The question of which two-qubit states are steerable [i.e., permit a demonstration of Einstein–Podolsky–Rosen (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 over arbitrary unit hemispheres for any positive matrix .
© 2015 Optical Society of America
Quantum systems can be correlated in ways that supercede classical descriptions. However, there are degrees of nonclassicality 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 cannot be expressed as a mixture of product states of the constituents. These are called nonseparable or entangled states. The product states appearing in such a mixture make up 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 [1,2], hence demonstrating (modulo loopholes ) the violation of local causality . This phenomenon—commonly known as Bell-nonlocality —is the only way for two spatially separated parties to verify the existence of entanglement if either of them, or their detectors, cannot be trusted . 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 nonseparability and Bell-nonlocality.
In between these types of nonclassical correlations lies EPR steering. The name is inspired by the seminal paper of Einstein, Podolsky, and Rosen (EPR) , and the follow-up by Schrödinger , which coined the term “steering” for the phenomenon EPR had noticed. Although introduced 80 years ago, as this Special Issue celebrates, the notion of EPR steering was only formalized eight years ago, by one of us and coworkers [9,10]. This formalization was that EPR steering is the only way to verify the existence of entanglement if one of the parties—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 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 nonlocality for quantum states coincide for pure states: any nonproduct pure state is nonseparable, EPR-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 [9,10].
Although the EPR-steerable set has been completely determined for certain classes of highly symmetric states (at least for the case in which Alice and Bob perform projective measurements) [9,10], 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 [12 –14]. However, no relation between the steering ellipsoid and EPR steerability has been determined.
In this paper, we investigate the EPR steerability of the class of two-qubit states whose reduced states are maximally mixed, the so-called T-states . 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 [11,16] (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 organized as follows: in Section 2 we discuss in detail the three notions of nonlocality, namely Bell-nonlocality, EPR steerability, and nonseparability. 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.
2. EPR STEERING AND LOCAL HIDDEN STATE MODELS
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, labeled by and , respectively, to obtain corresponding outcomes labeled by and . The measurement correlations are described by some set of joint probability distributions, , with and 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 and .17,18].
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 Section 1, 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
Comparing Eqs. (1) and (2), 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 . Moreover, while not as powerful as Bell-nonlocality in general, steerability is more robust to detection inefficiencies , and also enables the use of untrusted devices in quantum key distribution, albeit only on one side . By a similar argument, a separable quantum state shared by Alice and Bob, , is both Bell-local and nonsteerable. Moreover, the set of separable states is strictly smaller than the set of nonsteerable states .
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 the “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 make up 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. (2), reduces to the existence of a representation of the form
Determining EPR steerability in this case, in which Alice is permitted to measure any Hermitian observable, is surprisingly difficult, with the answer only known for certain special cases such as Werner states . 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” [12,14]. Properties of this ellipsoid are therefore reviewed in the following section.
3. QUANTUM STEERING ELLIPSOID
An arbitrary two-qubit state may be written in the standard form
If Alice makes a projective measurement on her qubit, and obtains an outcome corresponding to projector , Bob’s reduced state follows from Eq. (3a) as
To determine Bob’s possible steered states, note that the projector may be expanded in the Pauli basis as , with . This yields the corresponding steered state , with associated Bloch vector
The surface of the steering ellipsoid is defined to be the set of steered Bloch vectors, , and in Ref.  it is shown that interior points can be obtained from POVMs. The ellipsoid has center14]. Thus, the general equation of the steering ellipsoid surface is with being the displacement vector from the center .
Entangled states typically have large steering ellipsoids—the largest possible being the Bloch ball, which is generated by every pure entangled state . 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 . Thus, the separability of two-qubit states has a beautiful geometric characterization in terms of the quantum steering ellipsoid.
No similar characterization has been found for EPR steerability, to date. However, for nonseparable states, knowledge of the steering ellipsoid matrix , its center , and Bob’s Bloch vector uniquely determines the shared state up to a local unitary transformation on Alice’s system [14,21] and so is sufficient, in principle, to determine the EPR steerability of . In this paper we find a direct connection between EPR steerability and the quantum steering ellipsoid, for the case in which the Bloch vectors and vanish.
4. NECESSARY CONDITION FOR EPR STEERABILITY OF T-STATES
Let be a singular value decomposition of the spin correlation matrix , for some diagonal matrix and orthogonal matrices , . Noting that any is either a rotation or the product of a rotation with the parity matrix , it follows that can always be represented in the form , for proper rotations , where the diagonal matrix may now have negative entries.
The rotations and may be implemented by local unitary operations on the shared state , 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 . It follows that if the shared state has maximally mixed reduced states with , then it is completely described, up to local unitaries, by a diagonal ; i.e., one may consider15]. They are equivalent to mixtures of Bell states, and hence form a tetrahedron in the space parameterized by . Entangled T-states necessarily have , and the set of separable T-states forms an octahedron within the tetrahedron .
The T-state steering ellipsoid is centered at the origin, , and the ellipsoid matrix is simply , as follows from Eqs. (6) and (7) with . The semiaxes are for , and are aligned with the axes of the Bloch sphere. Thus, the equation of the ellipsoid surface in spherical coordinates is , with
B. Deterministic LHS Models for T-States
Without loss of generality, consider measurement by Alice of Hermitian observables on her qubit. Such observables can be equivalently represented via projections, , with . The probability of result “” and the corresponding steered Bloch vector are given by Eqs. (4) and (5) with , i.e.,3a), then from Eqs. (3a) and (3b), it follows there is an LHS model for Bob if and only if there is a representation of the form
Given LHS models for Bob for any two T-states, having spin correlation matrices and , it is trivial to construct an LHS model for the T-state corresponding to , for any , via the convexity property of nonsteerable states . Our strategy is to find deterministic LHS models for some set of T-states, for which the result “” is fully determined by knowledge of , i.e., . 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 are precisely those vectors that generate the surface of the quantum steering ellipsoid for the T-state . We make the ansatz that is proportional to some power of the function in Eq. (9) that defines this surface, i.e.,11) becomes . We note that this is automatically satisfied if is a hemisphere, as a consequence of the symmetry for the above form of .
Hence, under the assumptions that (i) is determined by the steering ellipsoid as per Eq. (12), and (ii) is a hemisphere for each unit vector , the only remaining constraint to be satisfied by a deterministic LHS model for a T-state is Eq. (11), i.e.,
Extensive numerical testing, with different values of the exponent , shows that this constraint can be satisfied by the choices13) for (see Appendix A). An explicit form for the corresponding normalization constant is also given in Appendix A, and it is further shown that the family of T-states satisfying Eq. (15) is equivalently defined by the condition
C. Necessary EPR-Steerability Condition
Equation (15) defines a surface in the space of possible matrices, plotted in Fig. 1(a) as a function of the semiaxes , and . 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 LHS models for Bob. Also shown is the boundary of the separable T-states ( ), in red, which is clearly a strict subset of the nonsteerable T-states. The green plane corresponds to the sufficient condition for EPR-steerable states, derived in Section 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. 1(a). 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.
D. Special Cases
When we can solve Eq. (15) explicitly, because the normalization constant simplifies. The corresponding equation of the semiaxis, in terms of , is given by1(b) displays this analytic EPR-steerable curve through the T-state subspace , showing more clearly the different correlation regions.
The symmetric situation corresponds to Werner states. Our deterministic LHS model is for in this case, which is known to represent the EPR-steerable boundary for Werner states . Thus, our model is certainly optimal for this class of states.
5. SUFFICIENT CONDITIONS FOR EPR STEERABILITY
In the previous section a strong necessary condition for the EPR steerability of T-states was obtained, corresponding to the boundary defined in Eq. (15) 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 . 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 . Both EPR-steering inequalities are further of interest in that they are asymmetric under the interchange of Alice and Bob’s roles.
A. Linear Asymmetric EPR-Steering Inequality
Suppose Alice and Bob share a two-qubit state with spin covariance matrix given byB that, if there is an LHS model for Bob, then the singular values , , and of the spin covariance matrix must satisfy the linear EPR-steering inequality
From , and using and for T-states, it follows immediately that one has the simple sufficient condition1(a) and 1(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.
B. Nonlinear Asymmetric EPR Steering in Equality
Suppose Alice and Bob share a two-qubit state as before, where Bob can measure the observables and on his qubit, with , for any , and Alice can measure corresponding Hermitian observables on her qubit, with outcomes labeled by . It may then be shown that any LHS model for Bob must satisfy the EPR-steering inequality 
As per the first part of Section 4.A, we may always choose a representation in which the spin correlation matrix is diagonal, i.e., , without loss of generality. Making the choices and in this representation, and are given by and the third component of in Eqs. (4) and (5), respectively, with . Hence, the above inequality simplifies to
For T-states, recalling that , the above inequality simplifies further, to the nonlinear inequality1(b) for the case . It is seen to be stronger than the linear condition in Eq. (20) 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.
6. RECAPITULATION AND FUTURE DIRECTIONS
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 nonseparable, 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 nonsteerable 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 .
Finally, our LHV–LHS models apply to the case in which 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 . 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 generalization.
APPENDIX A: DETAILS OF THE DETERMINISTIC LHS MODEL
The family of T-states described by our deterministic LHS model in Section 4.B corresponds to the surface defined by either of Eqs. (15) and (16). This is a consequence of the following theorem, proved further below.
Theorem 1. For any full-rank diagonal matrix and nonzero vector one has
Note that substitution of Eq. (14) into constraint Eq. (13) immediately yields Eq. (15) via the theorem (with ). Further, taking the dot product of the integral in the theorem with , multiplying by , and integrating over the unit sphere yields (reversing the order of integration)15) and (16) as desired. An explicit analytic formula for the normalization constant is given at the end of this appendix.
Proof. First, define ; that is,
The surface of integration is a hemisphere bounded by the great circle . In the simple case in which , the boundary curve has the parametric form for . Hence, the boundary curve in the generic case can be constructed by applying the orthogonal operator , which rotates from to , to the vector . That is,
For the purposes of integrating over the hemisphere, it is convenient to vary from 0 to and from 0 to its value on the boundary curve. From the above expression for the boundary, and using and , it follows that and . The last equation can be rearranged to read , and after squaring both sides this equation solves to give
The surface integral for in Eq. (A3) can now be written in the form
To evaluate the third component of , note that the integral over ,
The indeterminate sign here is fixed by examining the case in which and , for which and the integrand reduces to , which integrates to give . So, unsurprisingly, we choose the positive sign. This yields the third component of the surface integral to be
The integrals over in the remaining two components of in Eq. (A8) 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 and . Since the only dependence on and comes through the function , this eliminates the need to integrate over . In fact we only need to differentiate with respect to one of these parameters; choose . To see this, note thatA6) and (A7).
Consequently, there are three separate integrals that we need to evaluate, and these can be done in Mathematica (or by complex contour integration):A10) and integrate over to obtain A11) and (A12) simplify to . Combining this with Eq. (A9) and Eqs. (A11) and (A12), we have
Finally, the normalization constant in Eq. (15) may be analytically evaluated using Mathematica. Under the assumption that , denote . We find4.D. In general, we must leave it as an implicit equation in (that is, of the ’s).
APPENDIX B: EPR-STEERING INEQUALITY FOR SPIN COVARIANCE MATRIX
To demonstrate the linear EPR-steering inequality in Eq. (19), let denote some dichotomic observable that Alice can measure on her qubit, with outcomes labeled by , where is any unit vector. We will make a specific choice of below. Define the corresponding covariance function
If there is an LHS model for Bob, then, noting that one may take in Eq. (2) to be deterministic without loss of generality, there are functions such that , where , and the hidden state has corresponding Bloch vector .
Now, the Bloch sphere can be partitioned into two sets, and , for each value of . Hence, noting , is equal to
The integral in the last line can be evaluated for each value of by rotating the coordinates such that is aligned with the axis, yielding . Hence, the above inequality can be rewritten asB2) immediately yields the EPR-steering inequality 4.A). Since is invariant under such rotations, Eq. (19) follows.
Australian Research Council (ARC) (CE110001027); Engineering and Physical Sciences Research Council (EPSRC) (EP/K022512/1); European Union Seventh Framework Programme (316244, FP7/2007-2013).
S. J. thanks David Jennings for his early contributions to this project. S. J. is funded by EPSRC grant EP/K022512/1. This work was supported by the Australian Research Council Center of Excellence CE110001027 and the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. .
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