105) Θ(1 /d − (cid:104) λ | P b | λ (cid:105) ) (cid:35) . The physical intuition for this response function and de-tailed calculation are discussed in [19, Section E]. Withthis, direct computation gives R POVM ( W d ) ≥ d − d +1 d − d d + 1 . (9)Fig. 2 shows that this significantly improves thebound given by the original Barrett construction, inparticular it remains finite as d tends to infinity,lim d →∞ R POVM ( W d ) = 1 /e , where e is the Euler’s natu-ral constant. Note that constructing a LHS model for allPOVMs for certain Werner state, our results also implythat the respective Werner states do not violate any Bellinequality. Steering criteria for general states.—
We now showthat our methods can be used to analyse steerabilityof generic high-dimensional states, where Bob’s reducedstate is of full rank. Because steerability is invariant un-der local filtering on Bob’s side [28–30], we can assumeBob’s reduced state to be maximally mixed (by applyingan appropriate filtering).Then, one use the fact that the steerability from Aliceto Bob is non-increasing under local channels on Alice’sside [31]. Given two states ρ and τ , each with Bob’sreduced state maximally mixed, we define D ( ρ, τ ) to bethe maximum value of η such that ρ η in Eq. (2) can stillbe obtained from a certain local channel E on Alice’s sideacting on τ , namely ρ η = ( E ⊗ I )[ τ ], where I is the identitychannel. Slightly extending the result of [31], it directlyfollows that given an unsteerable state τ , i.e., R n ( τ ) ≥ R n ( ρ ) ≥ D ( ρ, τ ) . (10)Given τ , the computation of D ( ρ, τ ) is a standard opti-mization over the suitable channel E , which can be doneusing semidefinite programming [32]. By choosing τ to bean unsteerable Werner state, or an unsteerable isotropicstate, Eq. (10) gives a lower bound for R n ( ρ ) and conse-quently a way to prove the unsteerability of a generichigh-dimensional state.Interestingly, one can also turnthe logic of Eq. (10) around and prove steerability. Inthis case, one chooses ρ to be a state of which R n ( ρ ) isknown then D ( ρ, τ ) > R n ( ρ ) implies that R n ( τ ) < τ .Another way to prove steerability is to average thestate by random unitary such that it results in a Wernerstate or a isotropic state [6, 26]. During this process, thecritical radius cannot decrease, we thus find R n ( ρ ) ≤ min (cid:26) ( d + 1) R n ( W d )1 − dF W , ( d − R n ( S d ) d − F S − (cid:27) , (11)where F S = Tr( S d ρ ) and F W = Tr( F d ρ ), with F d beingthe swap operator between two systems of dimension d .Such an upper bound allows one to prove the steerabilityof the state. Conclusion.—
We showed that the number of out-comes of measurements is essential for their ability toreveal nonclassicality. While we concentrated on quan-tum steering as a prototype quantum correlation, we con-jecture that this also holds for Bell correlation. Thatis, there are states which admit a local hidden variablemodel for all dichotomic measurements, nevertheless vi-olating certain Bell inequalities for finite measurementsettings with sufficient number of outcomes.An immediate application and extension of this phe-nomenon is the development of methods to assess thequality of measurements in experiments. In fact, it canbe anticipated that the results obtained here can leadto novel protocols for the self-testing of measurementsin experiments. Furthermore, our results may be usedto characterize the resources needed for the simulationof measurements, as there are situations where all di-chotomic measurements can easily be simulated, whilethree-outcome measurements cannot.Moreover, we provided novel criteria for the steerabil-ity and unsteerability of general quantum states. Espe-cially the presented LHS model for Werner states im-proves the known models drastically. These results willbe useful for the applications of steering in informationprocessing, such as quantum key distribution in asym-metric scenarios [33], or the characterization of joint mea-sureability [7].We thank Travis Baker, S´ebastien Designolle, MatthiasKleinmann, M. Toan Nguyen, Jiangwei Shang, andRoope Uola for inspiring discussions. This work was sup-ported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation - 447948357) and the ERC(Consolidator Grant 683107/TempoQ). ∗ [email protected] † [email protected][1] W. H. Zurek, “Decoherence, einselection, and the quan-tum origins of the classical,” Rev. Mod. Phys. , 715–775 (2003).[2] M. A. Nielsen and I. L. Chuang, Quantum computationand quantum information (Cambridge University Press,2010).[3] M. Schlosshauer,
Decoherence and the quantum-to-classical transition (Springer, 2007).[4] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum- mechanical description of physical reality be consideredcomplete,” Phys. Rev. , 777 (1935).[5] J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,”Physics , 195 (1964).[6] R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,”Phys. Rev. A , 4277 (1989).[7] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G¨uhne,“Quantum steering,” Rev. Mod. Phys. , 015001 (2020).[8] D. Cavalcanti and P. Skrzypczyk, “Quantum steering: areview with focus on semidefinite programming,” Rep.Prog. Phys. , 024001 (2017).[9] E. Schr¨odinger, “Discussion of probability relations be-tween separated systems,” Proc. Cambridge Philos. Soc. , 555 (1935).[10] H. M. Wiseman, S. J. Jones, and A. C. Doherty,“Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. , 140402(2007).[11] R. Uola, C. Budroni, O. G¨uhne, and J.-P. Pellonp¨a¨a, “Aone-to-one mapping between steering and joint measur-ability problems,” Phys. Rev. Lett. , 230402 (2015).[12] M. T. Quintino, T. V´ertesi, and N. Brunner, “Joint mea-surability, Einstein-Podolsky-Rosen steering, and Bellnonlocality,” Phys. Rev. Lett. , 160402 (2014).[13] T. Moroder, O. Gittsovich, M. Huber, and O. G¨uhne,“Steering bound entangled states: A counterexample tothe stronger Peres conjecture,” Phys. Rev. Lett. ,050404 (2014).[14] T. V´ertesi and N. Brunner, “Disproving the Peres con-jecture by showing Bell nonlocality from bound entan-glement,” Nat. Commun. , 5297 (2014).[15] M. Kleinmann and A. Cabello, “Quantum correlationsare stronger than all nonsignaling correlations producedby n -outcome measurements,” Phys. Rev. Lett. ,150401 (2016).[16] M. Kleinmann, T. V´ertesi, and A. Cabello, “Proposedexperiment to test fundamentally binary theories,” Phys.Rev. A , 032104 (2017).[17] X.-M. Hu, B.-H. Liu, Y. Guo, G.-Y. Xiang, Y.-F. Huang,C.-F. Li, G.-C. Guo, M. Kleinmann, T. V´ertesi, andA. Cabello, “Observation of stronger-than-binary corre-lations with entangled photonic qutrits,” Phys. Rev. Lett. , 180402 (2018).[18] J. Barrett, “Nonsequential positive-operator-valued mea-surements on entangled mixed states do not always vio-late a bell inequality,” Phys. Rev. A , 042302 (2002).[19] Supplementary matterials.[20] H. C. Nguyen, H. V. Nguyen, and O. G¨uhne, “Ge-ometry of Einstein–Podolsky–Rosen correlations,” Phys.Rev. Lett. , 240401 (2019).[21] G. M. D’Ariano, P. Lo Presti, and P. Perinotti, “Classi-cal randomness in quantum measurements,” J. Phys. A:Math. Gen. , 5979 (2005).[22] S. Jevtic, M. J. W. Hall, M. R. Anderson, M. Zwierz,and H. M. Wiseman, “Einstein-Podolsky-Rosen steeringand the steering ellipsoid,” J. Opt. Soc. Am. B , A40(2015).[23] H. C. Nguyen and T. Vu, “Nonseparability and steer-ability of two-qubit states from the geometry of steeringoutcomes,” Phys. Rev. A , 012114 (2016).[24] H. C. Nguyen and T. Vu, “Necessary and sufficient condi-tion for steerability of two-qubit states by the geometry ofsteering outcomes,” Europhys. Lett. , 10003 (2016). [25] M. L. Almeida, S. Pironio, J. Barrett, G. T´oth, andA. Ac´ın, “Noise robustness of the nonlocality of entangledquantum states,” Phys. Rev. Lett. , 040403 (2007).[26] M. Horodecki and P. Horodecki, “Reduction criterion ofseparability and limits for a class of distillation proto-cols,” Phys. Rev. A , 4206 (1999).[27] H. C. Nguyen, A. Milne, T. Vu, and S. Jevtic, “Quan-tum steering with positive operator valued measures,” J.Phys. A , 355302 (2018).[28] M. T. Quintino, T. V´ertesi, D. Cavalcanti, R. Augusiak,M. Demianowicz, A. Ac´ın, and N. Brunner, “Inequiva-lence of entanglement, steering, and bell nonlocality forgeneral measurements,” Phys. Rev. A , 032107 (2015).[29] R. Uola, T. Moroder, and O. G¨uhne, “Joint measura-bility of generalized measurements implies classicality,” Phys. Rev. Lett. , 160403 (2014).[30] R. Gallego and L. Aolita, “Resource theory of steering,”Phys. Rev. X , 041008 (2015).[31] T. J. Baker, S. Wollmann, G. J. Pryde, and H. M. Wise-man, “Necessary conditions for steerability of two qubits,from consideration of local operations,” arXiv:1906.04693(2019).[32] S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).[33] C. Branciard, E. G. Cavalcanti, S. P. Walborn,V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasi-bility, and the connection with steering,” Phys. Rev. A , 010301 (2012). upplementary Materials: Some quantum measurements with three outcomes canreveal nonclassicality where all two-outcome measurements fail to do so H. Chau Nguyen ∗ and Otfried G¨uhne † Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany (Dated: December 10, 2020)
Appendix A: More on the geometry of the critical radius
In this section, we discuss in more detail the definition of the critical radius, Eq. (2) in the maintext. In particular,we clarify the geometrical motivation of the choice of the separable noisy state ( A ⊗ ρ B ) /d .Let us first remark that the critical radius in Ref. [1] was originally defined differently (as a measure of certaingeometrical aspect of the state rho itself); but one can show that it is equivalent to definition Eq. (2) in the maintextwith that particular choice of the separable noisy state ( A ⊗ ρ B ) /d . Within our current context, we can also motivatethis choice as follows. Let us consider the attempt to define the critical radius as R M ( ρ ) = max { η ≥ ρ η is unsteerable w.r.t. M } . (A1)with the more general form of ρ η = ηρ + (1 − η )( τ A ⊗ τ B ).Consider the family of sets of steering outcomes { Tr A [ ρ η ( M ⊗ ≤ M ≤ } with ρ η = ηρ + (1 − η ) τ A ⊗ τ B for0 ≤ η ≤
1. We then see that if τ B = ρ B , this family shares the same common point ρ B = Tr A ( ρ η ) for all η .The choice τ A = A /d is a bit subtler. As we also mentioned in the main text, the extreme points of the set { M : 0 ≤ M ≤ A } are projections, which organise in planes of operators with integer traces. Therefore the extremepoints of the set { Tr A [( M ⊗ B ) ρ η ] : 0 ≤ M ≤ A } for certain η also organise in a set of parallel planes (linear imagesof the planes of operators of integer traces). If τ A = A /d , this set of parallel planes remain constant for all the sets { Tr A [( M ⊗ B ) ρ η ] : 0 ≤ M ≤ A } as η varies.These two geometrical features have imprints on the computational aspect of the critical radius. We are to demon-strate this implication below and link the definition (A1) to the suggestion in Ref. [1, Supplementary X].The key step in making the definition (A1) in a computational form, at least in principle, is to use the results ofRef. [5, Theorem 1], which states that ρ η is unsteerable if and only if for some choice of a distribution µ over Bob’sBloch sphere (i.e., a choice of LHS ensemble), one has Z d µ ( σ ) max i Tr( Z i σ ) ≥ n X i =1 Tr[ ρ η ( E i ⊗ Z i )] (A2)for all measurements E = ( E , E , . . . , E n ) on Alice’s side and arbitrary choice of observables Z = ( Z , Z , . . . , Z n )on Bob’s side. Thus the critical radius can be formally written asmax η,µ η s . t . η ≥ , R d µ ( σ ) max i Tr( Z i σ ) ≥ P ni =1 Tr[ ρ η ( E i ⊗ Z i )] for all E, Z, R d µ ( σ ) = Tr A ( ρ η ) . (A3)By the choosing τ B = ρ B , the second constraint is independent of η . As a result, one can then rewrite the optimisationproblem asmax η,µ η s . t . η ≥ , R d µ ( σ ) { max i Tr( Z i σ ) − P ni =1 Tr( E i τ A ) Tr( Z i σ ) } ≥ η P ni =1 Tr[( ρ − τ A ⊗ ρ B )( E i ⊗ Z i )] for all E, Z, R d µ ( σ ) = ρ B . (A4) ∗ [email protected] † [email protected] The importance of the choice τ A = A /d becomes apparent when we assume that the measurement E is projective.Indeed, if τ A = A /d , and the measurement is rank-1 projective, then Tr( E i τ A ) = 1 /d for all i . Thus one can furtherrewrite the optimisation problem asmax η,µ η s . t . η ≥ R d µ ( σ ) { max i Tr( Z i σ ) − /d P ni =1 Tr( Z i , σ ) } ≥ η max E { P ni =1 Tr[( ρ − τ A ⊗ ρ B )( E i ⊗ Z i )] } for all Z R d µ ( σ ) = ρ B (A5)The seemingly minor difference between (A5) and (A4) is in fact very important. One way to see it is to considerapproximating the Bloch sphere by a set of discrete points σ . The effect of this discretisation is that in the secondconstraint of the optimisation problems (A4) and (A5), one only needs to consider a finite number of choices for Z ; werefer the readers to Ref. [1] for a more detailed discussion of this discretisation procedure. Then for the problem (A5),upon solving a finite number of optimisation over measurements E corresponding to different choices of Z in thesecond constraint, the computation of the critical radius as the optimal η is then a linear program (with finite numberof constraints). On the other hand, the problem (A4) cannot be brought into a linear program (with finite number ofconstraints) despite the finite number of choices of Z .From Eq. (A4), one also see that the definition of the critical radius directly link to that suggested in Ref. [5,Supplementary X]. Indeed, the problem (A4) can be rewritten as R − [ ρ ] = inf µ sup E,Z P ni =1 Tr[( ρ − τ A ⊗ ρ B )( E i ⊗ Z i )] R d µ ( σ ) { max i Tr( Z i σ ) − P ni =1 Tr( E i τ A ) Tr( Z i σ ) } , (A6)subject to R d µ ( σ ) σ = ρ B . In doing so, we note that R d µ ( σ ) { max i Tr( Z i σ ) − P ni =1 Tr( E i τ A ) Tr( Z i , σ ) } ≥
0. It isclear that Eq. (A6) is the same as the suggestion for an extension of critical radius as in Ref. [5, Supplementary X]with τ A = A /d A .While we have presented the motivation for the choice of the separable state ( A ⊗ ρ B ) /d in the definition ofthe critical radius, this does not rule out the possible choices. Further exploration of these possibilities can be aninteresting research direction. Appendix B: Integration over the high dimensional Bloch sphere
We will frequently have to work with integrals over the high dimensional Bloch sphere (i.e., the set of pure states).Here we describe how that can be done, following Refs. [2, 3] with small modifications.Specifically, we work with the Hilbert space of dimension d . Let Q be a projection of rank k , we are interested inthe following integration a n ( k, t ) = Z d ω ( λ ) h λ | Q | λ i n Θ( h λ | Q | λ i − t ) , (B1)where Θ is Heaviside’s step function and ω denotes the Haar measure over the pure states. Note that although theprojection Q appears in the integral on the right-hand side, we will see that the left-hand side only depends on itsrank k , which justifies the notation a n ( k, t ).We choose the basis {| i i} di =1 such that Q = P ki =1 | i ih i | . The pure state can be written as | λ i = P di =1 r i e iθ i | i i . TheHaar measure thus can be formally written asd ω ( λ ) = 1 Z d Y i =1 r i d r i d θ i δ ( d X i =1 r i − . (B2)The range of r i is [0 , + ∞ ) and the range of θ i is [0 , π ). The normalisation factor Z can be found by Z = d Y i =1 Z + ∞ r i d r i Z π d θ i δ ( d X i =1 r i − . (B3)Now note that the integrands in (B1) and (B3) do not depend on the phase θ i , thus the integration over the phase θ i can be carried out directly. Moreover, the integrals over r i can be simplified by changing the variable u i = r i .Eventually, we obtain a n ( k, t ) = I n ( Q, t ) I ( Q, , (B4)with I n ( Q, t ) = Z d uδ (1 − d X i =1 u i )Θ( k X i =1 u i − t )( k X i =1 u i ) n , (B5)where d u = d u d u . . . d u d and the integral is taken over the whole range [0 , + ∞ ) of u i .Let s p ( ξ ) = Z d x d x . . . d x p δ ( ξ − x − x − . . . − x p ) . (B6)Then by rescaling the integral variable, one can easily show that s p ( ξ ) = s p (1) ξ p − . (B7)Note that s p (1) is simply the area of the p − δ -function.With this notation, we then can integrate out u k +1 , u k +2 , . . . , u d in (B5) to get I n ( Q, t ) = s d − k (1) Z d u . . . d u k Θ( k X i =1 u i − t )(1 − k X i =1 u i ) d − k − ( k X i =1 u i ) n . (B8)To carry out this integral, we write I n ( Q, t ) = s d − k (1) Z d u . . . d u k Θ( k X i =1 u i − t )(1 − k X i =1 u i ) d − k − ( k X i =1 u i ) n Z d xδ ( x − k X i =1 u i ) . (B9)Upon changing the integral order, we have I n ( Q, t ) = s d − k (1) Z d x Z d u . . . d u k Θ( x − t )(1 − x ) d − k − x n δ ( x − u − u − · · · − u k ) (B10)= s d − k (1) s k (1) Z t d x (1 − x ) d − k − x k − n (B11)= s d − k (1) s k (1) β (1 − t, d − k, k + n ) , (B12)where β ( z, a, b ) = R z d ξξ a − (1 − ξ ) b − is Euler’s incomplete β -function. So a n ( k, t ) = β (1 − t, d − k, k + n ) β ( d − k, k ) , (B13)where β ( a, b ) = β (1 , a, b ) is Euler’s complete β -function. As we remarked in the paragraph following (B1), a n ( k, t )only depends on the rank k of the projection Q . Appendix C: The canonical cross-sections of the capacity of the uniform distribution
Generally, it has been shown [4, 5] that the extreme points of K ( ω ) are of the form K ( Z ) = Z d ω ( λ )Θ( h λ | Z | λ i ) | λ ih λ | , (C1)with varying operator Z . In particular, let us consider a special family of these extreme points where Z = Q − t Q is a (fixed) projection of rank k and varying t , K ( Q, t ) = Z d ω ( λ )Θ( h λ | Q | λ i − t ) | λ ih λ | . (C2)Let us now show that K ( Q, t ) is in the span of { , Q } . While this can be done directly by inspection, a moreelegant argument makes use of the concepts of von Neumann algebras [6]. Since Q is a projection, the span of { , Q } is also the von Neumann algebra generated by B and Q . To show that K ( Q, t ) is in the algebra, we show that it
Tr[ K ] T r( K P ) − T r( K ) k / d k=1k=2k=3 FIG. 1. The boundary of canonical cross-sections of the capacity of the uniform distribution specified in the plane spanned by { , Q } , with Tr( Q ) = k for dimension d = 4. The operators K on the boundary are specified by two orthogonal coordinatesTr( K ) and Tr[ K ( P − k/d )] = Tr( KP ) − Tr( K ) k/d . commutes with all unitaries in the commutant of the span of { , Q } [6]. That is, let U be an unitary operator thatcommutes with Q , we want to show that U also commute with K ( Q, t ). Indeed,
U K ( Q, t ) U † = Z d ω ( λ )Θ( h λ | Q | λ i − t ) U | λ ih λ | U † . (C3)Upon transforming | λ i = U | λ i and noting that the Haar measure is invariant under this transformation, and that h λ | U QU † | λ i = h λ | Q | λ i ) since U commutes with Q , we obtain an identical formula as equation (C2) for K ( Q, t ).Being in the span of { , Q } , K ( Q, t ) is characterised by two parameters Tr[ K ( Q, t )] = a ( k, t ) and Tr[ QK ( Q, t )] = a ( k, t ), with a ( k, t ) = β (1 − t, d − k, k ) β ( d − k, k ) , (C4) a ( k, t ) = β (1 − t, d − k, k + 1) β ( d − k, k ) , (C5)as defined in Eq. (B1) and Eq. (B13).As t varying from 0 to 1, K ( Q, t ) draws a curve starting at { , Q } . Asa consequence, this forms a half of the boundary of the cross-section of K ( ω ) in this plane; see Fig. 1. The other halfof boundary of the cross-section is formed by K ( − Q, t ) for t varying from 0 to 1. Appendix D: Critical radii for dichotomic measurements1. Werner states
Recall that the fully antisymmetric state of dimension d × d is defined by W d = 2 π − d ( d − , (D1)where π − is the projection onto the antisymmetric subspace of C d ⊗ C d . The Werner state acting in dimension d isobtained as a convex combination of the fully antisymmetric state W d with the maximally mixed state, W dη = ηW d + (1 − η ) d ⊗ d , (D2)and η is referred to as the mixing parameter.Suppose Alice makes a dichotomic measurement E = ( P, − P ), where P is a projection of rank r . For the outcome P , Bob’s system is steered to Tr A ( W dη P ⊗
1) = η − Pd ( d −
1) + (cid:20) η r − d − − η ) rd (cid:21) d , (D3)where r = rank( P ).Observe that this steering outcome belongs to the plane spanned by − P . We consider the cross-section ofthe capacity of the uniform distribution K ( ω ) in the corresponding plane, i.e., K ( − P, t ), with the border describedby equation (C5). We are interested in whether the conditional state (D3) is inside this cross-section. The conditionfor this to happen can be easily derived by identifying the critical value η c for the mixing parameter such that theconditional state (D3) is on the border of the capacity (C5), which is given by a ( d − r, t c ) = rd , (D4) a ( d − r, t c ) = η c d − rd ( d −
1) + (cid:20) x c r − d − − η c ) rd (cid:21) (cid:16) − rd (cid:17) . (D5)Solving t c from equation (D4), one can compute η c from equation (D5). Recall that η c is in fact precisely the definitionof the critical radii, R r ( W d ) = η c .One can derive a more explicit formula for R r ( W d ). Indeed, from equation (D5), we find η c = d ( d − r ( d − r ) (cid:20) a ( d − r, t c ) + r d − rd (cid:21) . (D6)Upon using the definition of a n ( k, t ) in equation (B13), the recursive relation for the incomplete β -function [7, page263], β ( z, a, b + 1) = ba + b β ( z, a, b ) + 1 a + b z a (1 − z ) b , (D7)and the definition of the complete β -function in terms of the Γ-function [7, page 259], β ( a, b ) = Γ( a )Γ( b )Γ( a + b ) , (D8)one arrives at R r ( W d ) = ( d − d + 1)Γ( r + 1)Γ( d − r + 1) (1 − t c ) r t d − rc . (D9)Although not given in a closed form for arbitrary r , R r ( W d ) can be easily computed in a computer. For all d ≤ ,we compute R r ( W d ) and find that it is always minimised at r = 1. Thus in all these cases we can identify R with R . For r = 1, equation (D4) can be solved explicitly for t c , and we arrive at R ( W d ) = ( d − [1 − (1 − /d ) / ( d − ] . (D10)
2. Isotropic states
Recall that the maximally entangled state on C d ⊗ C d is defined by S d = | ψ + ih ψ + | , (D11)where | ψ + i = √ d P dk =1 | k i ⊗ | k i for certain basis | k i . The isotropic state is then defined by S dη = ηS d + (1 − η ) d ⊗ d . (D12)The computation of R r ( S d ) follows similar steps as that for the Werner state. There is a remarkable difference,though. For the isotropic state, the steering outcome at Bob’s side corresponding to the projection outcome P atAlice’s side, Tr A [ S dη ( P ⊗ rd (cid:20) η ¯ Pr + (1 − η ) d (cid:21) , (D13)belongs to the canonical cross-section of the capacity of the uniform distribution indicated by K ( ¯ P , t ). Here ¯ P denotesthe complex conjugate of P . Thus here we need to consider the cross-section of K ( ω ) with the plan spanned by ¯ P and
1, in contrast to the case for the Werner states.Following the same steps in Section D 1, we proceed by identifying the critical value η c for the mixing parametersuch that the conditional state (D13) is on the border of the capacity (C5), which is given by a ( r, t c ) = rd , (D14) a ( r, t c ) = η c rd + (1 − η c ) r d . (D15)Solving t c from equation (D14), one can compute η c from equation (D15). Again, η c is in fact precisely the definitionof the critical radii, R r ( S d ) = η c .An explicit formula for R r ( S d ) can also be derived. From equation (D15), we find η c = a ( r, t c ) − r /d r/d (1 − r/d ) . (D16)Then using the definition of a n ( k, t ) in equation (B13), the recursive relation (D7) and the relation between β -function and Γ-function (D8), one obtains R r ( S d ) = Γ( d + 1)Γ( d − r + 1)Γ( r + 1) (1 − t c ) d − r t rc . (D17)Note the difference with the equation (D9) for the Werner state. For all d ≤ , we again find that R r is minimisedat r = 1. We thus have for all d ≤ , R ( S d ) = 1 − d − / ( d − . (D18) Appendix E: New local hidden state model for the Werner states with generalised measurements
In the following, we present the details of the derivation of the bound R POVM ( W d ) ≥ d − d +1 d − d d + 1 . (E1)This bound is the critical mixing pramameter η c such thatTr( W dη c E a ⊗
1) = Z d ω ( λ ) G a ( λ ) | λ ih λ | , (E2)with the response function G a ( λ ) = α a h λ | − P a d − | λ i Θ(1 /d − h λ | P a | λ i ) + α a d (1 − d X b =1 α b h λ | − P b d − | λ i Θ(1 /d − h λ | P b | λ i ) . (E3)Recall from the maintext that E a = α a P a , where P a are rank-1 projections. One can recognise that the firstterm in this response function is, upto a prefactor, given by the response functions for dichotomic measurementsΘ(1 /d − h λ | P a | λ i ). The second term is constructed such that the response function is automatically normalised, P na =1 G a ( λ ) = 1. It is easy to show that the function is positive, thus is a valid response function.We need to compute the operator on the right hand side of equation (E2). To do this, we note Z d ω ( λ ) G a ( λ ) | λ ih λ | = α a X a + α a d ( d − d X b =1 α b X b ) , (E4)where X a = Z d ω ( λ ) 1 d − h λ | Q a | λ i Θ[ h λ | Q a | λ i − (1 − /d )] | λ ih λ | , (E5)where Q a = B − P a . We again can show that X a is in the span of { , Q a } , which can be characterised byTr( X a ) = 1 d − a ( d − , − /d ) , (E6)Tr( X a Q a ) = 1 d − a ( d − , − /d ) . (E7)The critical value of η c where this construction of local hidden state model works is then η c = d d − a ( d − , − /d ) − da ( d − , − /d ) . (E8)With the explicit expressions of a ( d − , − /d ) and a ( d − , − /d ) one obtains equation (E1). [1] H. C. Nguyen, H. V. Nguyen, and O. G¨uhne, “Geometry of Einstein–Podolsky–Rosen correlations,” Phys. Rev. Lett. ,240401 (2019).[5] H. C. Nguyen, A. Milne, T. Vu, and S. Jevtic, “Quantum steering with positive operator valued measures,” J. Phys. A ,355302 (2018).[2] R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev.A , 4277 (1989).[3] J. Barrett, “Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a bellinequality,” Phys. Rev. A , 042302 (2002).[4] H. C. Nguyen and T. Vu, “Nonseparability and steerability of two-qubit states from the geometry of steering outcomes,”Phys. Rev. A , 012114 (2016).[6] A. Connes, Noncommutative Geometry (Academic Press, 1994).[7] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).(Dover, 1964).