Einstein-Podolsky-Rosen steering and the steering ellipsoid
Sania Jevtic, Michael J. W. Hall, Malcolm R. Anderson, Marcin Zwierz, Howard M. Wiseman
EEinstein-Podolsky-Rosen steering and the steering ellipsoid
Sania Jevtic, Michael J. W. Hall, Malcolm R. Anderson, Marcin Zwierz,
4, 2 and Howard M. Wiseman Mathematical Sciences, John Crank 501, Brunel University, Uxbridge UB8 3PH, United Kingdom Centre for Quantum Computation and Communication Technology (Australian Research Council),Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia Mathematical and Computing Sciences, Universiti Brunei Darussalam, Gadong BE 1410, Negara Brunei Darussalam Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warszawa, Poland
The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering)remains open. Here, a strong necessary condition is obtained for the steerability of two-qubitstates having maximally-mixed reduced states, via the construction of local hidden state models.It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions arealso obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantumsteering ellipsoid formalism, and explicitly evaluates the integral of n / ( n (cid:124) A n ) over arbitrary unithemispheres for any positive matrix A . I. INTRODUCTION
Quantum systems can be correlated in ways that su-persede classical descriptions. However, there are degreesof non-classicality for quantum correlations. For simplic-ity, we consider only bipartite correlations, with the two,spatially separated, parties being named Alice and Bobas usual.At the weaker end of the spectrum are quantum sys-tems whose states cannot be expressed as a mixture ofproduct-states of the constituents. These are called non-separable or entangled states. The product-states ap-pearing in such a mixture comprise a local hidden state(LHS) model for any measurements undertaken by Aliceand Bob.At the strongest end of the spectrum are quantumsystems whose measurement correlations can violate aBell inequality [1, 2], hence demonstrating (modulo loop-holes [3]) the violation of local causality [4]. Thisphenomenon—commonly known as Bell-nonlocality [5]—is the only way for two spatially separated parties to ver-ify the existence of entanglement if either of them, ortheir detectors, cannot be trusted [6]. We say that abipartite state is Bell-local if and only if there is a localhidden variable (LHV) model for any measurements Aliceand Bob perform. Here the ‘variables’ are not restrictedto be quantum states, hence the distinction between non-separability and Bell-nonlocality.In between these types of non-classical correlations liesEPR-steering. The name is inspired by the seminal pa-per of Einstein, Podolsky, and Rosen (EPR) [7], andthe follow-up by Schr¨odinger [8], which coined the term“steering” for the phenomenon EPR had noticed. Al-though introduced eighty years ago, as this Special Issuecelebrates, the notion of EPR-steering was only formal-ized eight years ago, by one of us and co-workers [9, 10].This formalization was that EPR-steering is the only wayto verify the existence of entanglement if one of the par-ties — conventionally Alice [9–11] — or her detectors,cannot be trusted. We say that a bipartite state is EPR-steerable if and only if it allows a demonstration of EPR- steering. A state is not EPR-steerable if and only if thereexists a hybrid LHV–LHS model explaining the Alice–Bob correlations. Since in this paper we are concernedwith steering, when we refer to a LHS model we meana LHS model for Bob only; it is implicit that Alice canhave a completely general LHV model.The above three notions of non-locality for quantumstates coincide for pure states: any non-product purestate is non-separable, EPS-steerable, and Bell-nonlocal.However for mixed states, the interplay of quantum andclassical correlations produces a far richer structure. Formixed states the logical hierarchy of the three conceptsleads to a hierarchy for the bipartite states: the set ofseparable states is a strict subset of the set of non-EPR-steerable states, which is a strict subset of the set ofBell-local states [9, 10].Although the EPR-steerable set has been completelydetermined for certain classes of highly symmetric states(at least for the case where Alice and Bob perform pro-jective measurements) [9, 10], until now very little wasknown about what types of states are steerable even forthe simplest case of two qubits. In this simplest case,the phenomenon of steering in a more general sense —i.e. within what set can Alice steer Bob’s state by mea-surements on her system — has been studied extensivelyusing the so-called steering ellipsoid formalism [12–14].However, no relation between the steering ellipsoid andEPR-steerability has been determined.In this manuscript, we investigate EPR-steerability ofthe class of two-qubit states whose reduced states aremaximally mixed, the so-called T-states [15]. We usethe steering ellipsoid formalism to develop a determin-istic LHS model for projective measurements on thesestates and we conjecture that this model is optimal. Fur-thermore we obtain two sufficient conditions for T-statesto be EPR-steerable, via suitable EPR-steering inequal-ities [11, 16] (including a new asymmetric steering in-equality for the spin covariance matrix). These sufficientconditions touch the necessary condition in some regionsof the space of T-states, and everywhere else the gap be-tween them is quite small.The paper is organised as follows. In section 2 we dis- a r X i v : . [ qu a n t - ph ] M a r cuss in detail the three notions of non-locality, namelyBell-nonlocality, EPR-steerability and non-separability.Section 3 introduces the quantum steering ellipsoid for-malism for a two-qubit state, and in section 4 we use thesteering ellipsoid to develop a deterministic LHS modelfor projective measurements on T-states. In section 5,two asymmetric steering inequalities for arbitrary two-qubit states are derived. Finally in section 6 we concludeand discuss further work. II. EPR-STEERING AND LOCAL HIDDENSTATE MODELS
Two separated observers, Alice and Bob, can use ashared quantum state to generate statistical correlationsbetween local measurement outcomes. Each observercarries out a local measurement, labelled by A and B re-spectively, to obtain corresponding outcomes labelled by a and b . The measurement correlations are described bysome set of joint probability distributions, { p ( a, b | A, B ) } ,with A and B ranging over the available measurements.The type of state shared by Alice and Bob may be clas-sified via the properties of these joint distributions, forall possible measurement settings A and B .The correlations of a Bell-local state have a local hid-den variable (LHV) model [1, 2], p ( a, b | A, B ) = (cid:88) λ P ( λ ) p ( a | A, λ ) p ( b | B, λ ) , (1)for some ‘hidden’ random variable λ with probability dis-tribution P ( λ ). Hence, the measured correlations may beunderstood as arising from ignorance of the value of λ ,where the latter locally determines the statistics of theoutcomes a and b and is independent of the choice of A and B . Conversely, a state is defined to be Bell -nonlocal if it has no LHV model. Such states allow, for example,the secure generation of a cryptographic key between Al-ice and Bob without trust in their devices [17, 18].In this paper, we are concerned with whether the stateis steerable ; that is, whether it allows for correlations thatdemonstrate EPR-steering. As discussed in the introduc-tion, EPR-steering by Alice is demonstrated if it is not the case that the correlations can be described by a hy-brid LHV–LHS model, wherein, p ( a, b | A, B ) = (cid:88) λ P ( λ ) p ( a | A, λ ) p Q ( b | B, λ ) , (2)where the local distributions p Q ( b | B, λ ) correspond tomeasurements on local quantum states ρ B ( λ ), i.e., p Q ( b | B, λ ) = tr[ ρ B ( λ ) F Bb ] . Here { F Bb } denotes the positive operator valued measure(POVM) corresponding to measurement B . The state issaid to be steerable by Alice if there is no such model.The roles of Alice and Bob may also be reversed in theabove, to define steerability by Bob. Comparing Eqs. (1) and (2), it is seen that all non-steerable states are Bell-local. Hence, all Bell-nonlocalstates are steerable, by both Alice and Bob. In fact,the class of steerable states is strictly larger [9]. More-over, while not as powerful as Bell-nonlocality in general,steerability is more robust to detection inefficiencies [19],and also enables the use of untrusted devices in quantumkey distribution, albeit only on one side [20]. By a simi-lar argument, a separable quantum state shared by Aliceand Bob, ρ = (cid:80) λ p ( λ ) ρ A ( λ ) ⊗ ρ B ( λ ), is both Bell-localand nonsteerable. Moreover, the set of separable statesis strictly smaller than the set of nonsteerable states [9].It is important that EPR-steerability of a quantumstate not be confused with merely the dependence of thereduced state of one observer on the choice of measure-ment made by another, which can occur even for sepa-rable states. The term ‘steering’ has been used with ref-erence to this phenomenon, in particular for the conceptof ‘steering ellipsoid’, which we will use in our analysis.EPR-steering, as defined above, is a special case of thisphenomenon, and is only possible for a subset of nonsep-arable states.We are interested in the EPR-steerability of states forall possible projective measurements. If Alice is doingthe steering, then it is sufficient for Bob’s measurementsto comprise some tomographically complete set of pro-jectors. It is straightforward to show in this case thatthe condition for Bob to have an LHS model, Eq. (2),reduces to the existence of a representation of the form p E ρ EB := tr A [ ρ E ⊗ ] = (cid:88) λ P ( λ ) p (1 | E, λ ) ρ B ( λ ) , (3a) p E = tr[ ρE ⊗ I ] = (cid:88) λ P ( λ ) p (1 | E, λ ) . (3b)Here E is any projector that can be measured by Alice; p E is the probability of result ‘ E = 1’ and p (1 | E, λ ) isthe corresponding probability given λ ; ρ EB is the reducedstate of Bob’s component corresponding to this result;and tr A [ · ] denotes the partial trace over Alice’s compo-nent. Note that this form, and hence EPR-steerability byAlice, is invariant under local unitary transformations onBob’s components.Determining EPR-steerability in this case, where Al-ice is permitted to measure any Hermitian observable,is surprisingly difficult, with the answer only known forcertain special cases such as Werner states [9]. However,in this paper we give a strong necessary condition forthe EPR-steerability of a large class of two-qubit states,which we conjecture is also sufficient. This condition isobtained via the construction of a suitable LHS model,which is in turn motivated by properties of the ‘quantumsteering ellipsoid’ [12, 14]. Properties of this ellipsoid aretherefore reviewed in the following section. III. THE QUANTUM STEERING ELLIPSOID
An arbitrary two-qubit state may be written in thestandard form ρ = 14 ⊗ + a · σ ⊗ + ⊗ b · σ + (cid:88) j,k T jk σ j ⊗ σ k . Here ( σ , σ , σ ) ≡ σ denote the Pauli spin operators,and a j = tr[ ρ σ j ⊗ ] , b j = tr[ ρ ⊗ σ j ] , T jk = tr[ ρ σ j ⊗ σ k ] . Thus, a and b are the Bloch vectors for Alice and Bob’squbits, and T is the spin correlation matrix.If Alice makes a projective measurement on her qubit,and obtains an outcome corresponding to projector E ,Bob’s reduced state follows from Eq. (3a) as ρ EB = tr A [ ρ E ⊗ ]tr[ ρ E ⊗ ] . We will also refer to ρ EB as Bob’s ‘steered state’.To determine Bob’s possible steered states, note thatthe projector E may be expanded in the Pauli basis as E = ( + e · σ ), with | e | = 1. This yields the corre-sponding steered state ρ EB = ( + b ( e ) · σ ), with asso-ciated Bloch vector b ( e ) = 12 p e ( b + T (cid:124) e ) , (4)where p e is the associated probability of result ‘ E = 1’, p e := tr[ ρ ( E ⊗ )] = 12 (1 + a · e ) , (5)called p E previously. In what follows we will refer to thevector e rather than its corresponding operator E .The surface of the steering ellipsoid is defined to bethe set of steered Bloch vectors, { b ( e ) : | e | = 1 } , and inRef. [14] it is shown that interior points can be obtainedfrom positive operator-valued measurements (POVMs).The ellipsoid has centre c = b − T (cid:124) a − a , (6)and the semiaxes s , s , s are the roots of the eigenvaluesof the matrix Q = 11 − a ( T (cid:124) − ba (cid:124) ) (cid:18) + aa (cid:124) − a (cid:19) ( T − ab (cid:124) ) . (7)The eigenvectors of Q give the orientation of the ellipsoidaround its centre [14]. Thus, the general equation of thesteering ellipsoid surface is x (cid:124) Q − x = 1 with x ∈ R being the displacement vector from the centre c .Entangled states typically have large steeringellipsoids—the largest possible being the Bloch ball, which is generated by every pure entangled state [14]. Incontrast, the volume of the steering ellipsoid is strictlybounded for separable states. Indeed, a two-qubitstate is separable if and only if its steering ellipsoid iscontained within a tetrahedron contained within theBloch sphere [14]. Thus, the separability of two-qubitstates has a beautiful geometric characterisation interms of the quantum steering ellipsoid.No similar characterisation has been found for EPR-steerability, to date. However, for non-separable states,knowledge of the steering ellipsoid matrix Q , its centre c ,and Bob’s Bloch vector b uniquely determines the sharedstate ρ up to a local unitary transformation on Alice’ssystem [14], [21] and so is sufficient, in principle, to de-termine the EPR-steerability of ρ . In this paper we finda direct connection between EPR-steerability and thequantum steering ellipsoid, for the case that the Blochvectors a and b vanish. IV. NECESSARY CONDITION FOREPR-STEERABILITY OF T-STATESA. T-states
Let T = O A (cid:101) DO (cid:124) B be a singular value decomposition ofthe spin correlation matrix T , for some diagonal matrix (cid:101) D ≥ O A , O B ∈ O(3). Notingthat any O ∈ O(3) is either a rotation or the productof a rotation with the parity matrix − I , it follows that T can always be represented in the form T = R A DR (cid:124) B ,for proper rotations R A , R B ∈ SO(3), where the diagonalmatrix D may now have negative entries.The rotations R A and R B may be implemented by lo-cal unitary operations on the shared state ρ , amount-ing to a local basis change. Hence, all properties of ashared two-qubit state, including steerability propertiesin particular, can be formulated in a representation inwhich the spin correlation matrix has the diagonal form T ≡ D = diag[ t , t , t ]. It follows that if the shared state ρ has maximally-mixed reduced states with a = b = ,then it is completely described, up to local unitaries, bya diagonal T , i.e. one may consider ρ = 14 ⊗ + (cid:88) j t j σ j ⊗ σ j (8)without loss of generality. Such states are called T-states [15]. They are equivalent to mixtures of Bell states,and hence form a tetrahedron in the space parameterisedby ( t , t , t ) [15]. Entangled T-states necessarily have t t t <
0, and the set of separable T-states forms anoctahedron within the tetrahedron [15].The T-state steering ellipsoid is centred at the origin, c = , and the ellipsoid matrix is simply Q = T (cid:124) T , asfollows from Eqs. (6) and (7) with a = b = . Thesemiaxes are s i = | t i | for i = 1 , ,
3, and are aligned withthe x, y, z -axes of the Bloch sphere. Thus, the equationof the ellipsoid surface in spherical coordinates ( r, θ, φ ) is r = 1 /f ( θ, φ ), with f ( θ, φ ) := sin θ cos φs + sin θ sin φs + cos θs . (9)We find a remarkable connection between this equationand the EPR-steerability of T-states in the following sub-section. B. Deterministic LHS models for T-states
Without loss of generality, consider measurement byAlice of Hermitian observables on her qubit. Such ob-servables can be equivalently represented via projections, E = ( + e . · σ ), with | e | = 1. The probability of result‘ E = 1’ and the corresponding steered Bloch vector aregiven by Eqs. (4) and (5) with a = b = , i.e., p e = 1 / , b ( e ) = T T e = T e . Hence, letting n ( λ ) denote the Bloch vector correspond-ing to ρ B ( λ ) in Eq. (3a), then from Eqs. (3a) and (3b),it follows there is an LHS model for Bob if and only ifthere is a representation of the form (cid:88) λ P ( λ ) p (1 | e , λ ) = 12 , (cid:88) λ P ( λ ) p (1 | e , λ ) n ( λ ) = 12 T e , for all unit vectors e . Noting further that n ( λ ) can al-ways be represented as some mixture of unit vectors, cor-responding to pure ρ B ( λ ), these conditions are equivalentto the existence of a representation of the form (cid:90) P ( n ) p (1 | e , n ) d n = 12 , (10) (cid:90) P ( n ) p (1 | e , n ) n d n = 12 T e , (11)with integration over the Bloch sphere. Thus, the unitBloch vector n labels both the local hidden state and thehidden variable.Given LHS models for Bob for any two T-states, hav-ing spin correlation matrices T and T , it is trivial toconstruct an LHS model for the T-state correspondingto T q = (1 − q ) T + qT , for any 0 ≤ q ≤
1, via the con-vexity property of nonsteerable states [11]. Our strategyis to find deterministic
LHS models for some set of T-states, for which the result ‘ E = 1’ is fully determinedby knowledge of n , i.e., p (1 | e , n ) ∈ { , } . LHS modelscan then be constructed for all convex combinations ofT-states in this set.To find deterministic LHS models, we are guided bythe fact that the steered Bloch vectors b ( e ) = T e areprecisely those vectors that generate the surface of thequantum steering ellipsoid for the T-state [14]. We makethe ansatz that P ( n ) is proportional to some power of the function f ( θ, φ ) in Eq. (9) that defines this surface,i.e., P ( n ) = N T [ f ( θ, φ )] m ≡ N T (cid:2) n (cid:124) T − n (cid:3) m/ (12)for n = (sin θ cos φ, sin θ sin φ, cos θ ), where N T is a nor-malisation constant. Further, denoting the region of theBloch sphere, for which p (1 | e , n ) = 1 by R [ e ], the con-dition in Eq. (10) becomes (cid:82) R [ e ] P ( n ) d n = . We notethis is automatically satisfied if R ( e ) is a hemisphere, asa consequence of the symmetry P ( n ) = P ( − n ) for theabove form of P ( n ).Hence, under the assumptions that (i) P ( n ) is deter-mined by the steering ellipsoid as per Eq. (12), and (ii) R [ e ] is a hemisphere for each unit vector e , the only re-maining constraint to be satisfied by a deterministic LHSmodel for a T-state is Eq. (11), i.e., N T (cid:90) R [ e ] (cid:2) n (cid:124) T − n (cid:3) m/ n d n = 12 T e , (13)for some suitable mapping e → R [ e ].Extensive numerical testing, with different values ofthe exponent m , show that this constraint can be satisfiedby the choices m = − , R [ e ] = { n : n T − e ≥ } , (14)for a two-parameter family of T-states. Assuming thenumerical results are correct, it is not difficult to show,using infinitesimal rotations of e about the z -axis, thatthis family corresponds to those T-states that satisfy2 πN T | det T | = 1 . (15)Fortunately, we have been able to confirm these re-sults analytically by explicitly evaluating the integral inEq. (13) for m = − N T is alsogiven in Appendix A, and it is further shown that thefamily of T-states satisfying Eq. (15) is equivalently de-fined by the condition (cid:90) √ n (cid:124) T n d n = 2 π. (16)This may be interpreted geometrically in terms of theharmonic mean radius of the ‘inverse’ ellipsoid x (cid:124) T x =1 being equal to 2. C. Necessary EPR-steerability condition
Equation (15) defines a surface in the space of possi-ble T matrices, plotted in Fig. 1(a) as a function of thesemiaxes s , s and s . As a consequence of the convex-ity of nonsteerable states (see above), all T-states corre-sponding to the region defined by this surface and thepositive octant have local hidden state models for Bob. s s FIG. 1. Correlation bounds for T-states, with s i = | t i | . Topfigure (a) : the red plane separates separable (left) and en-tangled (right) T-states. The sandwiched blue surface corre-sponds to the necessary condition for EPR-steerability gener-ated by our deterministic LHS model in Sec. 4B: all T-statesto the left of this surface are not EPR-steerable. We conjec-ture that this condition is also sufficient, i.e., that all states tothe right of the blue surface are EPR-steerable. For compar-ison, the green plane corresponds to the sufficient conditionfor EPR-steerability in Eq. (20) of section 5A: all T-statesto the right of this surface are EPR-steerable. Only a por-tion of the surfaces are shown, as they are symmetric underpermutations of s , s , s . Bottom figure (b) : Cross sec-tion through the top figure at s = s , where the necessarycondition can be determined analytically (see Sec. 4D). Theadditional black dashed curve corresponds to the non-linearsufficient condition for EPR-steerability in Eq. (22). Also shown is the boundary of the separable T-states( s + s + s ≤ s + s + s > forEPR-steerable states, derived in Sec. 5 below.It follows that a necessary condition for a T-state to beEPR-steerable by Alice is that it corresponds to a pointabove the sandwiched surface shown in Fig. 1(a). Notethat this condition is in fact symmetric between Alice andBob, since their steering ellipsoids are the same for T-states. Because of the elegant relation between our LHSmodel and the steering ellipsoid, and other evidence givenbelow, we conjecture that this condition is also sufficient for EPR-steerability. D. Special cases
When | t | = | t | we can solve Eq. (15) explicitly, be-cause the normalisation constant N T simplifies. Thecorresponding equation of the s semiaxis, in terms of u := s /s = s /s , is given by s = (cid:104) arctan( √ u − − u √ u − − (cid:105) − u < , (cid:104) − √ − u − u − ln | −√ − u − | √ − u − (cid:105) − u > , (17)and s = for u = 1. Fig. 1(b) displays this ana-lytic EPR-steerable curve through the T-state subspace | t | = | t | ⇔ s = s , showing more clearly the differentcorrelation regions.The symmetric situation s = s = s correspondsto Werner states. Our deterministic LHS model is for s = s = s = 1 / V. SUFFICIENT CONDITIONS FOREPR-STEERABILITY
In the previous section a strong necessary conditionfor the EPR-steerability of T-states was obtained, cor-responding to the boundary defined in Eq. (15) anddepicted in Fig. 1. While we have conjectured thatthis condition is also sufficient, it is not actually knownif all T-states above this boundary are EPR-steerable.Here we give two sufficient general conditions for EPR-steerability, and apply them to T-states.These conditions are examples of EPR-steering in-equalities, i.e., statistical correlation inequalities thatmust be satisfied by any LHS model for Bob [11]. Thus,violation of such an inequality immediately implies thatAlice and Bob must share an EPR-steerable resource.Our first condition is based on a new EPR-steeringinequality for the spin covariance matrix, and the secondon a known nonlinear EPR-steering inequality [16]. BothEPR-steering inequalities are further of interest in thatthey are asymmetric under the interchange of Alice andBob’s roles.
A. Linear asymmetric EPR-steering inequality
Suppose Alice and Bob share a two-qubit state withspin covariance matrix C given by C jk := (cid:104) σ j ⊗ σ k (cid:105) − (cid:104) σ j ⊗ (cid:105) (cid:104) ⊗ σ k (cid:105) = T jk − a j b k , (18)and that each can measure any Hermitian observable ontheir qubit. We show in Appendix B that, if there isan LHS model for Bob, then the singular values c , c , c of the spin covariance matrix must satisfy the linearEPR-steering inequality c + c + c ≤ (cid:112) − b . (19)From C = T − ab (cid:124) , and using a = b = and s j = | t j | for T-states, it follows immediately that one has thesimple sufficient condition s + s + s >
32 (20)for the EPR-steerability of T-states (by either Alice orBob). The boundary of T-states satisfying this condi-tion is plotted in Figs. 1 (a) and (b), showing that thecondition is relatively strong. In particular, it is a tan-gent plane to the necessary condition at the point corre-sponding to Werner states (which we already knew to bea point on the true boundary of EPR-steerable states).However, in some parameter regions a stronger conditioncan be obtained, as per below.
B. Nonlinear asymmetric EPR-steering inequality
Suppose Alice and Bob share a two-qubit state as be-fore, where Bob can measure the observables ⊗ σ , ⊗ σ φ on his qubit, with σ φ := σ cos φ + σ sin φ , forany φ ∈ [0 , π ], and Alice can measure correspondingHermitian observables A ⊗ , A φ ⊗ on her qubit, withoutcomes labelled by ±
1. It may then be shown thatany LHS model for Bob must satisfy the EPR-steeringinequality [16]1 π (cid:90) π/ − π/ (cid:104) A φ ⊗ σ φ (cid:105) dφ ≤ π (cid:20) p + (cid:113) − (cid:104) ⊗ σ (cid:105) + p − (cid:113) − (cid:104) ⊗ σ (cid:105) − (cid:21) , where p ± denotes the probability that Alice obtains re-sult A = ±
1, and (cid:104) ⊗ σ (cid:105) ± is Bob’s corresponding con-ditional expectation value for ⊗ σ for this result.As per the first part of Sec. 4A, we may always choosea representation in which the spin correlation matrix T is diagonal, i.e., T = diag[ t , t , t ], without loss ofgenerality. Making the choices A = σ and A φ = σ (sign t ) cos φ + σ (sign t ) sin φ in this representation,then p ± and (cid:104) ⊗ σ (cid:105) ± are given by p e and the thirdcomponent of b ( e ) in Eqs. (5) and (4), respectively, with e = (0 , , ± (cid:124) . Hence, the above inequality simplifies to | t | + | t | ≤ π (cid:104)(cid:112) (1 + a ) − ( t + b ) + (cid:112) (1 − a ) − ( t − b ) (cid:105) , (21)where a and b are the third components of Alice andBob’s Bloch vectors a and b . For T-states, recalling that s i ≡ | t i | , the above inequal-ity simplifies further, to the nonlinear inequality f ( s , s , s ) := s + s − π (cid:113) − s ≤ . Hence, since similar inequalities can be obtained by per-muting s , s , s , we have the sufficient conditionmax { f ( s , s , s ) , f ( s , s , s ) , f ( s , s , s ) } > s = s . It is seen to be stronger than the linearcondition in Eq. (20) if one semiaxis is sufficiently large.The region below both sufficient conditions is never farabove the smooth curve of our necessary condition, sup-porting our conjecture that the latter is the true bound-ary. VI. RECAPITULATION AND FUTUREDIRECTIONS
In this paper we have considered steering for the setof two-qubit states with maximally mixed marginals (‘T-states’), where Alice is allowed to make arbitrary projec-tive measurements on her qubit. We have constructed aLHV–LHS model (LHV for Alice, LHS for Bob), whichdescribes measurable quantum correlations for all sep-arable, and a large portion of non-separable, T-states.That is, this model reproduces the steering scenario, bywhich Alice’s measurement collapses Bob’s state to a cor-responding point on the surface of the quantum steeringellipsoid. Our model is constructed using the steering el-lipsoid, and coincides with the optimal LHV–LHS modelfor the case of Werner states. Furthermore, only a small(and sometimes vanishing) gap remains between the setof T-states that are provably non-steerable by our LHV–LHS model, and the set that are provably steerable bythe two steering inequalities that we derive. As such,we conjecture that this LHV–LHS model is in fact opti-mal for T-states. Proving this, however, remains an openquestion.A natural extension of this work is to consider LHV–LHS models for arbitrary two-qubit states. How canknowledge of their steering ellipsoids be incorporated intosuch LHV–LHS models? Investigations in this directionhave already begun, but the situation is far more com-plex when Alice and Bob’s Bloch vectors have nonzeromagnitude and the phenomenon of “one-way steering”may arise [24].Finally, our LHV–LHS models apply to the case whereAlice is restricted to measurements of Hermitian observ-ables. It would be of great interest to generalize theseto arbitrary POVM measurements. However, we notethat this is a very difficult problem even for the case oftwo-qubit Werner states [22]. Nevertheless, the steeringellipsoid is a depiction of all collapsed states, includingthose arising from POVMs (they give the interior pointsof the ellipsoid) and perhaps this can provide some intu-ition for how to proceed with this generalisation.
ACKNOWLEDGMENTS
SJ would like to thank David Jennings for his earlycontributions to this project. SJ is funded by EP-SRC grant EP/K022512/1. This work was supportedby the Australian Research Council Centre of ExcellenceCE110001027 and the European Union Seventh Frame-work Programme (FP7/2007-2013) under grant agree-ment n ◦ [316244]. Appendix A: Details of the deterministic LHS model
The family of T-states described by our deterministicLHS model in Sec. 4B corresponds to the surface definedby either of Eqs. (15) and (16). This is a consequence ofthe following theorem, proved further below.
Theorem 1.
For any full-rank diagonal matrix T andnonzero vector v one has (cid:90) n · v ≥ n d n ( n (cid:124) T − n ) = π | det T | T v | T v | . (A1)Note that substitution of Eq. (14) into constraint (13)immediately yields Eq. (15) via the theorem (with v = T − e ). Further, taking the dot product of the integralin the theorem with v , multiplying by N T , and integrat-ing v over the unit sphere, yields (reversing the order ofintegration) (cid:90) d n P ( n ) (cid:90) n · v ≥ d v v · n = π, whereas carrying out the same operationson the righthand side of the theorem yields πN T | det T | (cid:82) √ v (cid:124) T v d v . Equating these imme-diately implies the equivalence of Eqs. (15) and (16)as desired. An explicit analytic formula for the nor-malisation constant N T is given at the end of thisappendix. Proof.
First, define Q = T − ∈ GL (3 , R ); that is, Q = diag( a, b, c ) = ( t − , t − , t − ) , (A2)and q ( v ) := (cid:90) n · v ≥ n d n ( n (cid:124) Q n ) . (A3)Noting v in the theorem may be taken to be a unit vectorwithout loss of generality, we will parameterise the unitvectors n and v by n = (sin θ cos φ, sin θ sin φ, cos θ ) (cid:124) , (A4) v = (sin α cos β, sin α sin β, cos α ) (cid:124) , (A5) with θ, α ∈ [0 , π ] and φ, β ∈ [0 , π ). Thus, d n ≡ sin θ d θ d φ . Further, without loss of generality, it willbe assumed that v points into the northern hemisphere,so that cos α ≥
0. Then α ∈ [0 , π/
2] and β ∈ [0 , π ).The surface of integration is a hemisphere bounded bythe great circle n · v = 0. In the simple case where v = (0 , , T , the boundary curve has the parametricform ( x, y, z ) = (cos γ, sin γ,
0) for γ ∈ (0 , π ). Hence, theboundary curve in the generic case can be constructed byapplying the orthogonal operator R , that rotates v from(0 , , T to (sin α cos β, sin α sin β, cos α ) (cid:124) , to the vector(cos γ, sin γ, T . That is, R = cos β − sin β β cos β
00 0 1 cos α α − sin α α = cos α cos β − sin β sin α cos β cos α sin β cos β sin α sin β − sin α α , and the boundary curve has the form xyz = R cos γ sin γ = cos α cos β cos γ − sin β sin γ cos α sin β cos γ + cos β sin γ − sin α cos γ . For the purposes of integrating over the hemisphere,it is convenient to vary φ from 0 to 2 π and θ from 0 toits value χ ( φ ) on the boundary curve. From the aboveexpression for the boundary, and using z = cos θ and y/x = tan φ , it follows that cos χ = − sin α cos γ and(cos α sin β cos γ + cos β sin γ ) cos φ = (cos α cos β cos γ − sin β sin γ ) sin φ . The last equation be rearranged to readcos α sin( φ − β ) cos γ = cos( φ − β ) sin γ , and after squaringboth sides this equation solves to givecos γ = ± cos( φ − β )[cos ( φ − β ) + cos α sin ( φ − β )] / . Now, χ assumes its maximum value when φ = β , whichaccording to the relation cos χ = − sin α cos γ and thefact that α ∈ [0 , π/
2] should correspond to γ = 0. So wetake the upper sign in the last equation, yieldingcos χ = − sin α cos( φ − β )[cos ( φ − β ) + cos α sin ( φ − β )] / = − sin α cos( φ − β )[cos α + sin α cos ( φ − β )] / . (A6)It follows immediately thatsin χ = cos α [cos α + sin α cos ( φ − β )] / , (A7)with the choice of sign fixed by the fact that sin χ ≥ α ≥ q ( v ) in Eq. (A3) can now bewritten in the form: (cid:90) π (cid:90) χ ( φ )0 (sin θ cos φ, sin θ sin φ, cos θ ) T sin θ dθ dφ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) . (A8)To evaluate the the third component of q ( v ), note thatthe integral over θ , (cid:90) χ ( φ )0 sin θ cos θ d θ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) , can be evaluated explicitly by making the substitution w = sin θ , as (cid:82) ( A + Bw ) − dw = − B − ( A + Bw ) − forany B (cid:54) = 0, yielding12 c sin χa sin χ cos φ + b sin χ sin φ + c cos χ . After inserting the expressions for cos χ and sin χ derivedearlier, we have (cid:90) χ ( φ )0 sin θ cos θ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) dθ = 12 c cos αa cos α cos φ + b cos α sin φ + c sin α cos ( φ − β ) . We now need to integrate the last expression over φ . In-troducing new constants l = a cos α + c sin α cos β,m = b cos α + c sin α sin β,n = c sin α sin β cos β, the full surface integral simplifies to a form that maybe evaluated by Mathematica (or by contour integrationover the unit circle in the complex plane): (cid:90) π (cid:90) χ ( φ )0 sin θ cos θdθ dφ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) = cos α c (cid:90) π dφl cos φ + m sin φ + 2 n sin φ cos φ = ± cos α c π √ lm − n . The indeterminate sign here is fixed by examining thecase α = 0 and a = b = c , for which χ ( φ ) = π/ a − sin θ cos θ , which integratesto give πa − . So, unsurprisingly, we choose the positivesign. This yields the third component of surface integralto be[ q ( v )] = π cos αc [ ab cos α + c ( a sin β + b cos β ) sin α ] / . (A9)The integrals over θ in the remaining two componentsof q ( v ) in Eq. (A8) are unfortunately not so straightfor-ward. However, there is a simple trick that allows us tocalculate both surface integrals explicitly, and that is todifferentiate the integrals with respect to the parameters α and β . Since the only dependence on α and β comesthrough the function χ ( φ ), this eliminates the need to in-tegrate over θ . In fact we only need to differentiate withrespect to one of these parameters, choose α . To see this,note that ∂∂α (cid:90) π (cid:90) χ ( φ )0 (cos φ, sin φ ) sin θ dθ dφ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) = (cid:90) π (cos φ, sin φ ) sin χ ( a sin χ cos φ + b sin χ sin φ + c cos χ ) ∂χ∂α dφ, where ∂χ/∂α can be evaluated by making use of the equa-tions (A6) and (A7).In fact, − sin χ ∂χ∂α = ∂∂α (cid:18) − sin α cos( φ − β )[cos α + sin α cos ( φ − β )] / (cid:19) = − cos α cos( φ − β )[cos α + sin α cos ( φ − β )] / . Inserting the last two equations and the expressions forsin χ and cos χ into the integrals above, and using theconstants l, m and n defined earlier, then gives: ∂∂α (cid:90) π (cid:90) χ ( φ )0 (cos φ, sin φ ) sin θ d θ d φ ( a sin θ cos φ + b sin θ sin φ + c cos θ ) = cos α (cid:90) π (cos φ, sin φ ) cos( φ − β )[ a cos φ cos α + b sin φ cos α + c sin α cos ( φ − β )] d φ = cos α (cid:90) π (sin β sin φ cos φ + cos β cos φ, sin β sin φ + cos β sin φ cos φ )( l cos φ + m sin φ + 2 n sin φ cos φ ) d φ. (A10)Consequently, there are three separate integrals weneed to evaluate and these can be done in Mathemat- ica (or by complex contour integration): (cid:90) π (sin φ, cos φ, sin φ cos φ ) dφ ( l cos φ + m sin φ + 2 n sin φ cos φ ) = π ( l, m, − n )( lm − n ) / . Using the values we have for l, m, n we substitute theseback into equation (A10) and integrate over α to obtain[ q ( v )] = π (cid:90) cos α ( m cos β − n sin β )( lm − n ) / dα = a − π sin α cos β [ ab cos α + c sin α ( b cos β + a sin β )] / , (A11)and[ q ( v )] = π (cid:90) cos α ( l sin β − n cos β )( lm − n ) / dα = b − π sin α sin β [ ab cos α + c sin α ( b cos β + a sin β )] / . (A12)The absence of integration constants can be confirmed bynoting that these expressions vanish for α = 0 – i.e., when the vector v is aligned with the z -axis – as they shouldby symmetry. Note the denominators of Eqs. (A11) and(A12) simplify to abc ( v (cid:124) Q − v ). Combining this withEqs. (A9) and (A11)-(A12), we have q ( v ) = πQ − v (cid:112) abc ( v (cid:124) Q − v ) , (A13)and so setting Q = T − , the theorem follows as desired.Finally, the normalisation constant N T in Eq. (15) maybe analytically evaluated using Mathematica. Under theassumption that | t | > | t | > | t | , denote a = | t | , b = | t | , c = | t | . We find N − T = (cid:90) n · n =1 ( n (cid:124) T − n ) − d n = 2 πabc ( a + b )( b + c )( c − a ) × (cid:0) X + Y { b ( c − a ) E [ C ] + a ( b + c ) K [ C ] + ib ( c − a )( E [ A , B ] − E [ A , B ]) + ic ( a + b )( F [ A , B ] − F [ A , B ]) } (cid:1) , (A14)where F [ · , · ] , E [ · , · ] are the elliptic integrals the first andsecond kind, E [ · ] is the complete elliptic integral and K [ · ]is the complete elliptic integral of the first kind, and A = i arccsch (cid:18) a √ c − a (cid:19) , A = i ln (cid:18) b + c √ c − b (cid:19) ,B = a ( c − b ) b ( c − a ) , C = c ( b − a ) b ( c − a ) ,X = c ( c − a )[( a + c )( b + c )+ ab ] , Y = ( a + b + c ) (cid:112) c − a . Thus, the normalisation constant N T has a rather non-trivial form. It is highly unlikely that we can invert it toexpress the EPR-steerability condition 2 πN T | det T | = 1as c = g ( a, b ) where g is some function of a, b , other thanin the special cases considered in Sec. 4D. In general, wemust leave it as an implicit equation in a, b, c (that is, ofthe t j s). Appendix B: EPR-Steering inequality for spincovariance matrix
To demonstrate the linear EPR-steering inequality inEq. (19), let A v denote some dichotomic observable thatAlice can measure on her qubit, with outcomes labelledby ±
1, where v is any unit vector. We will make a specific choice of A v below. Define the corresponding covariancefunction C ( v ) := (cid:104) A v ⊗ v · σ (cid:105) − (cid:104) A v (cid:105) (cid:104) v · σ (cid:105) . (B1)If there is an LHS model for Bob then, noting that onemay take p ( a | x, λ )in Eq. (2) to be deterministic withoutloss of generality, there are functions α v ( λ ) = ± C ( v ) = (cid:80) λ p ( λ )[ α v ( λ ) − ¯ α v ] [ n ( λ ) − b ] · v , where¯ α v = (cid:80) λ p ( λ ) α v ( λ ), and the hidden state ρ B ( λ ) hascorresponding Bloch vector n ( λ ).Now, the Bloch sphere can be partitioned into two sets, S + ( λ ) = { v : [ n ( λ ) − b ] · v ≥ } and S − ( λ ) = { v :[ n ( λ ) − b ] · v < } , for each value of λ . Hence, noting − − ¯ α v ≤ α v ( λ ) − ¯ α v ≤ − ¯ α v , (cid:82) C ( v ) d v is equal to (cid:88) λ p ( λ ) (cid:40)(cid:90) S + ( λ ) d v [ α v ( λ ) − ¯ α v ] [ n ( λ ) − b ] · v + (cid:90) S − ( λ ) d v [ α v ( λ ) − ¯ α v ] [ n ( λ ) − b ] · v (cid:41) ≤ (cid:88) λ p ( λ ) (cid:40)(cid:90) S + ( λ ) d v [1 − ¯ α v ] [ n ( λ ) − b ] · v − (cid:90) S − ( λ ) d v [1 + ¯ α v ] [ n ( λ ) − b ] · v (cid:41) = (cid:88) λ p ( λ ) (cid:90) d v | [ n ( λ ) − b ] · v | − (cid:88) λ p ( λ ) (cid:90) d v ¯ α v [ n ( λ ) − b ] · v = (cid:88) λ p ( λ ) | n ( λ ) − b | (cid:90) d v | v · w ( λ ) | , where w ( λ ) denotes the unit vector in the n ( λ ) − b direc-tion, and the last line follows by interchanging the sum-mation and integration in the second term of the previousline.The integral in the last line can be evaluated for eachvalue of λ by rotating the coordinates such that w ( λ )is aligned with the z -axis, yielding (cid:82) d v | v · w ( λ ) | = (cid:82) d v | v | = (cid:82) π d φ (cid:82) π d θ sin θ | cos θ | = 2 π . Hence, theabove inequality can be rewritten as14 π (cid:90) d v C ( v ) ≤ (cid:88) λ p ( λ ) | n ( λ ) − b |≤ (cid:34)(cid:88) λ p ( λ ) | n ( λ ) − b | (cid:35) / ≤ √ − b · b , (B2) where the second and third lines follow using the Schwarzinequality and | n ( λ ) | ≤
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