document: quantum systems can be correlated in ways that supersede classical descriptions . 
   however , there are degrees of non - classicality for quantum correlations . for simplicity , we consider only bipartite correlations , with the two , spatially separated , parties being named alice and bob as usual .    at the weaker end of the spectrum are quantum systems whose states can not be expressed as a mixture of product - states of the constituents . 
   these are called non - separable or entangled states . 
   the product - states appearing in such a mixture comprise a local hidden state ( lhs ) model for any measurements undertaken by alice and bob .    at the strongest end of the spectrum 
   are quantum systems whose measurement correlations can violate a bell inequality  @xcite , hence demonstrating ( modulo loopholes  @xcite ) the violation of local causality  @xcite . 
   this phenomenon  commonly known as bell - nonlocality  @xcite  is the only way for two spatially separated parties to verify the existence of entanglement if either of them , or their detectors , can not be trusted  @xcite . 
   we say that a bipartite state is bell - local if and only if there is a local hidden variable ( lhv ) model for any measurements alice and bob perform . here 
   the ` variables ' are not restricted to be quantum states , hence the distinction between non - separability and bell - nonlocality .    in between these types of non - classical correlations lies epr - steering . 
   the name is inspired by the seminal paper of einstein , podolsky , and rosen ( epr )  @xcite , and the follow - up by schrdinger  @xcite , which coined the term `` steering '' for the phenomenon epr had noticed . although introduced eighty years ago , as this special issue celebrates , the notion of epr - steering was only formalized eight years ago , by one of us and co - workers  @xcite . 
   this formalization was that epr - steering is the only way to verify the existence of entanglement if one of the parties  conventionally alice  @xcite  or her detectors , can not 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 there exists a hybrid lhv  lhs model explaining the alice  bob correlations . since in this paper we are concerned with steering , when we refer to a lhs model we mean a lhs model for bob only ; it is implicit that alice can have a completely general lhv model . 
   the above three notions of non - locality for quantum states coincide for pure states : any non - product pure state is non - separable , eps - steerable , and bell - nonlocal . 
   however for mixed states , the interplay of quantum and classical correlations produces a far richer structure . for mixed states 
   the logical hierarchy of the three concepts leads to a hierarchy for the bipartite states : the set of separable states is a strict subset of the set of non - epr - steerable states , which is a strict subset of the set of bell - local states  @xcite . 
   although the epr - steerable set has been completely determined for certain classes of highly symmetric states ( at least for the case where alice and bob perform projective measurements )  @xcite , until now very little was known about what types of states are steerable even for the 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 measurements on her system  has been studied extensively using the so - called steering ellipsoid formalism @xcite . 
   however , no relation between the steering ellipsoid and epr - steerability has been determined .    in this manuscript 
   , we investigate epr - steerability of the class of two - qubit states whose reduced states are maximally mixed , the so - called t - states  @xcite . 
   we use the steering ellipsoid formalism to develop a deterministic lhs model for projective measurements on these states and we conjecture that this model is optimal . 
   furthermore we obtain two sufficient conditions for t - states to be epr - steerable , via suitable epr - steering inequalities  @xcite ( including a new asymmetric steering inequality for the spin covariance matrix ) . 
   these sufficient conditions touch the necessary condition in some regions of the space of t - states , and everywhere else the gap between them is quite small . 
   the paper is organised as follows . in section 2 
   we discuss in detail the three notions of non - locality , namely bell - nonlocality , epr - steerability and non - separability . 
   section 3 introduces the quantum steering ellipsoid formalism for a two - qubit state , and in section 4 we use the steering ellipsoid to develop a deterministic lhs model for projective measurements on t - states . in section 5 , 
   two asymmetric steering inequalities for arbitrary two - qubit states are derived . 
   finally in section 6 we conclude and discuss further work . 
   two separated observers , alice and bob , can use a shared quantum state to generate statistical correlations between local measurement outcomes . 
   each observer carries out a local measurement , labelled by @xmath1 and @xmath2 respectively , to obtain corresponding outcomes labelled by @xmath3 and @xmath4 . 
   the measurement correlations are described by some set of joint probability distributions , @xmath5 , with @xmath1 and @xmath2 ranging over the available measurements . 
   the type of state shared by alice and bob may be classified via the properties of these joint distributions , for all possible measurement settings @xmath1 and @xmath2 . 
   the correlations of a _ bell - local _ state have a local hidden variable ( lhv ) model @xcite , @xmath6 for some ` hidden ' random variable @xmath7 with probability distribution @xmath8 . 
   hence , the measured correlations may be understood as arising from ignorance of the value of @xmath7 , where the latter locally determines the statistics of the outcomes @xmath3 and @xmath4 and is independent of the choice of @xmath1 and @xmath2 . 
   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 alice and bob without trust in their devices  @xcite .    in this paper , 
   we are concerned with whether the state is _ steerable _ ; that is , whether it allows for correlations that demonstrate epr - steering . 
   as discussed in the introduction , epr - steering by alice is demonstrated if it is _ not _ the case that the correlations can be described by a hybrid lhv  lhs model , wherein , @xmath9 where the local distributions @xmath10 correspond to measurements on local quantum states @xmath11 , i.e. , @xmath12.\ ] ] here @xmath13 denotes the positive operator valued measure ( povm ) corresponding to measurement @xmath2 . 
   the state is said to be _ 
   steerable _ by alice if there is _ no _ such model . 
   the roles of alice and bob may also be reversed in the above , to define steerability by bob . 
   comparing eqs . 
   ( [ lhv ] ) and ( [ lhs ] ) , it is seen that all nonsteerable states are bell - local . 
   hence , all bell - nonlocal states are steerable , by both alice and bob . 
   in fact , the class of steerable states is strictly larger @xcite . 
   moreover , while not as powerful as bell - nonlocality in general , steerability is more robust to detection inefficiencies @xcite , and also enables the use of untrusted devices in quantum key distribution , albeit only on one side @xcite . by a similar argument , 
   a separable quantum state shared by alice and bob , @xmath14 , is both bell - local and nonsteerable . 
   moreover , the set of separable states is strictly smaller than the set of nonsteerable states  @xcite . 
   it is important that epr - steerability of a quantum state not be confused with merely the dependence of the reduced state of one observer on the choice of measurement made by another , which can occur even for separable states . 
   the term ` steering ' has been used with reference to this phenomenon , in particular for the concept of ` steering ellipsoid ' , which we will use in our analysis . 
   epr - steering , as defined above , is a special case of this phenomenon , and is only possible for a subset of nonseparable states . 
   we are interested in the epr - steerability of states for all possible _ projective _ measurements . if alice is doing the steering , then it is sufficient for bob s measurements to comprise some tomographically complete set of projectors . it is straightforward to show in this case that the condition for bob to have an lhs model , eq . 
   ( [ lhs ] ) , reduces to the existence of a representation of the form    @xmath15 = \sum_\lambda p(\lambda)\ , p(1|e,\lambda ) \,\rho_b(\lambda ) , \\ 
   \label{reduced_p } p_e & = \operatorname{tr } [ \rho e\otimes i ] = \sum_\lambda p(\lambda ) p(1|e,\lambda).\end{aligned}\ ] ]    here @xmath16 is any projector that can be measured by alice ; @xmath17 is the probability of result ` @xmath18 ' and @xmath19 is the corresponding probability given @xmath7 ; @xmath20 is the reduced state of bob s component corresponding to this result ; and @xmath21 $ ] denotes the partial trace over alice s component . 
   note that this form , and hence epr - steerability by alice , is invariant under local unitary transformations on bob s components . 
   determining epr - steerability in this case , where alice is permitted to measure any hermitian observable , is surprisingly difficult , with the answer only known for certain special cases such as werner states @xcite . 
   however , in this paper we give a strong necessary condition for the epr - steerability of a large class of two - qubit states , which we conjecture is also sufficient . 
   this condition is obtained via the construction of a suitable lhs model , which is in turn motivated by properties of the ` quantum steering ellipsoid '  @xcite . 
   properties of this ellipsoid are therefore reviewed in the following section . 
   an arbitrary two - qubit state may be written in the standard form @xmath22 here @xmath23 denote the pauli spin operators , and @xmath24,~ b_j=\operatorname{tr}[\rho\ ,   \i\otimes\sigma_j],~t_{jk}=\operatorname{tr}[\rho \,\sigma_j\otimes\sigma_k ] .\ ] ] thus , @xmath25 and @xmath26 are the bloch vectors for alice and bob s qubits , and @xmath27 is the spin correlation matrix .    if alice makes a projective measurement on her qubit , and obtains an outcome corresponding to projector @xmath16 , bob s reduced state follows from eq . 
   ( [ reduced ] ) as @xmath28 } { \operatorname{tr}[\rho \,e \otimes \i ] } .\ ] ] we will also refer to @xmath20 as bob s ` steered state ' .    to determine bob s possible steered states , note that the projector @xmath16 may be expanded in the pauli basis as @xmath29 , with @xmath30 . 
   this yields the corresponding steered state @xmath31 , with associated bloch vector @xmath32 where @xmath33 is the associated probability of result @xmath34 , @xmath35 = \frac 12 ( 1+{\boldsymbol}a\cdot{\boldsymbol}e),\ ] ] called @xmath17 previously . in what follows 
   we will refer to the vector @xmath36 rather than its corresponding operator @xmath16 . 
   the surface of the steering ellipsoid is defined to be the set of steered bloch vectors , @xmath37 , and in ref . 
   @xcite it is shown that interior points can be obtained from positive operator - valued measurements ( povms ) . 
   the ellipsoid has centre @xmath38 and the semiaxes @xmath39 are the roots of the eigenvalues of the matrix @xmath40 the eigenvectors of @xmath41 give the orientation of the ellipsoid around its centre @xcite . 
   thus , the general equation of the steering ellipsoid surface is @xmath42 with @xmath43 being the displacement vector from the centre @xmath44 . 
   entangled states typically have large steering ellipsoids  the largest possible being the bloch ball , which is generated by every pure entangled state @xcite . 
   in contrast , the volume of the steering ellipsoid is strictly bounded for separable states . indeed 
   , a two - qubit state is separable if and only if its steering ellipsoid is contained within a tetrahedron contained within the bloch sphere @xcite . 
   thus , the separability of two - qubit states has a beautiful geometric characterisation in terms 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 @xmath41 , its centre @xmath45 , and bob s bloch vector @xmath46 uniquely determines the shared state @xmath47 up to a local unitary transformation on alice s system @xcite , @xcite and so is sufficient , in principle , to determine the epr - steerability of @xmath47 . in this paper 
   we find a direct connection between epr - steerability and the quantum steering ellipsoid , for the case that the bloch vectors @xmath48 and @xmath46 vanish . 
   let @xmath49 be a singular value decomposition of the spin correlation matrix @xmath27 , for some diagonal matrix @xmath50 and orthogonal matrices @xmath51 . noting that any @xmath52 is either a rotation or the product of a rotation with the parity matrix @xmath53 
   , it follows that @xmath27 can always be represented in the form @xmath54 , for proper rotations @xmath55 , where the diagonal matrix @xmath56 may now have negative entries . 
   the rotations @xmath57 and @xmath58 may be implemented by local unitary operations on the shared state @xmath47 , amounting to a local basis change . 
   hence , all properties of a shared two - qubit state , including steerability properties in particular , can be formulated in a representation in which the spin correlation matrix has the diagonal form @xmath59 $ ] . 
   it follows that if the shared state @xmath47 has maximally - mixed reduced states with @xmath60 , then it is completely described , up to local unitaries , by a diagonal @xmath27 , i.e. one may consider @xmath61 without loss of generality . 
   such states are called t - states  @xcite . 
   they are equivalent to mixtures of bell states , and hence form a tetrahedron in the space parameterised by @xmath62 @xcite . entangled t - states necessarily have @xmath63 , and the set of separable t - states forms an octahedron within the tetrahedron @xcite . 
   the t - state steering ellipsoid is centred at the origin , @xmath64 , and the ellipsoid matrix is simply @xmath65 , as follows from eqs . 
   ( [ qse_centre ] ) and ( [ qse_mat ] ) with @xmath66 . the semiaxes are @xmath67 for @xmath68 , and are aligned with the @xmath69-axes of the bloch sphere . 
   thus , the equation of the ellipsoid surface in spherical coordinates @xmath70 is @xmath71 , with @xmath72 we find a remarkable connection between this equation and the epr - steerability of t - states in the following subsection .      without loss of generality , 
   consider measurement by alice of hermitian observables on her qubit . 
   such observables can be equivalently represented via projections , @xmath73 , with @xmath74 . 
   the probability of result ` @xmath18 ' and the corresponding steered bloch vector are given by eqs . 
   ( [ qsteerb ] ) and ( [ pe ] ) with @xmath60 , i.e. , @xmath75 hence , letting @xmath76 denote the bloch vector corresponding to @xmath77 in eq . 
   ( [ reduced ] ) , then from eqs . 
   ( [ reduced ] ) and , it follows there is an lhs model for bob if and only if there is a representation of the form @xmath78 for all unit vectors @xmath36 . noting further that @xmath76 can always be represented as some mixture of unit vectors , corresponding to pure @xmath11 , these conditions are equivalent to the existence of a representation of the form @xmath79 with integration over the bloch sphere . 
   thus , the unit bloch vector @xmath80 labels both the local hidden state and the hidden variable .    given lhs models for bob for any two t - states , having spin correlation matrices @xmath81 and @xmath82 , it is trivial to construct an lhs model for the t - state corresponding to @xmath83 , for any @xmath84 , via the convexity property of nonsteerable states @xcite . 
   our strategy is to find _ deterministic _ lhs models for some set of t - states , for which the result ` @xmath18 ' is fully determined by knowledge of @xmath80 , i.e. , @xmath85 . 
   lhs models can then be constructed for all convex combinations of t - states in this set .    to find deterministic lhs models , we are guided by the fact that the steered bloch vectors @xmath86 are precisely those vectors that generate the surface of the quantum steering ellipsoid for the t - state @xcite . 
   we make the ansatz that @xmath87 is proportional to some power of the function @xmath88 in eq . 
   ( [ ell_eqn ] ) that defines this surface , i.e. , @xmath89^m   \equiv n_t\,\left[{\boldsymbol}n^\intercal t^{-2}{\boldsymbol}n\right]^{m/2}\ ] ] for @xmath90 , where @xmath91 is a normalisation constant . 
   further , denoting the region of the bloch sphere , for which @xmath92 by @xmath93 $ ] , the condition in eq . 
   ( [ peint ] ) becomes @xmath94 } p({\boldsymbol}n ) { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 $ ] . 
   we note this is automatically satisfied if @xmath95 is a hemisphere , as a consequence of the symmetry @xmath96 for the above form of @xmath87 . 
   hence , under the assumptions that ( i ) @xmath87 is determined by the steering ellipsoid as per eq . 
   ( [ pform ] ) , and ( ii ) @xmath93 $ ] is a hemisphere for each unit vector @xmath36 , the only remaining constraint to be satisfied by a deterministic lhs model for a t - state is eq . 
   ( [ beint ] ) , i.e. , @xmath97 }   \left [ { \boldsymbol}n^\intercal t^{-2 } { \boldsymbol}n\right]^{m/2}\,{\boldsymbol}n { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 t{\boldsymbol}e,\ ] ] for some suitable mapping @xmath98 $ ] .    extensive numerical testing , with different values of the exponent @xmath99 , show that this constraint can be satisfied by the choices @xmath100= \{{\boldsymbol}n : { \boldsymbol}nt^{-1 } { \boldsymbol}e\geq 0\},\ ] ] for a two - parameter family of t - states . assuming the numerical results are correct , it is not difficult to show , using infinitesimal rotations of @xmath36 about the @xmath101-axis , that this family corresponds to those t - states that satisfy @xmath102 fortunately 
   , we have been able to confirm these results analytically by explicitly evaluating the integral in eq . 
   ( [ con ] ) for @xmath103 ( see appendix a ) . an explicit form for the corresponding normalisation constant @xmath91 
   is also given in appendix a , and it is further shown that the family of t - states satisfying eq . 
   ( [ surface ] ) is equivalently defined by the condition @xmath104 this may be interpreted geometrically in terms of the harmonic mean radius of the ` inverse ' ellipsoid @xmath105 being equal to @xmath106 
   .      equation ( [ surface ] ) defines a surface in the space of possible @xmath27 matrices , plotted in fig . 
   1(a ) as a function of the semiaxes @xmath107 and @xmath108 . as a consequence of the convexity of nonsteerable states 
   ( see above ) , all t - states corresponding to the region defined by this surface and the positive octant have local hidden state models for bob 
   . also shown is the boundary of the separable t - states ( @xmath109 @xcite ) , in red , which is clearly a strict subset of the nonsteerable t - states . 
   the green plane corresponds to the sufficient condition @xmath110 for epr - steerable states , derived in sec .  5 below . 
   it follows that a necessary condition for a t - state to be epr - steerable by alice is that it corresponds to a point above the sandwiched surface shown in fig . 
   note that this condition is in fact symmetric between alice and bob , since their steering ellipsoids are the same for t - states . because of the elegant relation between our lhs model and the steering ellipsoid , and other evidence given below , we conjecture that this condition is also _ sufficient _ for epr - steerability .    . 
   * top figure ( a ) * : the red plane separates separable ( left ) and entangled ( right ) t - states . 
   the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec . 
   4b : all t - states to the left of this surface are not epr - steerable . 
   we conjecture that this condition is also sufficient , i.e. , that all states to the right of the blue surface are epr - steerable . for comparison , the green plane corresponds to the sufficient condition for epr - steerability in eq . 
   ( [ linsuff ] ) of section 5a : all t - states to the right of this surface are epr - steerable . 
   only a portion of the surfaces are shown , as they are symmetric under permutations of @xmath111 . * 
   bottom figure ( b ) * : cross section through the top figure at @xmath112 , where the necessary condition can be determined analytically ( see sec . 
   the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq . 
   .,title="fig:",width=336 ] . 
   * top figure ( a ) * : the red plane separates separable ( left ) and entangled ( right ) t - states . 
   the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec . 
   4b : all t - states to the left of this surface are not epr - steerable . 
   we conjecture that this condition is also sufficient , i.e. , that all states to the right of the blue surface are epr - steerable . 
   for comparison , the green plane corresponds to the sufficient condition for epr - steerability in eq . 
   ( [ linsuff ] ) of section 5a : all t - states to the right of this surface are epr - steerable . 
   only a portion of the surfaces are shown , as they are symmetric under permutations of @xmath111 . * 
   bottom figure ( b ) * : cross section through the top figure at @xmath112 , where the necessary condition can be determined analytically ( see sec . 
   the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq . .,title="fig:",width=259 ]      when @xmath113 we can solve eq . 
   ( [ surface ] ) explicitly , because the normalisation constant @xmath91 simplifies . 
   the corresponding equation of the @xmath108 semiaxis , in terms of @xmath114 , is given by @xmath115 ^{-1 } & u < 1 ,   \\ \left[1 - \frac { \sqrt{1-u^{-2}}}{2(u^2 - 1)}\ln\frac{|1-\sqrt{1-u^{-2}}|}{1+\sqrt{1-u^{-2 } } } \right ] ^{-1 } & u > 1 , \end{array } \right.\end{aligned}\ ] ] and @xmath116 for @xmath117 . 
   1(b ) displays this analytic epr - steerable curve through the t - state subspace @xmath118 , showing more clearly the different correlation regions . 
   the symmetric situation @xmath119 corresponds to werner states . 
   our deterministic lhs model is for @xmath120 in this case , which is known to represent the epr - steerable boundary for werner states @xcite . 
   thus , our model is certainly optimal for this class of states . 
   in the previous section a strong necessary condition for the epr - steerability of t - states was obtained , corresponding to the boundary defined in eq . 
   ( [ surface ] ) and depicted in fig .  1 . 
   while we have conjectured that this condition is also sufficient , it is not actually known if 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 inequalities , i.e. , statistical correlation inequalities that must be satisfied by any lhs model for bob @xcite . 
   thus , violation of such an inequality immediately implies that alice and bob must share an epr - steerable resource . 
   our first condition is based on a new epr - steering inequality for the spin covariance matrix , and the second on a known nonlinear epr - steering inequality @xcite . 
   both epr - steering inequalities are further of interest in that they are asymmetric under the interchange of alice and bob s roles . 
   suppose alice and bob share a two - qubit state with spin covariance matrix @xmath121 given by @xmath122 and that each can measure any hermitian observable on their qubit . 
   we show in appendix  [ app : cov ] that , if there is an lhs model for bob , then the singular values @xmath123 , 
   @xmath124 , @xmath125 of the spin covariance matrix must satisfy the linear epr - steering inequality @xmath126 from @xmath127 , and using @xmath60 and @xmath128 for t - states , it follows immediately that one has the simple _ sufficient _ condition @xmath129 for the epr - steerability of t - states ( by either alice or bob ) . 
   the boundary of t - states satisfying this condition is plotted in figs . 
   1 ( a ) and  ( b ) , showing that the condition is relatively strong . in particular 
   , it is a tangent plane to the necessary condition at the point corresponding to werner states ( which we already knew to be a point on the true boundary of epr - steerable states ) . 
   however , in some parameter regions a stronger condition can be obtained , as per below . 
   suppose alice and bob share a two - qubit state as before , where bob can measure the observables @xmath130 , @xmath131 on his qubit , with @xmath132 , for any @xmath133 $ ] , and alice can measure corresponding hermitian observables @xmath134 on her qubit , with outcomes labelled by @xmath135 . 
   it may then be shown that any lhs model for bob must satisfy the epr - steering inequality  @xcite @xmath136,\end{aligned}\ ] ] where @xmath137 denotes the probability that alice obtains result @xmath138 , and @xmath139 is bob s corresponding conditional expectation value for @xmath130 for this result . 
   as per the first part of sec .  4a , we may always choose a representation in which the spin correlation matrix @xmath27 is diagonal , i.e. , @xmath140 $ ] , without loss of generality . making the choices 
   @xmath141 and @xmath142 in this representation , then @xmath143 and @xmath139 are given by @xmath33 and the third component of @xmath144 in eqs . 
   ( [ pe ] ) and ( [ qsteerb ] ) , respectively , with @xmath145 . 
   hence , the above inequality simplifies to @xmath146,\end{aligned}\ ] ] where @xmath147 and @xmath148 are the third components of alice and bob s bloch vectors @xmath48 and @xmath46 .    for t - states , recalling that @xmath149 , the above inequality simplifies further , to the nonlinear inequality @xmath150 hence , since similar inequalities can be obtained by permuting @xmath151 , we have the _ sufficient _ condition @xmath152 for the epr - steerability of t - states . 
   the boundary of t - states satisfying this condition is plotted in fig . 
   1(b ) for the case @xmath112 . 
   it is seen to be stronger than the linear condition in eq . 
   ( [ linsuff ] ) if one semiaxis is sufficiently large . 
   the region below both sufficient conditions is never far above the smooth curve of our necessary condition , supporting our conjecture that the latter is the true boundary . 
   in this paper we have considered steering for the set of two - qubit states with maximally mixed marginals ( ` t - states ' ) , where alice is allowed to make arbitrary projective measurements on her qubit . 
   we have constructed a lhv  lhs model ( lhv for alice , lhs for bob ) , which describes measurable quantum correlations for all separable , and a large portion of non - separable , t - states . 
   that is , this model reproduces the steering scenario , by which alice s measurement collapses bob s state to a corresponding point on the surface of the quantum steering ellipsoid . 
   our model is constructed using the steering ellipsoid , and coincides with the optimal lhv  lhs model for the case of werner states . 
   furthermore , only a small ( and sometimes vanishing ) gap remains between the set of t - states that are provably non - steerable by our lhv  lhs model , and the set that are provably steerable by the two steering inequalities that we derive . 
   as such , we conjecture that this lhv 
   lhs model is in fact optimal for t - states . 
   proving this , however , remains an open question . 
   a natural extension of this work is to consider lhv 
   lhs models for arbitrary two - qubit states 
   . how can knowledge of their steering ellipsoids be incorporated into such lhv  lhs models ? 
   investigations in this direction have already begun , but the situation is far more complex when alice and bob s bloch vectors have nonzero magnitude and the phenomenon of `` one - way steering '' may arise @xcite .    finally , our lhv 
   lhs models apply to the case where alice is restricted to measurements of hermitian observables 
   . it would be of great interest to generalize these to arbitrary povm measurements . 
   however , we note that this is a very difficult problem even for the case of two - qubit werner states @xcite 
   . nevertheless , the steering ellipsoid is a depiction of all collapsed states , including those arising from povms ( they give the interior points of the ellipsoid ) and perhaps this can provide some intuition for how to proceed with this generalisation . 
   sj would like to thank david jennings for his early contributions to this project . 
   sj is funded by epsrc grant ep / k022512/1 . 
   this work was supported by the australian research council centre of excellence ce110001027 and the european union seventh framework programme ( fp7/2007 - 2013 ) under grant agreement n@xmath153 [ 316244 ] . 
   the family of t - states described by our deterministic lhs model in sec . 
   4b corresponds to the surface defined by either of eqs . 
   ( [ surface ] ) and ( [ surface2 ] ) . 
   this is a consequence of the following theorem , proved further below .    for any full - rank diagonal matrix @xmath27 and nonzero vector @xmath154 
   one has @xmath155    note that substitution of eq . 
   ( [ re ] ) into constraint  ( [ con ] ) immediately yields eq . 
   ( [ surface ] ) via the theorem ( with @xmath156 ) . 
   further , taking the dot product of the integral in the theorem with @xmath157 , multiplying by @xmath91 , and integrating @xmath157 over the unit sphere , yields ( reversing the order of integration ) @xmath158 whereas carrying out the same operations on the righthand side of the theorem yields @xmath159 . equating these immediately implies the equivalence of eqs . 
   ( [ surface ] ) and ( [ surface2 ] ) as desired . 
   an explicit analytic formula for the normalisation constant @xmath91 is given at the end of this appendix . 
   first , define @xmath160 ; that is , @xmath161 and @xmath162 noting @xmath157 in the theorem may be taken to be a unit vector without loss of generality , we will parameterise the unit vectors @xmath80 and @xmath157 by @xmath163 with @xmath164 $ ] and @xmath165 . 
   thus , @xmath166 . 
   further , without loss of generality , it will be assumed that @xmath167 points into the northern hemisphere , so that @xmath168 . then @xmath169 $ ] and @xmath170 . 
   the surface of integration is a hemisphere bounded by the great circle @xmath171 . in the simple case where 
   @xmath172 , the boundary curve has the parametric form @xmath173 for @xmath174 . 
   hence , the boundary curve in the generic case can be constructed by applying the orthogonal operator @xmath175 , that rotates @xmath167 from @xmath176 to @xmath177 , to the vector @xmath178 . 
   that is , @xmath179 and the boundary curve has the form @xmath180    for the purposes of integrating over the hemisphere , it is convenient to vary @xmath181 from @xmath182 to @xmath183 and @xmath184 from @xmath182 to its value @xmath185 on the boundary curve . from the above expression for the boundary , and using @xmath186 and @xmath187 , it follows that @xmath188 and @xmath189 . 
   the last equation be rearranged to read @xmath190 , and after squaring both sides this equation solves to give @xmath191^{1/2}}.\]]now , @xmath192 assumes its maximum value when @xmath193 , which according to the relation @xmath188 and the fact that @xmath194 $ ] should correspond to @xmath195 . 
   so we take the upper sign in the last equation , yielding @xmath196^{1/2}}\\ & = \frac{-\sin \alpha\cos ( \phi -\beta)}{[\cos ^{2}\alpha+\sin ^{2}\alpha\cos ^{2}(\phi -\beta)]^{1/2 } } .\end{aligned}\ ] ] it follows immediately that @xmath197^{1/2}},\end{aligned}\ ] ] with the choice of sign fixed by the fact that @xmath198 and ( by assumption ) @xmath168 .    the surface integral for @xmath199 in eq . 
   ( [ qv ] ) can now be written in the form : @xmath200    to evaluate the the third component of @xmath199 , note that the integral over @xmath184 , @xmath201can be evaluated explicitly by making the substitution @xmath202 , as @xmath203 for any @xmath204 , yielding @xmath205 after inserting the expressions for @xmath206 and @xmath207 derived earlier , we have @xmath208 we now need to integrate the last expression over @xmath181 . 
   introducing new constants @xmath209 the full surface integral simplifies to a form that may be evaluated by mathematica ( or by contour integration over the unit circle in the complex plane ) : @xmath210the indeterminate sign here is fixed by examining the case @xmath211 and @xmath212 , for which @xmath213 and the integrand reduces to @xmath214 , which integrates to give @xmath215 . 
   so , unsurprisingly , we choose the positive sign . 
   this yields the third component of surface integral to be @xmath216_3 = \frac{\pi \cos \alpha}{c[ab\cos ^{2}\alpha+c(a\sin ^{2}\beta+b\cos ^{2}\beta)\sin ^{2}\alpha]^{1/2}}.\end{aligned}\ ] ]    the integrals over @xmath184 in the remaining two components of @xmath199 in eq . 
   ( [ qvnew ] ) are unfortunately not so straightforward . 
   however , there is a simple trick that allows us to calculate both surface integrals explicitly , and that is to differentiate the integrals with respect to the parameters @xmath217 and @xmath218 . 
   since the only dependence on @xmath217 and @xmath218 comes through the function @xmath219 , this eliminates the need to integrate over @xmath184 . 
   in fact we only need to differentiate with respect to one of these parameters , choose @xmath217 . to see this , note that @xmath220where @xmath221 can be evaluated by making use of the equations and . 
   in fact , @xmath222^{1/2}}\right)\end{aligned}\ ] ] @xmath223^{3/2}}.\end{aligned}\ ] ] inserting the last two equations and the expressions for @xmath224 and @xmath225 into the integrals above , and using the constants @xmath226 and @xmath227 defined earlier , then gives :    @xmath228^{2}}{\mathop{}\!\mathrm{d}}\phi   \nonumber \\ 
   \label{xy1 } ~&=&\cos ^{2}\alpha\int_{0}^{2\pi } \frac{(\sin \beta\sin \phi \cos \phi + \cos \beta\cos ^{2}\phi , \sin \beta\sin ^{2}\phi + \cos \beta\sin 
   \phi \cos \phi ) } { ( l \cos ^{2}\phi + m \sin ^{2}\phi + 2n \sin \phi \cos \phi ) ^{2}}{\mathop{}\!\mathrm{d}}\phi .\end{aligned}\ ] ]    consequently , there are three separate integrals we need to evaluate and these can be done in mathematica ( or by complex contour integration ) : @xmath229 using the values we have for @xmath230 we substitute these back into equation and integrate over @xmath217 to obtain @xmath231_1 = \pi\int \frac{\cos ^{2}\alpha(m\cos \beta - n\sin \beta )   } { ( lm - n^2)^{3/2}}d\alpha\end{aligned}\ ] ] @xmath232^{1/2}},\end{aligned}\ ] ] and @xmath233_2 = \pi\int \frac{\cos ^{2}\alpha(l\sin \beta -n\cos \beta ) } { ( lm - n^2)^{3/2}}d\alpha\\ \label{qv2 } & = \frac{b^{-1}\pi \sin\alpha\sin\beta}{[ab \cos^2\alpha + c\sin^2\alpha(b\cos^2\beta + a\sin^2\beta)]^{1/2}}.\end{aligned}\ ] ] the absence of integration constants can be confirmed by noting that these expressions vanish for @xmath211  i.e. , when the vector @xmath234 is aligned with the @xmath101-axis  as they should by symmetry . 
   note the denominators of eqs . and simplify to @xmath235 . combining this with eqs . 
   ( [ qv3 ] ) and ( [ qv1])-([qv2 ] ) , we have @xmath236 and so setting @xmath237 , the theorem follows as desired .    finally , the normalisation constant @xmath91 in eq . 
   ( [ surface ] ) may be analytically evaluated using mathematica . under the assumption that @xmath238 , denote @xmath239 . 
   we find    @xmath240+a(b+c)k[c]+ib(c - a)(e[a_1,b]-e[a_2,b])+ic(a+b)(f[a_1,b]-f[a_2,b])\right\}\big),\end{aligned}\ ] ] 
   to demonstrate the linear epr - steering inequality in eq . 
   ( [ lin ] ) , let @xmath253 denote some dichotomic observable that alice can measure on her qubit , with outcomes labelled by @xmath135 , where @xmath157 is any unit vector . 
   we will make a specific choice of @xmath253 below . 
   define the corresponding covariance function @xmath254 if there is an lhs model for bob then , noting that one may take @xmath255 in eq . 
   ( [ lhs ] ) to be deterministic without loss of generality , there are functions @xmath256 such that @xmath257\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v$ ] , where @xmath258 , and the hidden state @xmath11 has corresponding bloch vector @xmath76 . 
   now , the bloch sphere can be partitioned into two sets , @xmath259\cdot{\boldsymbol}v \geq 0\}$ ] and @xmath260\cdot{\boldsymbol}v < 0\}$ ] , for each value of @xmath7 . 
   hence , noting @xmath261 , @xmath262 is equal to @xmath263\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & \left . 
   +   \int_{s_-(\lambda ) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , [ \alpha_{{\boldsymbol}v}(\lambda ) - \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\end{aligned}\ ] ] @xmath264\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & -\left . 
   \int_{s_-(\lambda ) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , [ 1 + \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\\ & = & \sum_\lambda p(\lambda ) \int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , |[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v|\end{aligned}\ ] ] @xmath265\cdot{\boldsymbol}v\\ & = &   \sum_\lambda p(\lambda ) |{\boldsymbol}n(\lambda)-{\boldsymbol}b|\,\int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , |{\boldsymbol}v\cdot { \boldsymbol}w(\lambda)|,\end{aligned}\ ] ] where @xmath266 denotes the unit vector in the @xmath267 direction , and the last line follows by interchanging the summation and integration in the second term of the previous line . 
   the integral in the last line can be evaluated for each value of @xmath7 by rotating the coordinates such that @xmath268 is aligned with the @xmath101-axis , yielding @xmath269 . 
   hence , the above inequality can be rewritten as @xmath270^{1/2 } \\ 
   \label{cav } & \leq & \frac{1}{2}\sqrt{1-{\boldsymbol}b\cdot{\boldsymbol}b},\end{aligned}\ ] ] where the second and third lines follow using the schwarz inequality and @xmath271 , respectively . 
   note , by the way , that the first inequality is tight for the case @xmath272\cdot{\boldsymbol}v\right)$ ] . 
   now , making the choice @xmath273 with @xmath274 , one has from eqs . 
   ( [ cjk ] ) and ( [ cv ] ) that @xmath275 combining with eq . 
   ( [ cav ] ) immediately yields the epr - steering inequality @xmath276 finally , this inequality may similarly be derived in a representation in which local rotations put the spin covariance matrix @xmath121 in diagonal form , with coefficients given up to a sign by the singular values of @xmath121 ( similarly to the spin correlation matrix @xmath27 in sec . 
   4a ) . since @xmath277 is invariant under such rotations , eq . 
   ( [ lin ] ) follows . 
   a.  j. bennet , d.  a. evans , d.  j. saunders , c.  branciard , e.  g. cavalcanti , h.  m. wiseman , and g.  j. pryde , `` arbitrarily loss - tolerant einstein - podolsky - rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole , '' phys . 
   x * 2 * , 031003 ( 2012 ) .    c.  branciard , e.  g. cavalcanti , s.  p. walborn , v.  scarani , and h.  m. wiseman , `` one - sided device - independent quantum key distribution : security , feasibility , and the connection with steering , '' phys . 
   a * 85 * , 010301 ( 2012 ) .    for ellipsoids of separable states , 
   there is a further ambiguity in the ` chirality ' of alice s local basis , that is , we may determine @xmath278 up to a local unitary and a partial transpose on alice s system @xcite .
summary: 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 - qubit states 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 are also obtained , via asymmetric epr - steering inequalities . 
   our work uses ideas from the quantum steering ellipsoid formalism , and explicitly evaluates the integral of @xmath0 over arbitrary unit hemispheres for any positive matrix @xmath1 .