Abstract

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 n/(nAn)2 over arbitrary unit hemispheres for any positive matrix A.

© 2015 Optical Society of America

1. INTRODUCTION

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 [3]) the violation of local causality [4]. This phenomenon—commonly known as Bell-nonlocality [5]—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 [6]. 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) [7], and the follow-up by Schrödinger [8], 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 [911]—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 [1214]. 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 [15]. 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 A and B, respectively, to obtain corresponding outcomes labeled by a and b. The measurement correlations are described by some 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 classified via the properties of these joint distributions, for all possible measurement settings A and B.

The correlations of a Bell-local state have a LHV model [1,2],

p(a,b|A,B)=λP(λ)p(a|A,λ)p(b|B,λ),
for some “hidden” random variable λ with probability distribution P(λ). Hence, the measured correlations may be understood as arising from ignorance of the value of λ, where the latter locally determines the statistics of the outcomes 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 Alice and Bob without trust in their devices [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

p(a,b|A,B)=λP(λ)p(a|A,λ)pQ(b|B,λ),
where the local distributions pQ(b|B,λ) correspond to measurements on local quantum states ρB(λ), i.e.,
pQ(b|B,λ)=tr[ρB(λ)FbB].
Here, {FbB} denotes the positive operator valued measure (POVM) corresponding to measurement B. 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. (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 [9]. Moreover, 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 quantum key distribution, albeit only on one side [20]. By a similar argument, a separable quantum state shared by Alice and Bob, ρ=Σλp(λ)ρA(λ)ρB(λ), is both Bell-local and nonsteerable. Moreover, the set of separable states is strictly smaller than the set of nonsteerable states [9].

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

pEρBE:=trA[ρE1]=λP(λ)p(1|E,λ)ρB(λ),
pE=tr[ρEI]=λP(λ)p(1|E,λ).
Here E is any projector that can be measured by Alice, pE is the probability of result “E=1” and p(1|E,λ) is the corresponding probability given λ, ρBE is the reduced state of Bob’s component corresponding to this result, and trA[·] 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, 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 [9]. 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

ρ=14(11+a·σ1+1b·σ+j,kTjkσjσk).
Here (σ1,σ2,σ3)σ denote the Pauli spin operators, and
aj=tr[ρσj1],bj=tr[ρ1σj],Tjk=tr[ρσjσk].
Thus, a and b are the Bloch vectors for Alice and Bob’s qubits, 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

ρBE=trA[ρE1]tr[ρE1].
We will also refer to ρBE as Bob’s “steered state.”

To determine Bob’s possible steered states, note that the projector E may be expanded in the Pauli basis as E=12(1+e·σ), with |e|=1. This yields the corresponding steered state ρBE=12(1+b(e)·σ), with associated Bloch vector

b(e)=12pe(b+Te),
where pe is the associated probability of result “E=1”,
petr[ρ(E1)]=12(1+a·e),
called pE previously. In what follows we will refer to the vector e rather than its corresponding operator E.

The surface of the steering ellipsoid is defined to be the set of steered Bloch vectors, {b(e):|e|=1}, and in Ref. [14] it is shown that interior points can be obtained from POVMs. The ellipsoid has center

c=bTa1a2,
and the semiaxes s1,s2,s3 are the roots of the eigenvalues of the matrix
Q=11a2(Tba)(1+aa1a2)(Tab).
The eigenvectors of Q give the orientation of the ellipsoid around its center [14]. Thus, the general equation of the steering ellipsoid surface is xQ1x=1 with xR3 being the displacement vector from the center c.

Entangled states typically have large steering ellipsoids—the largest possible being the Bloch ball, which is generated by every pure entangled state [14]. 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 [14]. 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 Q, its center c, and Bob’s Bloch vector b uniquely determines the shared state ρ up to a local unitary transformation on Alice’s system [14,21] and so is sufficient, in principle, to determine the EPR steerability of ρ. In this paper we find a direct connection between EPR steerability and the quantum steering ellipsoid, for the case in which the Bloch vectors a and b vanish.

4. NECESSARY CONDITION FOR EPR STEERABILITY OF T-STATES

A. T-States

Let T=OAD˜OB be a singular value decomposition of the spin correlation matrix T, for some diagonal matrix D˜0 and orthogonal matrices OA, OBO(3). Noting that any OO(3) is either a rotation or the product of a rotation with the parity matrix I, it follows that T can always be represented in the form T=RADRB, for proper rotations RA,RBSO(3), where the diagonal matrix D may now have negative entries.

The rotations RA and RB may be implemented by local unitary operations on the shared state ρ, 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 TD=diag[t1,t2,t3]. It follows that if the shared state ρ has maximally mixed reduced states with a=b=0, then it is completely described, up to local unitaries, by a diagonal T; i.e., one may consider

ρ=14(11+jtjσjσj)
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 parameterized by (t1,t2,t3) [15]. Entangled T-states necessarily have t1t2t3<0, and the set of separable T-states forms an octahedron within the tetrahedron [15].

The T-state steering ellipsoid is centered at the origin, c=0, and the ellipsoid matrix is simply Q=TT, as follows from Eqs. (6) and (7) with a=b=0. The semiaxes are si=|ti| for i=1,2,3, and are aligned with the x,y,z axes of the Bloch sphere. Thus, the equation of the ellipsoid surface in spherical coordinates (r,θ,ϕ) is r=1/f(θ,ϕ), with

f(θ,ϕ)2sin2θcos2ϕs12+sin2θsin2ϕs22+cos2θs32.
We find a remarkable connection between this equation and the EPR steerability of T-states in the following subsection.

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, E=12(1+e·σ), with |e|=1. The probability of result “E=1” and the corresponding steered Bloch vector are given by Eqs. (4) and (5) with a=b=0, i.e.,

pe=1/2,b(e)=Te=Te.
Hence, letting n(λ) denote the Bloch vector corresponding to ρB(λ) in Eq. (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
λP(λ)p(1|e,λ)=12,λP(λ)p(1|e,λ)n(λ)=12Te,
for all unit vectors e. Noting further that n(λ) can always be represented as some mixture of unit vectors, corresponding to pure ρB(λ), these conditions are equivalent to the existence of a representation of the form
P(n)p(1|e,n)d2n=12,
P(n)p(1|e,n)nd2n=12Te,
with integration over the Bloch sphere. Thus, the unit Bloch vector n labels both the LHS and the hidden variable.

Given LHS models for Bob for any two T-states, having spin correlation matrices T0 and T1, it is trivial to construct an LHS model for the T-state corresponding to Tq=(1q)T0+qT1, for any 0q1, via the convexity property of nonsteerable states [11]. Our strategy is to find deterministic LHS models for some set of T-states, for which the result “E=1” is fully determined by knowledge of n, i.e., p(1|e,n){0,1}. 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 b(e)=Te are precisely those vectors that generate the surface of the quantum steering ellipsoid for the T-state [14]. We make the 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)=NT[f(θ,ϕ)]mNT[nT2n]m/2
for n=(sinθcosϕ,sinθsinϕ,cosθ), where NT is a normalization constant. Further, denoting the region of the Bloch sphere, for which p(1|e,n)=1 by R[e], the condition in Eq. (11) becomes R[e]P(n)d2n=12. We note that this is automatically satisfied if R(e) is a hemisphere, as a consequence of the symmetry P(n)=P(n) for the above form of P(n).

Hence, under the assumptions that (i) P(n) is determined by the steering ellipsoid as per Eq. (12), and (ii) R[e] is a hemisphere for each unit vector e, the only remaining constraint to be satisfied by a deterministic LHS model for a T-state is Eq. (11), i.e.,

NTR[e][nT2n]m/2nd2n=12Te,
for some suitable mapping eR[e].

Extensive numerical testing, with different values of the exponent m, shows that this constraint can be satisfied by the choices

m=4,R[e]={n:nT1e0},
for a two-parameter family of T-states. Assuming the numerical results are correct, it is not difficult to show, using infinitesimal rotations of e about the z axis, that this family corresponds to those T-states that satisfy
2πNT|detT|=1.
Fortunately, we have been able to confirm these results analytically by explicitly evaluating the integral in Eq. (13) for m=4 (see Appendix A). An explicit form for the corresponding normalization constant NT 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
nT2nd2n=2π.
This may be interpreted geometrically in terms of the harmonic mean radius of the “inverse” ellipsoid xT2x=1 being equal to 2.

C. Necessary EPR-Steerability Condition

Equation (15) defines a surface in the space of possible T matrices, plotted in Fig. 1(a) as a function of the semiaxes s1,s2, and s3. 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 (s1+s2+s31 [15]), in red, which is clearly a strict subset of the nonsteerable T-states. The green plane corresponds to the sufficient condition s1+s2+s3>32 for EPR-steerable states, derived in Section 5 below.

 figure: Fig. 1.

Fig. 1. Correlation bounds for T-states, with si=|ti|. (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 Section 4.B: 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. (20) of Section 5.A: 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 s1,s2,s3. (b) Cross section through the top figure at s1=s2, where the necessary condition can be determined analytically (see Section 4.D). The additional black dashed curve corresponds to the nonlinear sufficient condition for EPR steerability in Eq. (22).

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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 |t1|=|t2| we can solve Eq. (15) explicitly, because the normalization constant NT simplifies. The corresponding equation of the s3 semiaxis, in terms of us3/s1=s3/s2, is given by

s3={[1+arctan(u21)u2u21]1u<1,[11u22(u21)ln|11u2|1+1u2]1u>1,
and s3=12 for u=1. Figure 1(b) displays this analytic EPR-steerable curve through the T-state subspace |t1|=|t2|s1=s2, showing more clearly the different correlation regions.

The symmetric situation s1=s2=s3 corresponds to Werner states. Our deterministic LHS model is for s1=s2=s3=1/2 in this case, which is known to represent the EPR-steerable boundary for Werner states [10]. 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 [11]. 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 [16]. 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 C given by

Cjkσjσkσj11σk=Tjkajbk,
and that each can measure any Hermitian observable on their qubit. We show in Appendix B that, if there is an LHS model for Bob, then the singular values c1, c2, and c3 of the spin covariance matrix must satisfy the linear EPR-steering inequality
c1+c2+c3321b2.

From C=Tab, and using a=b=0 and sj=|tj| for T-states, it follows immediately that one has the simple sufficient condition

s1+s2+s3>32
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 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 1σ3 and 1σϕ on his qubit, with σϕσ1cosϕ+σ2sinϕ, for any ϕ[0,2π], and Alice can measure corresponding Hermitian observables A31,Aϕ1 on her qubit, with outcomes labeled by ±1. It may then be shown that any LHS model for Bob must satisfy the EPR-steering inequality [16]

1ππ/2π/2Aϕσϕdϕ2π[p+11σ3+2+p11σ32],
where p± denotes the probability that Alice obtains result A3=±1, and 1σ3± is Bob’s corresponding conditional expectation value for 1σ3 for this result.

As per the first part of Section 4.A, we may always choose a representation in which the spin correlation matrix T is diagonal, i.e., T=diag[t1,t2,t3], without loss of generality. Making the choices A3=σ3 and Aϕ=σ1(signt1)cosϕ+σ2(signt2)sinϕ in this representation, p± and 1σ3± are given by pe and the third component of b(e) in Eqs. (4) and (5), respectively, with e=(0,0,±1). Hence, the above inequality simplifies to

|t1|+|t2|2π[(1+a3)2(t3+b3)2+(1a3)2(t3b3)2],
where a3 and b3 are the third components of Alice and Bob’s Bloch vectors a and b.

For T-states, recalling that si|ti|, the above inequality simplifies further, to the nonlinear inequality

f(s1,s2,s3)s1+s24π1s320.
Hence, since similar inequalities can be obtained by permuting s1,s2,s3, we have the sufficient condition
max{f(s1,s2,s3),f(s2,s3,s1),f(s3,s1,s2)}>0
for the EPR steerability of T-states. The boundary of T-states satisfying this condition is plotted in Fig. 1(b) for the case s1=s2. 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 [22].

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 [23]. 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 T and nonzero vector v one has

n·v0nd2n(nT2n)2=π|detT|T2v|Tv|.

Note that substitution of Eq. (14) into constraint Eq. (13) immediately yields Eq. (15) via the theorem (with v=T1e). Further, taking the dot product of the integral in the theorem with v, multiplying by NT, and integrating v over the unit sphere yields (reversing the order of integration)

d2nP(n)n·v0d2vv·n=π,
whereas carrying out the same operations on the right-hand side of the theorem yields πNT|detT|vT2vd2v. Equating these immediately implies the equivalence of Eqs. (15) and (16) as desired. An explicit analytic formula for the normalization constant NT is given at the end of this appendix.

Proof. First, define Q=T2GL(3,R); that is,

Q=diag(a,b,c)=(t12,t22,t32),
q(v)n·v0nd2n(nQn)2.
Noting that v in the theorem may be taken to be a unit vector without loss of generality, we will parameterize the unit vectors n and v by
n=(sinθcosϕ,sinθsinϕ,cosθ),
v=(sinαcosβ,sinαsinβ,cosα),
with θ,α[0,π] and ϕ,β[0,2π). Thus, d2nsinθdθdϕ. Further, without loss of generality, it will be assumed that v points into the northern hemisphere, so that cosα0. Then α0,π/2] and β0,2π).

The surface of integration is a hemisphere bounded by the great circle n·v=0. In the simple case in which v=(0,0,1)T, the boundary curve has the parametric form (x,y,z)=(cosγ,sinγ,0) for γ(0,2π). Hence, the boundary curve in the generic case can be constructed by applying the orthogonal operator R, which rotates v from (0,0,1)T to (sinαcosβ,sinαsinβ,cosα), to the vector (cosγ,sinγ,0)T. That is,

R=(cosβsinβ0sinβcosβ0001)(cosα0sinα010sinα0cosα)=(cosαcosβsinβsinαcosβcosαsinβcosβsinαsinβsinα0cosα),
and the boundary curve has the form
(xyz)=R(cosγsinγ0)=(cosαcosβcosγsinβsinγcosαsinβcosγ+cosβsinγsinαcosγ).

For the purposes of integrating over the hemisphere, it is convenient to vary ϕ from 0 to 2π and θ from 0 to its value χ(ϕ) on the boundary curve. From the above expression for the boundary, and using z=cosθ and y/x=tanϕ, it follows that cosχ=sinαcosγ and (cosαsinβcosγ+cosβsinγ)cosϕ=(cosαcosβcosγsinβsinγ)sinϕ. The last equation can be rearranged to read cosαsin(ϕβ)cosγ=cos(ϕβ)sinγ, and after squaring both sides this equation solves to give

cosγ=±cos(ϕβ)[cos2(ϕβ)+cos2αsin2(ϕβ)]1/2.
Now, χ assumes its maximum value when ϕ=β, which according to the relation cosχ=sinαcosγ and the fact that α[0,π/2] should correspond to γ=0. So we take the upper sign in the last equation, yielding
cosχ=sinαcos(ϕβ)[cos2(ϕβ)+cos2αsin2(ϕβ)]1/2=sinαcos(ϕβ)[cos2α+sin2αcos2(ϕβ)]1/2.
It follows immediately that
sinχ=cosα[cos2α+sin2αcos2(ϕβ)]1/2,
with the choice of sign fixed by the fact that sinχ0 and (by assumption) cosα0.

The surface integral for q(v) in Eq. (A3) can now be written in the form

02π0χ(ϕ)(sinθcosϕ,sinθsinϕ,cosθ)Tsinθdθdϕ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2.

To evaluate the third component of q(v), note that the integral over θ,

0χ(ϕ)sinθcosθdθ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2,
can be evaluated explicitly by making the substitution w=sin2θ, as (A+Bw)2dw=B1(A+Bw)1 for any B0, yielding
12csin2χasin2χcos2ϕ+bsin2χsin2ϕ+ccos2χ.
After inserting the expressions for cosχ and sinχ derived earlier, we have
0χ(ϕ)sinθcosθ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2dθ=12ccos2αacos2αcos2ϕ+bcos2αsin2ϕ+csin2αcos2(ϕβ).
We now need to integrate the last expression over ϕ. Introducing new constants
l=acos2α+csin2αcos2β,m=bcos2α+csin2αsin2β,n=csin2αsinβcosβ,
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):
02π0χ(ϕ)sinθcosθdθdϕ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2=cos2α2c02πdϕlcos2ϕ+msin2ϕ+2nsinϕcosϕ=±cos2α2c2πlmn2.

The indeterminate sign here is fixed by examining the case in which α=0 and a=b=c, for which χ(ϕ)=π/2 and the integrand reduces to a2sinθcosθ, which integrates to give πa2. So, unsurprisingly, we choose the positive sign. This yields the third component of the surface integral to be

[q(v)]3=πcosαc[abcos2α+c(asin2β+bcos2β)sin2α]1/2.

The integrals over θ in the remaining two components of q(v) 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 α and β. Since the only dependence on α and β comes through the function χ(ϕ), this eliminates the need to integrate over θ. In fact we only need to differentiate with respect to one of these parameters; choose α. To see this, note that

α02π0χ(ϕ)(cosϕ,sinϕ)sin2θdθdϕ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2=02π(cosϕ,sinϕ)sin2χ(asin2χcos2ϕ+bsin2χsin2ϕ+ccos2χ)2χαdϕ,
where χ/α can be evaluated by making use of the Eqs. (A6) and (A7).

In fact,

sinχχα=α(sinαcos(ϕβ)[cos2α+sin2αcos2(ϕβ)]1/2)=cosαcos(ϕβ)[cos2α+sin2αcos2(ϕβ)]3/2.
Inserting the last two equations and the expressions for sinχ and cosχ into the integrals above, and using the constants l, m, and n defined earlier, then gives
α02π0χ(ϕ)(cosϕ,sinϕ)sin2θdθdϕ(asin2θcos2ϕ+bsin2θsin2ϕ+ccos2θ)2=cos2α02π(cosϕ,sinϕ)cos(ϕβ)[acos2ϕcos2α+bsin2ϕcos2α+csin2αcos2(ϕβ)]2dϕ=cos2α02π(sinβsinϕcosϕ+cosβcos2ϕ,sinβsin2ϕ+cosβsinϕcosϕ)(lcos2ϕ+msin2ϕ+2nsinϕcosϕ)2dϕ.

Consequently, there are three separate integrals that we need to evaluate, and these can be done in Mathematica (or by complex contour integration):

02π(sin2ϕ,cos2ϕ,sinϕcosϕ)dϕ(lcos2ϕ+msin2ϕ+2nsinϕcosϕ)2=π(l,m,n)(lmn2)3/2.
Using the values we have for l,m,n we substitute these back into Eq. (A10) and integrate over α to obtain
[q(v)]1=πcos2α(mcosβnsinβ)(lmn2)3/2dα=a1πsinαcosβ[abcos2α+csin2α(bcos2β+asin2β)]1/2,
[q(v)]2=πcos2α(lsinβncosβ)(lmn2)3/2dα=b1πsinαsinβ[abcos2α+csin2α(bcos2β+asin2β)]1/2.
The absence of integration constants can be confirmed by noting that these expressions vanish for α=0—i.e., when the vector v is aligned with the z axis—as they should by symmetrical. Note the denominators of Eqs. (A11) and (A12) simplify to abc(vQ1v). Combining this with Eq. (A9) and Eqs. (A11) and (A12), we have
q(v)=πQ1vabc(vQ1v),
and so setting Q=T2, the theorem follows as desired.

Finally, the normalization constant NT in Eq. (15) may be analytically evaluated using Mathematica. Under the assumption that |t3|>|t2|>|t1|, denote a=|t1|,b=|t2|,c=|t3|. We find

NT1=n·n=1(nT2n)2d2n=2πabc(a+b)(b+c)(c2a2)×(X+Y{b(ca)E[C]+a(b+c)K[C]+ib(ca)(E[A1,B]E[A2,B])+ic(a+b)(F[A1,B]F[A2,B])}),
where F[·,·],E[·,·] are the elliptic integrals of the first and second kind, E[·] is the complete elliptic integral and K[·] is the complete elliptic integral of the first kind, and
A1=iarccsch(ac2a2),A2=iln(b+cc2b2),B=a2(c2b2)b2(c2a2),C=c2(b2a2)b2(c2a2),X=c(ca)[(a+c)(b+c)+ab],Y=(a+b+c)c2a2.
Thus, the normalization constant NT has a rather nontrivial form. It is highly unlikely that we can invert it to express the EPR-steerability condition 2πNT|detT|=1 as c=g(a,b), where g is some function of a, b, other than in the special cases considered in Section 4.D. In general, we must leave it as an implicit equation in a,b,c (that is, of the tj’s).

APPENDIX B: EPR-STEERING INEQUALITY FOR SPIN COVARIANCE MATRIX

To demonstrate the linear EPR-steering inequality in Eq. (19), let Av denote some dichotomic observable that Alice can measure on her qubit, with outcomes labeled by ±1, where v is any unit vector. We will make a specific choice of Av below. Define the corresponding covariance function

C(v)Avv·σAvv·σ.

If there is an LHS model for Bob, then, noting that one may take p(a|x,λ) in Eq. (2) to be deterministic without loss of generality, there are functions αv(λ)=±1 such that C(v)=Σλp(λ)[αv(λ)α¯v][n(λ)b]·v, where α¯v=Σλp(λ)αv(λ), and the hidden state ρB(λ) has corresponding Bloch vector n(λ).

Now, the Bloch sphere can be partitioned into two sets, S+(λ)={v:[n(λ)b]·v0} and S(λ)={v:[n(λ)b]·v<0}, for each value of λ. Hence, noting 1α¯vαv(λ)α¯v1α¯v, C(v)d2v is equal to

λp(λ){S+(λ)d2v[αv(λ)α¯v][n(λ)b]·v+S(λ)d2v[αv(λ)α¯v][n(λ)b]·v}λp(λ){S+(λ)d2v[1α¯v][n(λ)b]·vS(λ)d2v[1+α¯v][n(λ)b]·v}=λp(λ)d2v|[n(λ)b]·v|λp(λ)d2vα¯v[n(λ)b]·v=λp(λ)|n(λ)b|d2v|v·w(λ)|,
where w(λ) denotes the unit vector in the n(λ)b 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 λ by rotating the coordinates such that w(λ) is aligned with the z axis, yielding d2v|v·w(λ)|=d2v|v3|=02πdϕ0πdθsinθ|cosθ|=2π. Hence, the above inequality can be rewritten as

14πd2vC(v)12λp(λ)|n(λ)b|12[λp(λ)|n(λ)b|2]1/2121b·b,
where the second and third lines follow using the Schwarz inequality and |n(λ)|1, respectively. Note, by the way, that the first inequality is tight for the case in which αv(λ)=sign([n(λ)b]·v).

Now, making the choice Av=u·σ with ujsign(Cjj)vj, one has from Eqs. (18) and (B1) that

d2vC(v)=j,kCjksign(Cjj)d2vvjvk=j,kCjksign(Cjj)4π3δjk=4π3j|Cjj|.
Combining with Eq. (B2) immediately yields the EPR-steering inequality
j|Cjj|321b·b.
Finally, this inequality may similarly be derived in a representation in which local rotations put the spin covariance matrix C in diagonal form, with coefficients given up to a sign by the singular values of C (similarly to the spin correlation matrix T in Section 4.A). Since b·b=b2 is invariant under such rotations, Eq. (19) follows.

FUNDING INFORMATION

Australian Research Council (ARC) (CE110001027); Engineering and Physical Sciences Research Council (EPSRC) (EP/K022512/1); European Union Seventh Framework Programme (316244, FP7/2007-2013).

ACKNOWLEDGMENTS

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. [316244].

REFERENCES AND NOTES

1. J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

2. R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

3. J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014). [CrossRef]  

4. J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

5. H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014). [CrossRef]  

6. B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000). [CrossRef]  

7. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

8. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935). [CrossRef]  

9. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007). [CrossRef]  

10. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007). [CrossRef]  

11. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009). [CrossRef]  

12. F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).

13. M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011). [CrossRef]  

14. S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014). [CrossRef]  

15. R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996). [CrossRef]  

16. S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011). [CrossRef]  

17. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006). [CrossRef]  

18. J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005). [CrossRef]  

19. 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. Rev. X 2, 031003 (2012).

20. 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. Rev. A 85, 010301 (2012). [CrossRef]  

21. For ellipsoids of separable states, there is a further ambiguity in the “chirality” of Alice’s local basis; that is, we may determine ρ up to a local unitary and a partial transpose on Alice’s system [14].

22. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014). [CrossRef]  

23. R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014). [CrossRef]  

24. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

References

  • View by:

  1. J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].
  2. R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).
  3. J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
    [Crossref]
  4. J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].
  5. H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
    [Crossref]
  6. B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
    [Crossref]
  7. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
  8. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
    [Crossref]
  9. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
    [Crossref]
  10. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
    [Crossref]
  11. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
    [Crossref]
  12. F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).
  13. M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
    [Crossref]
  14. S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
    [Crossref]
  15. R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
    [Crossref]
  16. S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011).
    [Crossref]
  17. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
    [Crossref]
  18. J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
    [Crossref]
  19. 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. Rev. X 2, 031003 (2012).
  20. 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. Rev. A 85, 010301 (2012).
    [Crossref]
  21. For ellipsoids of separable states, there is a further ambiguity in the “chirality” of Alice’s local basis; that is, we may determine ρ up to a local unitary and a partial transpose on Alice’s system [14].
  22. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
    [Crossref]
  23. R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
    [Crossref]
  24. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

2014 (5)

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

2012 (2)

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. Rev. 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. Rev. A 85, 010301 (2012).
[Crossref]

2011 (2)

S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011).
[Crossref]

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

2009 (1)

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

2007 (2)

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

2006 (1)

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

2005 (1)

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

2001 (1)

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

2000 (1)

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

1996 (1)

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

1976 (1)

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

1964 (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Acín, A.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Barrett, J.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Bell, J. S.

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

Bennet, A. J.

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. Rev. X 2, 031003 (2012).

Bowles, J.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Branciard, C.

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. Rev. 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. Rev. A 85, 010301 (2012).
[Crossref]

Brunner, N.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Cavalcanti, E. G.

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. Rev. A 85, 010301 (2012).
[Crossref]

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. Rev. X 2, 031003 (2012).

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Doherty, A. C.

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Du, J.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Evans, D. A.

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. Rev. X 2, 031003 (2012).

Gisin, N.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Hardy, L.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Horodecki, M.

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

Horodecki, R.

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

Jennings, D.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Jevtic, S.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Jiang, F.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Jones, S. J.

S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

Kent, A.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Larsson, J.-Å.

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

Masanes, L.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Pryde, G. J.

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. Rev. X 2, 031003 (2012).

Pusey, M.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Quintino, M. T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Reid, M. D.

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Rudolph, T.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Saunders, D. J.

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. Rev. X 2, 031003 (2012).

Scarani, V.

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. Rev. A 85, 010301 (2012).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Shi, M.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Sun, C.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Terhal, B. M.

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

Verstraete, F.

F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).

Vértesi, T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Walborn, S. P.

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. Rev. A 85, 010301 (2012).
[Crossref]

Werner, R. F.

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Wiseman, H. M.

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

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. Rev. A 85, 010301 (2012).
[Crossref]

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. Rev. X 2, 031003 (2012).

S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Wolf, M. M.

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Epistemological Lett. (1)

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

J. Phys. A (3)

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

New. J. Phys. (1)

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Phys. Lett. A (1)

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Phys. Rev. A (5)

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: paths to an Einstein–Podolsky–Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011).
[Crossref]

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. Rev. A 85, 010301 (2012).
[Crossref]

Phys. Rev. Lett. (5)

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Phys. Rev. X (1)

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. Rev. X 2, 031003 (2012).

Physics (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

Proc. Cambridge Philos. Soc. (1)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Quantum Inform. Comput. (1)

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Other (3)

F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).

For ellipsoids of separable states, there is a further ambiguity in the “chirality” of Alice’s local basis; that is, we may determine ρ up to a local unitary and a partial transpose on Alice’s system [14].

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

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Figures (1)

Fig. 1.
Fig. 1. Correlation bounds for T-states, with s i = | t i | . (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 Section 4.B: 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. (20) of Section 5.A: 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 s 1 , s 2 , s 3 . (b) Cross section through the top figure at s 1 = s 2 , where the necessary condition can be determined analytically (see Section 4.D). The additional black dashed curve corresponds to the nonlinear sufficient condition for EPR steerability in Eq. (22).

Equations (63)

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p ( a , b | A , B ) = λ P ( λ ) p ( a | A , λ ) p ( b | B , λ ) ,
p ( a , b | A , B ) = λ P ( λ ) p ( a | A , λ ) p Q ( b | B , λ ) ,
p Q ( b | B , λ ) = tr [ ρ B ( λ ) F b B ] .
p E ρ B E := tr A [ ρ E 1 ] = λ P ( λ ) p ( 1 | E , λ ) ρ B ( λ ) ,
p E = tr [ ρ E I ] = λ P ( λ ) p ( 1 | E , λ ) .
ρ = 1 4 ( 1 1 + a · σ 1 + 1 b · σ + j , k T j k σ j σ k ) .
a j = tr [ ρ σ j 1 ] , b j = tr [ ρ 1 σ j ] , T j k = tr [ ρ σ j σ k ] .
ρ B E = tr A [ ρ E 1 ] tr [ ρ E 1 ] .
b ( e ) = 1 2 p e ( b + T e ) ,
p e tr [ ρ ( E 1 ) ] = 1 2 ( 1 + a · e ) ,
c = b T a 1 a 2 ,
Q = 1 1 a 2 ( T b a ) ( 1 + a a 1 a 2 ) ( T a b ) .
ρ = 1 4 ( 1 1 + j t j σ j σ j )
f ( θ , ϕ ) 2 sin 2 θ cos 2 ϕ s 1 2 + sin 2 θ sin 2 ϕ s 2 2 + cos 2 θ s 3 2 .
p e = 1 / 2 , b ( e ) = T e = T e .
λ P ( λ ) p ( 1 | e , λ ) = 1 2 , λ P ( λ ) p ( 1 | e , λ ) n ( λ ) = 1 2 T e ,
P ( n ) p ( 1 | e , n ) d 2 n = 1 2 ,
P ( n ) p ( 1 | e , n ) n d 2 n = 1 2 T e ,
P ( n ) = N T [ f ( θ , ϕ ) ] m N T [ n T 2 n ] m / 2
N T R [ e ] [ n T 2 n ] m / 2 n d 2 n = 1 2 T e ,
m = 4 , R [ e ] = { n : n T 1 e 0 } ,
2 π N T | det T | = 1 .
n T 2 n d 2 n = 2 π .
s 3 = { [ 1 + arctan ( u 2 1 ) u 2 u 2 1 ] 1 u < 1 , [ 1 1 u 2 2 ( u 2 1 ) ln | 1 1 u 2 | 1 + 1 u 2 ] 1 u > 1 ,
C j k σ j σ k σ j 1 1 σ k = T j k a j b k ,
c 1 + c 2 + c 3 3 2 1 b 2 .
s 1 + s 2 + s 3 > 3 2
1 π π / 2 π / 2 A ϕ σ ϕ d ϕ 2 π [ p + 1 1 σ 3 + 2 + p 1 1 σ 3 2 ] ,
| t 1 | + | t 2 | 2 π [ ( 1 + a 3 ) 2 ( t 3 + b 3 ) 2 + ( 1 a 3 ) 2 ( t 3 b 3 ) 2 ] ,
f ( s 1 , s 2 , s 3 ) s 1 + s 2 4 π 1 s 3 2 0 .
max { f ( s 1 , s 2 , s 3 ) , f ( s 2 , s 3 , s 1 ) , f ( s 3 , s 1 , s 2 ) } > 0
n · v 0 n d 2 n ( n T 2 n ) 2 = π | det T | T 2 v | T v | .
d 2 n P ( n ) n · v 0 d 2 v v · n = π ,
Q = diag ( a , b , c ) = ( t 1 2 , t 2 2 , t 3 2 ) ,
q ( v ) n · v 0 n d 2 n ( n Q n ) 2 .
n = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) ,
v = ( sin α cos β , sin α sin β , cos α ) ,
R = ( cos β sin β 0 sin β cos β 0 0 0 1 ) ( cos α 0 sin α 0 1 0 sin α 0 cos α ) = ( cos α cos β sin β sin α cos β cos α sin β cos β sin α sin β sin α 0 cos α ) ,
( x y z ) = R ( cos γ sin γ 0 ) = ( cos α cos β cos γ sin β sin γ cos α sin β cos γ + cos β sin γ sin α cos γ ) .
cos γ = ± cos ( ϕ β ) [ cos 2 ( ϕ β ) + cos 2 α sin 2 ( ϕ β ) ] 1 / 2 .
cos χ = sin α cos ( ϕ β ) [ cos 2 ( ϕ β ) + cos 2 α sin 2 ( ϕ β ) ] 1 / 2 = sin α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 .
sin χ = cos α [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 ,
0 2 π 0 χ ( ϕ ) ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) T sin θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 .
0 χ ( ϕ ) sin θ cos θ d θ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 ,
1 2 c sin 2 χ a sin 2 χ cos 2 ϕ + b sin 2 χ sin 2 ϕ + c cos 2 χ .
0 χ ( ϕ ) sin θ cos θ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 d θ = 1 2 c cos 2 α a cos 2 α cos 2 ϕ + b cos 2 α sin 2 ϕ + c sin 2 α cos 2 ( ϕ β ) .
l = a cos 2 α + c sin 2 α cos 2 β , m = b cos 2 α + c sin 2 α sin 2 β , n = c sin 2 α sin β cos β ,
0 2 π 0 χ ( ϕ ) sin θ cos θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = cos 2 α 2 c 0 2 π d ϕ l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ = ± cos 2 α 2 c 2 π l m n 2 .
[ q ( v ) ] 3 = π cos α c [ a b cos 2 α + c ( a sin 2 β + b cos 2 β ) sin 2 α ] 1 / 2 .
α 0 2 π 0 χ ( ϕ ) ( cos ϕ , sin ϕ ) sin 2 θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = 0 2 π ( cos ϕ , sin ϕ ) sin 2 χ ( a sin 2 χ cos 2 ϕ + b sin 2 χ sin 2 ϕ + c cos 2 χ ) 2 χ α d ϕ ,
sin χ χ α = α ( sin α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 ) = cos α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 3 / 2 .
α 0 2 π 0 χ ( ϕ ) ( cos ϕ , sin ϕ ) sin 2 θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = cos 2 α 0 2 π ( cos ϕ , sin ϕ ) cos ( ϕ β ) [ a cos 2 ϕ cos 2 α + b sin 2 ϕ cos 2 α + c sin 2 α cos 2 ( ϕ β ) ] 2 d ϕ = cos 2 α 0 2 π ( sin β sin ϕ cos ϕ + cos β cos 2 ϕ , sin β sin 2 ϕ + cos β sin ϕ cos ϕ ) ( l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ ) 2 d ϕ .
0 2 π ( sin 2 ϕ , cos 2 ϕ , sin ϕ cos ϕ ) d ϕ ( l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ ) 2 = π ( l , m , n ) ( l m n 2 ) 3 / 2 .
[ q ( v ) ] 1 = π cos 2 α ( m cos β n sin β ) ( l m n 2 ) 3 / 2 d α = a 1 π sin α cos β [ a b cos 2 α + c sin 2 α ( b cos 2 β + a sin 2 β ) ] 1 / 2 ,
[ q ( v ) ] 2 = π cos 2 α ( l sin β n cos β ) ( l m n 2 ) 3 / 2 d α = b 1 π sin α sin β [ a b cos 2 α + c sin 2 α ( b cos 2 β + a sin 2 β ) ] 1 / 2 .
q ( v ) = π Q 1 v a b c ( v Q 1 v ) ,
N T 1 = n · n = 1 ( n T 2 n ) 2 d 2 n = 2 π a b c ( a + b ) ( b + c ) ( c 2 a 2 ) × ( X + Y { b ( c a ) E [ C ] + a ( b + c ) K [ C ] + i b ( c a ) ( E [ A 1 , B ] E [ A 2 , B ] ) + i c ( a + b ) ( F [ A 1 , B ] F [ A 2 , B ] ) } ) ,
A 1 = i arccsch ( a c 2 a 2 ) , A 2 = i ln ( b + c c 2 b 2 ) , B = a 2 ( c 2 b 2 ) b 2 ( c 2 a 2 ) , C = c 2 ( b 2 a 2 ) b 2 ( c 2 a 2 ) , X = c ( c a ) [ ( a + c ) ( b + c ) + a b ] , Y = ( a + b + c ) c 2 a 2 .
C ( v ) A v v · σ A v v · σ .
λ p ( λ ) { S + ( λ ) d 2 v [ α v ( λ ) α ¯ v ] [ n ( λ ) b ] · v + S ( λ ) d 2 v [ α v ( λ ) α ¯ v ] [ n ( λ ) b ] · v } λ p ( λ ) { S + ( λ ) d 2 v [ 1 α ¯ v ] [ n ( λ ) b ] · v S ( λ ) d 2 v [ 1 + α ¯ v ] [ n ( λ ) b ] · v } = λ p ( λ ) d 2 v | [ n ( λ ) b ] · v | λ p ( λ ) d 2 v α ¯ v [ n ( λ ) b ] · v = λ p ( λ ) | n ( λ ) b | d 2 v | v · w ( λ ) | ,
1 4 π d 2 v C ( v ) 1 2 λ p ( λ ) | n ( λ ) b | 1 2 [ λ p ( λ ) | n ( λ ) b | 2 ] 1 / 2 1 2 1 b · b ,
d 2 v C ( v ) = j , k C j k sign ( C j j ) d 2 v v j v k = j , k C j k sign ( C j j ) 4 π 3 δ j k = 4 π 3 j | C j j | .
j | C j j | 3 2 1 b · b .

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