If a measurement is made on one half of a bipartite system, then, conditioned on the outcome, the other half has a new reduced state. If these reduced states defy classical explanation—that is, if shared randomness cannot produce these reduced states for all possible measurements—the bipartite state is said to be steerable. Determining which states are steerable is a challenging problem even for low dimensions. In the case of two-qubit systems, a criterion is known for T-states (that is, those with maximally mixed marginals) under projective measurements. In the current work, we introduce the concept of keyring models—a special class of local hidden state models. When the measurements made correspond to real projectors, these allow us to study steerability beyond T-states. Using keyring models, we completely solve the steering problem for real projective measurements when the state arises from mixing a pure two-qubit state with uniform noise. We also give a partial solution in the case when the uniform noise is replaced by independent depolarizing channels.

In his 1964 paper1 Bell made the fundamental observation that measurement correlations exhibited by some entangled quantum states cannot be explained by any local causal model. Specifically, if ρAB is the state of a bipartite system shared by Alice and Bob, and Alice is given a private input qQ and Bob is given a private input sS, then it is possible for Alice and Bob to measure ρAB and produce output messages aA and bB such that the conditional probability distribution P(abqs) cannot be simulated by any local hidden variable (LHV) model.

This can be interpreted as a fundamental confirmation of the models for nonlocality used in quantum physics, and it also has important applications in information processing. Device-independent quantum cryptography is based on the observation that if two untrusted input-output devices exhibit nonlocal correlations, their internal processes must be quantum. With correctly chosen protocols and mathematical proof, this observation allows a classical user to manipulate the devices to perform cryptographic tasks and at the same time verify their security.2,3

In 2007, the related notion of quantum steering was distilled,4 in which, rather than having Bob make a measurement, we directly consider the subnormalized marginal states ρ̃Bq,a that he holds when Alice receives input q and produces output a. A local hidden state (LHS) model attempts to generate these using shared randomness. Denoting the shared randomness by a random variable λ, distributed according to probability distribution μ(λ), Bob can output quantum state σλ, while Alice outputs a according to a probability distribution Pq,λ(a).

Suppose when Alice gets input q she performs a POVM {Eaq}aA so that ρ̃Bq,a=TrA((EaqIB)ρAB). A LHS model produces a faithful simulation if ρ̃Bq,a=λPq,λ(a)σλdμ(λ) for all q and a. If such a model exists, then we say that the state ρAB is unsteerable for the family of measurements {{Eaq}aA}qQ. If a LHS model exists for all possible measurements Alice could do (i.e., all Positive Operator Valued Measures (POVMs)), we say ρAB is unsteerable. Conversely, if there exists a set of measurements for which no LHS model exists, then ρAB is said to be steerable.

One can think of steering as an analog of non-locality for the case where one party (Bob) trusts his measurement device (and hence in principle could do tomography to determine his marginal state after being told Alice’s measurement and outcome). It is hence a useful intermediate between entanglement witnessing (both measurement devices trusted) and Bell violations (neither trusted) and has applications such as one-sided device-independent quantum cryptography5 and (sub)channel discrimination.6 Exhibiting new steerable states offers an expanded toolbox for such problems.

The steering decision problem is to determine whether or not a given state is steerable. This problem has proved to be difficult even for 2-qubit systems. To understand why this is so, consider a two-qubit state ρAB. If Alice were to measure {00,11} on input q = 0 and {++,} on input q = 1 (where ±=(0±1)/2), then it is possible for Bob to obtain one of four subnormalized states which we denote as ρ̃B0,ρ̃B1,ρ̃B+,ρ̃B (where, for example, ρ̃B0=TrA[(00IB)ρ]). Determining whether a LHS model exists for these four states is a search over a finite-dimensional space and is not difficult (see Refs. 7 and 8 for techniques for searching for LHS models). Next suppose Alice additionally performs the measurement {π/4π/4,5π/45π/4} for input q = 2, where

θcosθ20+sinθ21,
(1)

leading to states ρ̃Bπ/4,ρ̃B5π/4. There is no guarantee that a local hidden state model that simulates the previous four states will simulate this new pair as well (generally, the states ρ̃Bπ/8,ρ̃B5π/8 are not in the convex hull of the former states). A new search for local hidden state models is required, and the search space increases exponentially with each new measurement. Thus a direct approach—even when just dealing with measurements of the form {θθ,θ+πθ+π}—is unlikely to be feasible.

Previous work on steering has achieved success by exploiting the symmetries of certain classes of states. For the class of Werner states9ρAB(η)η[0,1] given by

ρAB(η)=ηΦ+Φ++(1η)I/4,
(2)

where Φ+=12(00+11), an exact classification of P-steerability (i.e., steerability for all projective measurements) has been performed (see  Appendix A for a summary of results on Werner states). More recently a complete classification of P-steerability for T-states (i.e., states for which ρA and ρB are maximally mixed) has been given.10–12 (Note that the requirement on ρB can be dropped—see Lemma 16 below.) In both cases, the methods depend critically on the symmetry of the states. For 2-qubit states outside the family of T-states, partial results on steerability exist (e.g., Refs. 13 and 14) but a full classification is not known.

In the current work, we develop new techniques to decide steerability in the case where ρA is not maximally mixed. We study Real Projective (RP)-steerability (i.e., steerability by the family of all measurements of the form {θθ,θ+πθ+π}) for real two-qubit states. [A state is real if its matrix elements are real in the {0,1} basis.] To illustrate our techniques, we give a complete classification of RP-steerability for the class of states, ρAB(α,η), formed by mixing partially entangled pure states with uniform noise, i.e.,

ρAB(α,η)=ηϕαϕα+(1η)I/4,
(3)

where ϕα=cosα00+sinα11. The classification is shown in Fig. 1, where the shaded/unshaded region represents the states that are unsteerable/steerable for real projective measurements. As a special case we recover the existing result15,16 that Werner states are RP-steerable if and only if η > 2/π (see Theorem 17).

FIG. 1.

RP-unsteerability for the states ρAB(α, η). In the shaded region, the states are unsteerable under real projective measurements, while above it they are steerable. (Note that the shaded region extends to η = 0.)

FIG. 1.

RP-unsteerability for the states ρAB(α, η). In the shaded region, the states are unsteerable under real projective measurements, while above it they are steerable. (Note that the shaded region extends to η = 0.)

Close modal

Our criterion also applies to a larger class of real 2-qubit states, specifically, all states whose steering ellipse is tilted at an angle less than π/4—see Theorem 14 and Corollary 15 for the formal statements. To achieve this classification, we introduce the concept of keyring models, which are a geometrically motivated class of local hidden state models for one-dimensional families of measurements. We explain these in more detail in Subsection I A.

Our approach invites generalizations. In its current form we have a criterion for steerability among all real 2-qubit states whose steering ellipse is tilted at an angle less than π/4. With additional work one may be able to go further to identify the set of all RP-steerable real 2-qubit states. Additionally, the keyring approach could be applied in more general scenarios where steering is attempted with any one-dimensional family of measurements.

Studying the behavior of qubit states under real projective measurements is a natural problem for experimental setups in which measurements in one plane of the Bloch sphere are easier than the most general measurements. However, another future goal would be to extend our methods to arbitrary complex measurements on 2-qubit states. This looks more challenging—steering with a 2-dimensional family of measurements is considerably harder than with a 1-dimensional family of measurements—but if it can be accomplished, it would be an important step towards a complete criterion for steering among arbitrary 2-qubit states.

Keyring models can also be used to construct a class of LHV models if we also use a (classical) function on Bob’s side to map his input and the hidden state to his output. They can hence be applied to the related problem of classically simulating bipartite correlations and may, for example, be useful for shedding new light on the problem of identifying the smallest detector efficiency for observing Bell inequality violations. We hope to find further applications of keyring models in this direction.

The difficulty in establishing steerability over all measurements is the need to rule out all LHS models. Our proof begins with the observation that in the case where ρAB is a real 2-qubit state and where the set of measurements comprises real projective measurements (i.e., those of the form {θθ,θ+πθ+π}), a more tractable (though still infinite dimensional) class of LHS models suffices. Specifically, we consider a class of LHS models that we call “keyring models,” which we now define.

Let RP1 denote the set of all real one-dimensional projectors on C2 (i.e., the set {θθ}θ[0,2π)). A keyring model is a pair (μ,{fθ}θ), where μ is a probability distribution on RP1, and fθ:RP1[0,1] is a two-step function—that is, roughly speaking, a function that takes two possible values and switches between them at two elements of RP1 (see Definition 4). The word “keyring” refers to the configuration of the two switching points on RP1 as θ varies. An example configuration is shown in Fig. 2. (This definition is related to the local hidden state models of Refs. 10–12 and 15, which are based on functions on RP2 that are supported on half-spheres. One key difference in the definition of a keyring model is that there is no uniformity in the positioning of the switching points of the functions fθ—they need not be diametrically opposite.)

FIG. 2.

An example configuration of endpoints in a keyring distribution. For every point in RP1 there are two associated endpoints in RP1. Here three pairs of endpoints are illustrated with (xα, yα) being the end points for α, for example.

FIG. 2.

An example configuration of endpoints in a keyring distribution. For every point in RP1 there are two associated endpoints in RP1. Here three pairs of endpoints are illustrated with (xα, yα) being the end points for α, for example.

Close modal

We show that ρAB is RP-steerable if and only if it can be simulated by a keyring model. Denoting the subnormalized reduced states on Bob’s side by ρ̃B(θ)=TrA((θθIB)ρAB), this is equivalent to the requirement

ρ̃B(θ)=xRP1xfθ(x)dμ
(4)

for all θ. From this we can conclude that that if the circumference of the steering ellipse {ρ̃B(θ)} is greater than 2, i.e.,

RP1ddθρ̃B(θ)1dθ>2,
(5)

where ∥⋅∥1 is the trace norm, then the state ρAB has no local hidden state model (see Proposition 6).

At this point, our proof diverges from that of Refs. 10–12 and 15, since the converse of the above statement is not true in our case: if (5) fails to hold, there could still be no local hidden state model. However, the following stronger condition guarantees the existence of a local hidden state model:

RP1ddθρ̃B(θ)dθ2ρB,
(6)

where |X|=XX is the absolute value of the operator. Moreover, the state ρAB is steerable if and only if

ρAB(IAY)ρAB(IAY)
(7)

is steerable for all positive definite Y (see Lemma 16), and by substituting in ρAB for ρAB in (5) and (6) we obtain an infinite family of criteria for RP-steerability and RP-unsteerability. We thus need to find a Y such that one of (5) and (6) holds for ρAB.

The most technically difficult part of our proof then shows that there must exist a positive definite density matrix Y such that

Y1RP1Yddθρ̃B(θ)YdθY1
(8)

is a scalar multiple of ρB. This compels (8) to either be greater than or less than or equal to ρB, and thus we achieve a criterion for steering which is both necessary and sufficient. We prove this by demonstrating that if we let Y tend to any projector P in RP1, then (8) must tend to an operator proportional to the orthogonal projector P^. Any continuous map from a 2-dimensional disc to itself which rotates the boundary of the disc must be an onto map, and this gives the desired result. (The proof of the aforementioned limit assertion is surprisingly subtle–it turns out that the rate at which the normalization of (8) approaches P^ is only logarithmic.)

Theorem 14 gives a formal statement of our main result. To apply the criteria (e.g., to obtain Fig. 1), we use numerical computations to find the appropriate operators Y from a given state ρAB.

For any Hilbert space H, let A(H) denote the set of all Hermitian operators on H, P(H) be the set of positive semidefinite operators on H, P>(H) be the set of positive definite operators on H, D(H) denote the set of all density operators on H, and D>(H) denote the set of all positive definite density operators on H. Let RA(H),RP(H), etc., denote the respective subsets of real operators (an operator X is real if iXjR for all i, j, where {i} is the standard basis). If A,BP(H), we write AB to mean that ABP(H) and AB for the complement of this. For an operator X on H, we use |X|XX and ∥X1 ≔ Tr|X|, the latter being the trace norm of X. If Tr(X) ≠ 0, we use X to denote the normalized version of X, i.e., XX/Tr(X). In addition, if Y is also an operator on H, then we use ⟨X, Y⟩ ≔ Tr(XY).

Throughout this paper, we take HA=HB=C2 to be qubit systems possessed by Alice and Bob and use RP1RD(H) to denote the set of one-dimensional real projectors on C2.

1. The steering ellipse

Any operator λRA(C2) can be expressed uniquely in terms of real numbers n, r1, r3 as

λ=12(nI+r1σ1+r3σ3),
(9)

where σ1=01+10 and σ3=0011 are the usual Pauli operators. Note that λRP1 if and only if n = 1 and r12+r32=1.

The tilt of λ, denoted as Tilt(λ), is the quantity r12+r32/|n| (if n = 0, then the tilt is ). The tilt angle of λ is arctan(Tilt(λ)). If we think of (n, r1, r3) as 3-dimensional Cartesian coordinates, then the tilt angle of λ is the angle that it forms with the (1, 0, 0) axis. We use these coordinates when we sketch steering ellipses later in this work. Note that an operator is positive semidefinite if and only if n ≥ 0 and its tilt is less than or equal to 1. It is useful to note that

λ1=|n|if Tilt(λ)1,r12+r32if Tilt(λ)>1.
(10)

Let ρABRD(C2C2). Then, the steering ellipse of ρAB on B is the function ρ̃B:RP1P(C2) given by

ρ̃B(θ)TrAθθIBρAB,
(11)

where θ is defined in (1). Note that {θ,θ+π} form an orthonormal basis, so ρB=ρ̃B(θ)+ρ̃B(θ+π) for any θ. (In the more general case of arbitrary projective measurements, the states on Bob’s side are a two-parameter family that defines an ellipsoid rather than an ellipse. Note also that the term “steering ellipsoid” is used to refer to the set of normalized states in the literature,10,17 while our steering ellipse comprises subnormalized states.)

Definition 1.

Let ρABRD(C2C2). Then, the tilt of the steering ellipse of ρAB is equal to the tilt of any nonzero vector that is normal to the 2-dimensional affine space that contains the steering ellipse of ρAB. (If the steering ellipse does not span a 2-dimensional affine space (i.e., it is degenerate), then we say that its tilt is equal to .)

Note that if the tilt of the steering ellipse is less than or equal to 1, then no element of the steering ellipse is strictly greater (in the positive semidefinite sense) than any other. This is a consequence of Lemma 20 in Appendix B 3.

2. Local hidden state models

In this section, we give a definition of a local hidden state model. It is not the most general definition possible, but it suffices for our purposes because of the form of the steering problem we are considering, as we now explain.

In the most general sense, a local hidden state model for a set of real 2-qubit subnormalized states ρ̃Bq,aqQ,aA is a probability distribution μ on D(C2) and set of functions fq,a:D(C2)[0,1]q,a with aAfq,a(x)=1 such that

ρ̃Bq,a=xD(C2)xfq,a(x)dμ.
(12)

(To connect with the earlier description, fq,a(x) is the probability that Alice gives the outcome a for measurement q when the hidden variable takes the value x.) However, via the map D(C2)RD(C2) given by x(x+x¯)/2, we may assume that μ, fa,q have support RD(C2), and by decomposing each operator in RD(C2) into a convex combination of one-dimensional projectors, we may further assume that μ, fq,a have support RP1. We are thus led to the following definition.

Definition 2.
A local hidden state model for a set ρ̃Bq,aqQ,aARP(C2) is a pair (μ,{fq,a}q,a) such that μ is a probability distribution on RP1, fq,a:RP1[0,1] with aAfq,a(x)=1 for all q, and
ρ̃Bq,a=xRP1xfq,a(x)dμ.
(13)
In the case of steering for real 2-qubit states under real projective measurements, it suffices to consider whether we can find (μ,{fθ}θ) with fθ:RP1[0,1] such that
ρ̃B(θ)=xRP1xfθ(x)dμ.
(14)
(Here fθ(x) is the probability that Alice gives the outcome corresponding to the first projector for the measurement {θθ,θ+πθ+π} when the hidden variable has value x.) If such a (μ,{fθ}θ) can be found, this constitutes a LHS model for the set {ρ̃B(θ)}θ[0,2π) and we say that ρAB is RP-unsteerable. Conversely, if no such model exists, we say that ρAB is RP-steerable.

Remark 3.

The property of having a LHS model is convex, i.e., if ρAB and ρAB have LHS models (for some set of measurements), then so does pρAB+(1p)ρAB for all 0 ≤ p ≤ 1 (and the same set of measurements).

In this section, we formalize the class of keyring models. We begin with some preliminary definitions. Drawing from Ref. 12, if μ is a probability distribution on RP1, let Box(μ) denote the convex set of all operators of the form

xRP1xf(x)dμ,
(15)

where f:RP1[0,1]. Note that Box(μ)RA(C2) with Tr(z) ≤ 1 for z ∈ Box(μ) and that an ellipse has a local hidden state model if and only if it is contained in Box(μ) for some probability distribution μ.

Note that there is a natural identification between RP1 and the unit circle S1R2 which is given by 12(I+r1σ1+r3σ3)(r1,r3) with r12+r32=1. We say that a sequence s1,s2,s3RP1 is a clockwise sequence if the images of s1, s2, s3 form a clockwise sequence in S1 and a counterclockwise sequence if the images of s1, s2, s3 form a counterclockwise sequence in S1. (If any of the points s1, s2, s3 are the same, then we will say that the sequence is both clockwise and counterclockwise.) We say that a sequence t1,,tnRP1 is clockwise (respectively, counterclockwise) if every 3-term subsequence of t1, t2, …, tn, t1 is clockwise (respectively, counterclockwise).

For any x,yRP1, let [x, y] denote the set of all zRP1 such that x, y, z is a clockwise sequence. Let (x,y)=RP1\[y,x]. Note that as implied by the notation, [x, y] is a closed set and (x, y) is open.

Definition 4.
A function f:RP1[0,1] is a two-step function if there are (not necessarily distinct) elements x,yRP1 and q ∈ [0, 1/2] such that
f(z)=1qif  z(x,y),qif  z(y,x),
(16)
with qf(x) ≤ 1 − q and qf(y) ≤ 1 − q. We refer to q as the bias of the function and to x, y as the endpoints of the function. If q < 1/2, then we refer specifically to x as the left endpoint and to y as the right endpoint.

A keyring model for a set {σa}aRP(C2) of subnormalized states is a local hidden state model (μ, { fa}) in which the functions are all two-step functions (see Fig. 2). The next proposition, which is proven in Appendix B 1, shows that any set that has a local hidden state model also has a keyring model. Hence when considering our steering problem it suffices to restrict the set of local hidden state models to keyring models.

Proposition 5.
Let μ be a probability distribution onRP1. Any element of z ∈ Box(μ) can be written as
z=xRP1xg(x)dμ,
(17)
where g is a two-step function. If z is on the boundary of Box(μ), then such a function g exists with bias q = 0.
Next we will use these techniques to prove a geometric fact about steerability. Let us say that the length of a piecewise differentiable curve S:[0,1]RA(C2) is its length under the trace norm,
01ddtS(t)1dt.
(18)

Proposition 6.

LetρABRD(C2C2)be a two-qubit state whose steering ellipse has tilt <1 and whose steering ellipse{ρ̃B(θ)}θhas a local hidden state model. Then, the length of{ρ̃B(θ)}θis no more than 2.

Note that using (10), the length of this curve is the Euclidean length of the projection of the ellipse onto the n = 0 plane in Bloch representation. It can be calculated using
02πddθr1(θ)2+ddθr3(θ)2dθ.
To prove Proposition 6, we first consider LHS models in which the distribution μ is supported on a finite set of points of RP1. Any probability distribution μ on RP1 can be approximated to an arbitrary degree of accuracy by a probability distribution which is supported on a finite set of points in the sense that for any ϵ > 0 there exists a finitely supported distribution μ′ such that for all two-step functions f we have
xRP1xf(x)dμxRP1xf(x)dμ1ϵ.

The next lemma shows that if μ is a probability distribution with finite support, then certain slices of Box(μ) must have circumference ≤2 under the trace norm.

Lemma 7.
Let μ be a probability distribution onRP1with finite support such thatxRP1xdμ=ρ, and letHRP>(C2). Then, the set
MBox(μ)M,H=(1/2)ρ,H
(19)
is enclosed by a curve of length ≤2.

This is proven in Appendix B 2.

Proof of Proposition 6.
Let (μ, {fθ}) be a keyring local hidden state model for the steering ellipse of ρAB. Let H be a positive definite operator that is normal to the steering ellipse of ρAB (such an operator exists because the tilt of the steering ellipse of ρAB is less than 1 by assumption). Because it is normal to the ellipse, ρ̃B(θ),H=u (independent of θ). Choose a sequence μ1, μ2, … of probability distributions on RP1 with finite support which converges to μ. Then, due to Lemma 7, the sets
MBox(μi)M,H=(1/2)ρB,H
(20)
are each enclosed by some curve of circumference ≤2. They furthermore converge to the set
MBox(μ)M,H=(1/2)ρB,H.
(21)
Because ρ̃B,H=ρ̃B(θ),H+ρ̃B(θ+π),H=2u, this set contains ρ̃B(θ). The desired result follows.

Proposition 6 gives a criterion for steerability that is sufficient but not necessary. In order to develop a criterion that is both necessary and sufficient, we will need to work not with the circumference of the steering ellipse, but with the following operator whose trace is equal to the circumference of the steering ellipse:

02πddθρ̃B(θ)dθ.
(22)

It is easiest to work with cases in which (22) is a scalar multiple of ρB. Our goal in the current section is to show that for any 2-qubit state ρAB whose steering ellipse has tilt <1, there is a YRD>(C2) such that operator (22) for ρAB=(IAY)ρAB(IAY) is a scalar multiple of ρB. This will enable the proof of our main result in Sec. V.

Subsections IV A and IV B will contain technical preparations. First we prove a concentration result for a particular type of integral.

Proposition 8.
LetURncontain the origin in its interior,F:URP(C2)be a continuous function such that F(0) ≠ 0, and letG:UR0be twice differentiable with G(x) = 0 if and only ifx = 0. Then,
lim(x2,,xn)0aaF(x)G(x)dx1=F(0).
(23)

Proof.
Since G is twice differentiable and x = 0 is a minimum of G, we have G(x)C|x|2 for some constant C > 0. Thus,
aadx1G(x)1Caadx1x12+y2
(24)
=1Cloga2+y2+aa2+y2a
(25)
=1Clog(1/y2)+2log(a2+y2+a),
(26)
where y2=i=2nxi2. Since y → 0 as (x2, …, xn) → (0, …, 0), this tends to . On the other hand, for any δ ∈ (0, a),
lim(x2,,xn)0[a,a]\(δ,δ)dx1G(x)=[a,a]\(δ,δ)dx1G(x1,0,,0)<,
(27)
since we assumed that G(x) has only one zero. Thus as (x2, …, xn) → 0, the integral of F(x)/G(x) on (−δ, δ) dominates the integral of the same quantity on [−a, a] \ (−δ, δ). The quantity on the left side of Eq. (23) is therefore in the convex hull of F((−δ, δ)). Since this holds true for any δ > 0, Eq. (23) follows.

Throughout this section, X and Y denote 2 × 2 real symmetric matrices. For any such matrix Y=deef, let Y^=feed denote the adjugate matrix. (The adjugate matrix has the same eigenspaces as Y, with the two eigenvalues interchanged.) Note that YY^=det(Y)I.

Definition 9.
If X,YRA(C2) and Y is invertible, let
XY=Y1YXYY1.
(28)

Note that if X is positive semidefinite, then its trace-norm and absolute value are easily computed: ∥X1 = Tr(X) and |X| = X. The next propositions compute these values in the case where X is neither positive semidefinite nor negative semidefinite.

Proposition 10.
If X is such that X 0 and X 0, then
X1=Tr(X2XX^),
(29)
X=X2XX^X1.
(30)

Proof.

Direct computation.

Proposition 11.
If X is such that X 0 and X 0 and Y is invertible, then
XY=XY2X(detX)Y^2YXY1.
(31)

Proof.

See Appendix B 5.

Note that YXY12 is a polynomial in the entries of X and Y [via (29)] and is therefore infinitely differentiable as a function of X and Y.

We now apply the results from Subsections IV A and IV B. Suppose that ρAB is a two-qubit state and that its steering ellipse {ρ̃B(θ)θR} has tilt less than 1. Define a function X:RRA(C2) so that

X(θ)=ddθρ̃B(θ)=TrA((D(θ)I)ρAB),
(32)

where D(θ)=12(sinθ00+cosθ(01+10)+sinθ11).

Note that because the steering ellipse {ρ̃B(θ)} has tilt <1, for every θ the operator X(θ) is neither positive semidefinite nor negative semidefinite (cf. Corollary 21).

Let PRP1. Let Y:R2RP(C2) be given by

Y(r1,r3)=P+r1σ1+r3σ3.
(33)

The function θ Tr(PX(θ)) varies sinusoidally and has exactly two zeros in [0, 2π). Without loss of generality, we will assume that the zeros are θ = 0 and θ = π. We wish to compute

lim(r1,r3)(0,0)π/2π/2XYdθ
(34)
=lim(r1,r3)(0,0)π/2π/2XY2X(detX)Y^2YXY12dθ.
(35)

The function PX(θ)P12=(Tr(PX(θ)))2 on the interval [−π/2, π/2] has a zero only at θ = 0. By Proposition 8 [with G(θ,r1,r3)=YXY12 and F(θ, r1, r3) equal to the numerator of the integrand in (35)], we obtain the following:

lim(r1,r3)(0,0)π/2π/2XYdθ=X(0)P2X(0)det(X(0))P^2
(36)
=2det(X(0))P^2
(37)
=P^.
(38)

Exploiting symmetry, the same equality holds when we replace the upper and lower integral limits with −π/2 and 3π/2 (or equivalently, with 0 and 2π). We therefore have the following.

Theorem 12.
Let ρABbe a two-qubit state whose steering ellipse{ρ̃B(θ)θR}has tilt less than 1. Then, for anyPRP1,
limYP02πddθρ̃B(θ)Ydθ=P^=IP,
(39)
where the limit is taken over all positive definite density operators Y.
As a consequence of Theorem 12, the function RD>(C2)RD(C2) given by
Y02πddθρ̃B(θ)Ydθ
(40)
extends continuously to a map RD(C2)RD(C2) which has the effect of mapping each element of RP1 to its orthogonal complement (see, for example, Theorem D on Page 78 of Ref. 18). By Lemma 22 in Appendix B 4, the function given by (40) is onto. In particular, its image contains ρB. We therefore have the following.

Lemma 13.
LetρABD(C2C2)be a two-qubit state whose steering ellipse has tilt <1 . Then, there existsYRD>(C2)such that
02πddθρ̃B(θ)Ydθ
(41)
is a scalar multiple of ρB.
Note that if we use ρAB=(IAY)ρAB(IAY) in Lemma 13, then we have that
02πddθρ̃B(θ)dθ
(42)
is a scalar multiple of ρB, which was our original goal.

Now we are ready to prove a criterion for RP-steerability that is both necessary and sufficient. The next theorem and corollary contain our main result.

Theorem 14.
LetρABD(C2C2)be a two-qubit state whose steering ellipse has tilt <1. Then, ρABis RP-unsteerable if and only if there existsYP>(C2) such that
YρBY0πYddθρ̃BθYdθ0.
(43)

Corollary 15.
LetρABD(C2C2)be a two-qubit state whose steering ellipse has tilt <1. Then ρABis RP-steerable if and only if there existsYP>(C2)such that
YρBY0πYddθρ̃BθYdθ0,
(44)
with the left-hand-side not equal to 0.
Note that (43) can be rewritten as
ρB0πddθρ̃BθYdθ0.
(45)

The following result found in Ref. 19 will be important for the proofs that follow.

Lemma 16.

If ρABhas a LHS model (for any set of measurements), then so does(IM)(ρAB)for any positive linear mapM.

In particular, for any invertible Hermitian operator Y, ρAB is RP-steerable if and only if (IAY)ρAB(IAY) is RP-steerable.

Proof of Theorem 14.

For any Hermitian operator X, define |X|± ≔ (|X| ± X)/2, and X±=TrX±.

Case 1: Suppose
ρBρ0πddθρ̃Bθdθ,
(46)
and define
σλddλρ̃Bλ++ρBρ2π.
(47)
Because ρ̃B(λ+π)=ρBρ̃B(λ), the operator (d/dλ)ρ̃B(λ+π) is the negation of the operator (d/dλ)ρ̃B(λ), and so the following equality also holds:
σλ=ddλρ̃Bλ+π+ρBρ2π.
(48)
We proceed to construct a local hidden state model from {σλ}λ. We have the following:
02πσλdλ=02πddλρ̃Bλ+dλ+ρBρ
(49)
=0πddλρ̃Bλ+dλ+π2πddλρ̃Bλ+dλ+(ρBρ)
(50)
=0πddλρ̃Bλ+dλ+0πddλρ̃Bλdλ+(ρBρ)
(51)
=0πddλρ̃Bλdλ+(ρBρ)
(52)
=ρ+ρBρ
(53)
=ρB.
(54)
For any θ ∈ [0, π] let gθ:RP1[0,1] be equal to zero on the interval [θ, θ + π] and equal to 1 elsewhere, and define gθ for θ ∈ (π, 2π] by gθ = 1 − gθπ. Then,
02πgθ(λ)σλdλ=1202π(2gθ(λ)1)σλdλ+02πσλdλ=12θθ+πmod2πσλdλ+θ+πmod2πθ+2πmod2πσλdλ+ρB=12θθ+πmod2π  ddλρ̃B(λ)+ddλρ̃B(λ)dλ+ρB=12θθ+πmod2πddλρ̃B(λ)dλ+ρB=12ρ̃B(θ+π)+ρ̃B(θ)+ρB=ρ̃B(θ).
Thus {ρ̃B(θ)}θ has a local hidden state model.
Case 2: Suppose that there exists YRP>(C2) such that
YρBY0πYddθρ̃BθYdθ.
(55)
In this case, the state
ρAB¯=(IY)ρAB(IY)
(56)
satisfies the conditions of Case 1. Since M:XY1XY1 is a positive map, by Lemma 16, a local hidden state model exists for ρAB.
Case 3: Suppose that for all YP>(C),
YρBYIY0πYddθρ̃BθYdθ.
(57)
By Lemma 13, we can find Y such that IY is a scalar multiple of Y ρBY (this is why Corollary 15 follows from Theorem 14). Thus we have
YρBY=c0πYddθρBθYdθ
(58)
for some c < 1. Letting γAB=(IY)ρAB(IY), we have
γB=c0πddθγBθdθ
(59)
which in particular means
0πddθγBθ1dθ(1/c)Tr(γB)>1.
(60)
By symmetry, replacing the upper limit (π) in the integral above has the effect of doubling its value; thus,
02πddθγBθ1dθ>2,
(61)
which implies by Proposition 6 that γ (and therefore ρ) has no local hidden variable model.

It is interesting to see what this criteria gives for Werner states, i.e., the family ρAB(η)=ηΦ+Φ++(1η)I/4 where η ∈ [0, 1] and Φ+=12(00+11).

Theorem 17.

States of the form ρAB(η) are RP-unsteerable forη2πand are RP-steerable forη>2π.

Proof.
The steering ellipses for these states are ρ̃B(θ)=141+ηcosθηsinθηsinθ1ηcosθ and have zero tilt for all η (since all these states have the same trace, the difference between any two states on the ellipse is orthogonal to I/2). The derivative with respect to θ is ddθ(ρ̃B(θ))=η4sinθcosθcosθsinθ which has |ddθ(ρ̃B(θ))|=η4I. Hence,
ρB0πddθρ̃Bθdθ=I/2πη4I.
Applying Theorem 14 and Corollary 15 with Y=I we have that Werner states are RP-unsteerable if πη412, i.e., η2π0.637 and are RP-steerable if η>2π.

Note that this boundary was already known15,16 and that it is possible to get close to this bound with small numbers of measurements.15,20

Consider the family ρAB(α,η)ηϕαϕα+(1η)I/4, where ϕαcosα00+sinα11 for 0απ4. The steering ellipses for these states are ρ̃Bα,η(θ)=ηcos2(α)cos2θ2η4+1412ηcos(α)sin(α)sin(θ)12ηcos(α)sin(α)sin(θ)ηsin2(α)sin2θ2η4+14 and are plotted in the Bloch representation in Fig. 3.

FIG. 3.

(a) Steering ellipses in the Bloch representation for η = 1, α = π/4 (blue), 0.65 (brown), 0.35 (purple), 0.1 (red), and 0 (green); (b) α = 0.35 and η = 1 (blue), 34 (brown), 12 (purple), 14 (red), and η = 0.01 (green). The small yellow circle on the right marks the origin.

FIG. 3.

(a) Steering ellipses in the Bloch representation for η = 1, α = π/4 (blue), 0.65 (brown), 0.35 (purple), 0.1 (red), and 0 (green); (b) α = 0.35 and η = 1 (blue), 34 (brown), 12 (purple), 14 (red), and η = 0.01 (green). The small yellow circle on the right marks the origin.

Close modal

One can verify that for Aα=sin2α00cos2α, Tr(Aαρ̃Bα,η(θ))=18(2η(1cos(4α))), which is independent of θ. Aα is hence normal to the steering ellipse and so the tilt of the ellipse is cos(2α) ≤ 1 and approaches 1 as α approaches 0.

Remark 18.

The tilt is independent of η and hence the steering ellipse for any two-qubit pure state has tilt at most 1.

We have ρB(α,η)=12(1+ηcos(2α))00+12(1ηcos(2α))11.

The derivative of the steering ellipse with respect to θ is
ddθρ̃Bα,η(θ)=η2cos2(α)sinθcos(α)sin(α)cos(θ)cos(α)sin(α)cos(θ)sin2(α)sinθ.
(62)

For α=π4, the case is as before. To investigate other values of α, we note that by Remark 3, if ρAB(α, η) has a LHS model, then so does ρAB(α, η′) for η′ < η. Thus, for each α there is a critical value η¯(α) such that ρAB(α, η) is RP-steerable for η>η¯(α) and is RP-unsteerable for ηη¯(α). We search for this critical value numerically.

Since Y has real entries, is positive, and multiplying by a constant does not affect whether (43) holds, we can take Y to have Tr(Y) = 1 and parameterize it in terms of two parameters r1 and r3 using a plane of the Bloch sphere via Y=12(I+r1σ1+r3σ3). To do the search we use the following subroutines:

  1. For fixed α and η, this searches over r1, r3 to find the largest value of the minimum eigenvalue of the expression on the left of (43). This uses gradient ascent with decreasing step-size, terminating when no improvement can be found for some minimal step-size or when r1, r3 are found such that the minimum eigenvalue is positive [i.e., (43) is satisfied]. The output is either the largest value found or the first positive value found.

  2. This is analogous to Subroutine 1, except it searches for the smallest value of the maximum eigenvalue of the expression on the left of (43), terminating either when a negative value is obtained or when no improvement can be found for some minimal step-size.

  3. For fixed α, this uses binary search to find the largest η for which Subroutine 1 returns a positive value, for some number of search steps.

  4. For fixed α, this uses binary search to find the smallest η for which Subroutine 2 returns a negative value, for some number of search steps.

Subroutine 3 hence gives a certified lower bound on η¯(α) and Subroutine 4 gives a certified upper bound. By varying the step-sizes and number of steps, in principle, we can make the gap between these as small as we like (in practice, the limits of machine precision provide a cutoff).

Note that if Subroutine 1 has a negative output, we cannot strictly rule out that there exists a Y such that condition (43) holds: in principle a smaller step-size might reveal a suitable Y. This is why we use Subroutine 2 in parallel.

The result is given in Fig. 1 (although the plot only shows η > 0.6, the region extends to η = 0).

Consider a source that generates an entangled state that is sent to two parties via two depolarizing channels with parameters ηA and ηB, i.e., these channels take

S(C2C2)S(C2C2):ρABρ^AB(EηAEηB)(ρAB),

where Eη:S(C2)S(C2) is given by Eη(ρ)=ηρ+(1η)I/2.

For ρAB=Φ+Φ+, this channel leads to Werner states (with parameter ηAηB instead of η). The states are hence RP-unsteerable if and only if ηAηB2π0.637.

More generally, for ρAB=ϕαϕα, we call the state after the channel ρ^AB(α,ηA,ηB) and note that

ρ^B=12(1+ηBcos(2α))00+(1ηBcos(2α))11

is independent of ηA. The steering ellipse for such a state is

ρ̃Bα,ηA,ηB(θ)=141+ηAcos(2α)cos(θ)+ηB(ηAcos(θ)+cos(2α))  ηAηBsin(2α)sin(θ)ηAηBsin(2α)sin(θ)  1+ηAcos(2α)cos(θ)ηB(ηAcos(θ)+cos(2α)).

For Aα,ηA,ηB=ηBcos(2α)2ηB00ηB+cos(2α)2ηB, we have Tr(Aα,ηA,ηBρ̃Bα,ηA,ηB(θ))=12sin2(2α), which is independent of θ. Hence Aα,ηA,ηB is the normal to the steering ellipse, and the ellipse has tilt cos(2α)ηB. This is less than 1 for ηB > cos(2α), so we can use Theorem 14 and Corollary 15 provided this holds.

The derivative of the steering ellipse is

ddθρ̃Bα,ηA,ηB(θ)=ηA4(ηB+cos(2α))sin(θ)ηBsin(2α)cos(θ)ηBsin(2α)cos(θ)(ηBcos(2α))sin(θ).
(63)

Since this is proportional to ηA, the amount of noise on Alice’s side (the untrusted side), the case of noise only on Bob’s side is representative of the general case.

We first make two observations for special cases, before proceeding with the general case:

  1. If there is no noise on Bob’s side (i.e., the trusted side), i.e., if ηB = 1, then ddθρ̃Bα,ηA,1(θ) is identical to that in (62) and the tilt of the steering ellipse of ρAB(α, ηA, 1) is cos(2α) ≤ 1, so we obtain the same result.

  2. If the state is maximally entangled, i.e., α=π4, then the situation is exactly the same as for a Werner state with η = ηAηB. In other words, ηAηB2π is a necessary and sufficient condition for RP-unsteerability of a state of the form ρAB(π/4, ηA, ηB).

We study the general case numerically, using similar techniques to before. The results are shown in Fig. 4.

FIG. 4.

Plot of the regions where a LHS model exists for all real projective measurements for ηA = 1 (blue), ηA = 0.9 (orange) and ηA = 0.8 (green), ηA = 0.7 (red) (although not shown, all regions extend downwards to ηB = 0), together with the purple curve ηB = cos(2α) which we need to be above to use Theorem 14 and Corollary 15. In the case ηA=2π (not shown), the state is RP-unsteerable for all ηB and α. For ηA = 0.9, 0.8, and 0.7, we have a complete classification: above each of the corresponding regions, the state is RP-steerable. In the case ηA = 1, the classification is incomplete for α ⪅ 0.37. Here, if ηB ≤ cos(2α), we are unable to decide whether or not the states are RP-steerable (while for ηB > cos(2α) we know the states are RP-steerable).

FIG. 4.

Plot of the regions where a LHS model exists for all real projective measurements for ηA = 1 (blue), ηA = 0.9 (orange) and ηA = 0.8 (green), ηA = 0.7 (red) (although not shown, all regions extend downwards to ηB = 0), together with the purple curve ηB = cos(2α) which we need to be above to use Theorem 14 and Corollary 15. In the case ηA=2π (not shown), the state is RP-unsteerable for all ηB and α. For ηA = 0.9, 0.8, and 0.7, we have a complete classification: above each of the corresponding regions, the state is RP-steerable. In the case ηA = 1, the classification is incomplete for α ⪅ 0.37. Here, if ηB ≤ cos(2α), we are unable to decide whether or not the states are RP-steerable (while for ηB > cos(2α) we know the states are RP-steerable).

Close modal

The left-hand side of (43) becomes easier to satisfy for lower ηA and so the region of RP-unsteerability increases as ηA is lowered. In other words, if ρ^AB(α,ηA,ηB) is RP-unsteerable, then so is ρ^AB(α,ηA,ηB) for ηAηA. At ηA=2π, the state is RP-unsteerable for all ηB and α.

Note that the regions shown in the above plot extend below the purple curve, although the condition on the tilt of the steering ellipse ceases to be satisfied there. To extend to this region we use the fact that more noise (lower ηB) makes a LHS model easier to construct. This is stated in the following lemma.

Lemma 19.

Ifρ^AB(α,ηA,ηB)has a LHS model (for any set of measurements), then so doesρ^AB(α,ηA,ηB)for allηB<ηB.

Proof.

This follows from Remark 3 and the fact that ρ^AB(α,ηA,ηB) is equal to ηBηBρ^AB(α,ηA,ηB)+ηBηBηBρ^A(α,ηA,ηB)I/2, i.e., is a convex combination of ρ^AB(α,ηA,ηB) and ρ^A(α,ηA,ηB)I/2, both of which have LHS models.

Hence, although we cannot use Theorem 14 throughout the α-ηB plane, we can nevertheless establish steerability of all states of the form ρ^AB(α,1,ηB) for α ⪆ 0.37 (for example). Furthermore, the numerics point to the existence of a critical value around 0.92 such that for values of ηA below this we can always use our criteria (graphically, the boundary of the region in which a LHS model exists always lies above ηB = cos(2α) for ηA ⪅ 0.92).

We are grateful to Kim Winick for numerous helpful discussions, to Emanuel Knill, Sania Jevtic, Stephen Jordan, and Chau Nguyen for useful feedback on an earlier version of the manuscript, and to Nicholas Brunner, Daniel Cavalcanti, and Ivan Supic for pointers to the literature. R.C. is supported by the EPSRC’s Quantum Communications Hub (Grant No. EP/M013472/1) and by an EPSRC First (Grant No. EP/P016588/1). C.A.M. and Y.S. were supported in part by US NSF Grant Nos. 1500095, 1526928, and 1717523. Y.S. was also supported in part by University of Michigan.

Werner states [cf. (2)] are separable if and only if η13,9 are steerable if η>12,4 and are non-local if η > 1/KG(3),21 where KG(3) is Grothendieck’s constant of order 3,22 which is known to satisfy 1.426 < KG(3) < 1.464 so that 0.683 < 1/KG(3) < 0.701.23,24 They are local for projective measurements if η ≤ 1/KG(3)21 and are local for all measurements for η ≤ 0.45524 and also have a LHS model for all measurements for η ≤ 5/12.19,25 For 1/3<η5/12, the states are non-separable and unsteerable. For 12<η1KG(3), the states are local for projective measurements and steerable. It is unknown whether these states are local for all measurements anywhere in this range, which would show steerability non-locality; however, this non-implication is known using another family of states.19 

The above is summarized in Fig. 5.

FIG. 5.

Summary of known results for Werner states. The approximation taken for 1/KG(3) is the mean of the known upper and lower bounds.

FIG. 5.

Summary of known results for Werner states. The approximation taken for 1/KG(3) is the mean of the known upper and lower bounds.

Close modal

1. Proof of Proposition 5

This proof uses similar methods to those in Ref. 12.

The proof will be divided into two cases: (1) the case where z lies on the boundary of Box(μ) and (2) the case where z lies in the interior of Box(μ).

(1) In the case where z lies on the boundary of Box(μ), because Box(μ) is convex, there must exist HRA(C2) such that the function xx,H on Box(μ) is maximized at z. We subdivide into three cases depending on H.

Case 1a: The element z is on the boundary and H > 0.

The operator

ρ=xRP1xdμ
(B1)

is greater than or equal to z, so ρz,H0. But this quantity cannot exceed 0 by assumption, so ρz,H=0, which yields ρ = z. Since the constant function RP1{1} satisfies the definition of a two-step function, we are done.

Case 1b: The element z is on the boundary and H ≱ 0.

In this case, there are unique distinct elements y,wRP1 such that y,H=w,H=0, x,H>0 for all x(y,w) and x,H<0 for all x(w,y). Choose a function f:RP1[0,1] such that

z=xRP1xf(x)dμ
(B2)

(such a function must exist because z ∈ Box(μ)). Let g be the two-step function

g(x)=1   if    x(y,w),0   if    x(w,y),f(y)   if    x=y,f(w)   if    x=w
(B3)

and let

r=xRP1xg(x)dμ.
(B4)

Since r ∈ Box(μ), r,Hz,H. Hence we have

0rz,H=x(y,w)(1f(x))xdμ,Hx(w,y)f(x)xdμ,H0,
(B5)

where the final inequality follows because any operator x(y,w) has a positive inner product with H and any operator x(w,y) has a negative inner product with H. It follows that z = r, which completes this case.

Case 1c: The element z is on the boundary and H is positive semidefinite and rank-one.

Let yRP1 be the unique element such that H,y=0. Let

g(x)=1if xy,f(y)ifx=y,
(B6)

where f:RP1[0,1] is a function such that (B2) holds. By similar reasoning as in Case 1b, this function also computes z.

Case 2: The element z is in the interior of Box(μ).

Let

c=xRP1(1/2)xdμ.
(B7)

Since z is interior it can be written as z = tc + (1 − t)b, where t ∈ [0, 1] and b is an element on the boundary of Box(μ). Let g be a two-step function which computes b, which must exist from the first part of the proof. Then, the function t/2 + (1 − t)g computes z.

2. Proof of Lemma 7

We will construct an explicit curve which is the boundary of (19). Let S = {s1, …, sn} be the support of μ, where the points 00,s1,,sn are in clockwise order, and define ρ̃mi=1mμ(si)si.

For any t[0,H,ρ], define a two-step function ht:RP1[0,1] as follows: if

tρm,H,ρm+1,H,
(B8)

then

ht(x)=1 for x00,sm+1,
(B9)
ht(sm+1)=tρm,Hμ(sm+1)sm+1,H,
(B10)

and ht is zero elsewhere. Note that by construction,

xRP1ht(x)x,Hdμ=t.
(B11)

Also define a zero-bias two-step function h¯t:RP1[0,1] by

h¯t=ht+ρ,H/2htif t<ρ,H/2,1ht+htρ,H/2otherwise
(B12)

so that for any t,

xRP1h¯t(x)x,Hdμ=ρ,H/2.
(B13)

Let

G(t)=xRP1h¯t(x)xdμ.
(B14)

The points in the image of G(t) are in region (19) by construction, and since they are obtained from zero-bias two-level functions, they lie on the boundary of Box(μ) (see Proposition 5). The image of G is the boundary of (19).

Note that for any fixed i, the function th¯t(si) is bitonic (in the sense that it only increases once and decreases once, modulo ρ,H) and thus

0ρ,Hddth¯t(si)dt2.
(B15)

Therefore, the length of the curve G satisfies

0ρ,HddtG(t)1dt=0ρ,HddtxRP1h¯t(x)xdμ1dt
(B16)
=0ρ,Hddti=1nh¯t(si)siμ(si)1dt
(B17)
0ρ,Hi=1nddth¯t(si)μ(si)dt
(B18)
i=1n2μ(si)=2,
(B19)

as desired.

3. Tilt of the derivative of the steering ellipse

Lemma 20.

Supposeλ,μRA(C2)with Tr(λμ) = 0 and Tilt(μ) < 1. Then Tilt(λ) > 1.

Proof.

Suppose λ=12(nI+r1σ1+r3σ3) and μ=12(mI+s1σ1+s3σ3) and write (r1, r3) = rer and (s1, s3) = ses, where er, es are unit vectors and r, s ≥ 0.

The condition Tr(λμ) = 0 can be written as −r.s = nm. Tilt(μ) < 1 is equivalent to s2 < m2. It follows that (r.s)2 = n2m2 > n2s2. This rearranges to (r.es)2>n2, from which it follows that r2 > n2, i.e., Tilt(λ) > 1.

Corollary 21.

LetρABRD(C2C2)be a two-qubit state whose steering ellipse{ρ̃B(θ)}has tilt smaller than 1. ThenTilt(ddθρ̃B(θ))>1for all θ.

4. A topological lemma

Lemma 22.

LetD={zCz1}and letS1={zCz=1}. Let F: DD be a continuous function such that for any zS1, F(z) = −z. Then, F is onto.

Proof.

Suppose, for the sake of contradiction, that yD\F(D). Let G: DS1 be the (unique) function defined by the condition that for any zD, F(z) lies on the line segment from y to G(z). Note that the function G also satisfies G(z) = −z for zS1. The family of functions Hα:S1S1α[0,1] given by Hα(z) = G(αz) is a continuous deformation between the negation map on S1 and the constant map which takes S1 to G(0). This is impossible, since these maps represent different elements of the fundamental group of S1. Thus, by contradiction, the original map F must be onto.

5. Proof of Proposition 11

By Proposition 10, we have

XY=Y1YXYY1
(B20)
=Y1(YXY)(YXY)Y1Y1(YXY)(ŶX^Ŷ)Y1YXY1
(B21)
=XY2XXdet(Y)X^ŶY1YXY1
(B22)
=XY2XXX^Ŷ(det(Y)Y1)YXY1
(B23)
=XY2Xdet(X)ŶŶYXY1,
(B24)

which is equal to the desired formula.

1.
J. S.
Bell
, “
On the Einstein-Podolsky-Rosen paradox
,” in
Speakable and Unspeakable in Quantum Mechanics
(
Cambridge University Press
,
1987
), Chap. 2.
2.
D.
Mayers
and
A.
Yao
, “
Quantum cryptography with imperfect apparatus
,” in
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
(
IEEE Computer Society
,
1998
), pp.
503
509
.
3.
J.
Barrett
,
L.
Hardy
, and
A.
Kent
, “
No signalling and quantum key distribution
,”
Phys. Rev. Lett.
95
,
010503
(
2005
).
4.
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
).
5.
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
).
6.
M.
Piani
and
J.
Watrous
, “
Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering
,”
Phys. Rev. Lett.
114
,
060404
(
2015
).
7.
D.
Cavalcanti
,
L.
Guerini
,
R.
Rabelo
, and
P.
Skrzypczyk
, “
General method for constructing local hidden variable models for entangled quantum states
,”
Phys. Rev. Lett.
117
,
190401
(
2016
).
8.
F.
Hirsch
,
M. T.
Quintino
,
T.
Vértesi
,
M. F.
Pusey
, and
N.
Brunner
, “
Algorithmic construction of local hidden variable models for entangled quantum states
,”
Phys. Rev. Lett.
117
,
190402
(
2016
).
9.
R. F.
Werner
, “
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
,”
Phys. Rev. A
40
,
4277
4281
(
1989
).
10.
S.
Jevtic
,
M. J. W.
Hall
,
M. R.
Anderson
,
M.
Zwierz
, and
H. M.
Wiseman
, “
Einstein–Podolsky–Rosen steering and the steering ellipsoid
,”
J. Opt. Soc. Am. B
32
,
A40
A49
(
2015
).
11.
H. C.
Nguyen
and
T.
Vu
, “
Nonseparability and steerability of two-qubit states from the geometry of steering outcomes
,”
Phys. Rev. A
94
,
012114
(
2016
).
12.
H. C.
Nguyen
and
T.
Vu
, “
Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes
,”
Europhys. Lett.
115
,
10003
(
2016
).
13.
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
).
14.
J.
Bowles
,
F.
Hirsch
,
M. T.
Quintino
, and
N.
Brunner
, “
Sufficient criterion for guaranteeing that a two-qubit state is unsteerable
,”
Phys. Rev. A
93
,
022121
(
2016
).
15.
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
).
16.
R.
Uola
,
K.
Luoma
,
T.
Moroder
, and
T.
Heinosaari
, “
Adaptive strategy for joint measurements
,”
Phys. Rev. A
94
,
022109
(
2016
).
17.
S.
Jevtic
,
M.
Pusey
,
D.
Jennings
, and
T.
Rudolph
, “
Quantum steering ellipsoids
,”
Phys. Rev. Lett.
13
,
020402
(
2014
).
18.
G. F.
Simmons
,
Introduction to Topology and Modern Analysis
, reprint ed. (
Krieger Publishing Company
,
1983
).
19.
M. T.
Quintino
 et al, “
Inequivalence of entanglement, steering, and Bell nonlocality for general measurements
,”
Phys. Rev. A
92
,
032107
(
2015
).
20.
J.
Bavaresco
 et al, “
Most incompatible measurements for robust steering tests
,”
Phys. Rev. A
96
,
022110
(
2017
).
21.
A.
Acín
,
N.
Gisin
, and
B.
Toner
, “
Grothendieck’s constant and local models for noisy entangled quantum states
,”
Phys. Rev. A
73
,
062105
(
2006
).
22.
A.
Grothendieck
, “
Résumé de la théorie métrique des produits tensoriels topologiques
,”
Bol. Soc. Mat. São Paulo
8
,
1
79
(
1953
).
23.
S.
Brierley
,
M.
Navascués
, and
T.
Vértesi
, “
Convex separation from convex optimization for large-scale problems
,” e-print arXiv:1609.05011 (
2016
).
24.
F.
Hirsch
,
M. T.
Quintino
,
T.
Vértesi
,
M.
Navascués
, and
N.
Brunner
, “
Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant KG(3)
,”
Quantum
1
,
3
(
2017
).
25.
J.
Barrett
, “
Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality
,”
Phys. Rev. A
65
,
042302
(
2002
).