Keyring models: An approach to steerability

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 ρ_AB is the state of a bipartite system shared by Alice and Bob, and Alice is given a private input $q∈Q$ and Bob is given a private input $s∈S$⁠, then it is possible for Alice and Bob to measure ρ_AB and produce output messages $a∈A$ and $b∈B$ such that the conditional probability distribution P(ab∣qs) 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 σ_λ, while Alice outputs a according to a probability distribution P_q,λ(a).

Suppose when Alice gets input q she performs a POVM ${Eaq}a∈A$ so that $ρ̃Bq,a=TrA((Eaq⊗IB)ρAB)$⁠. 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 ρ_AB is unsteerable for the family of measurements ${{Eaq}a∈A}q∈Q$⁠. If a LHS model exists for all possible measurements Alice could do (i.e., all Positive Operator Valued Measures (POVMs)), we say ρ_AB is unsteerable. Conversely, if there exists a set of measurements for which no LHS model exists, then ρ_AB 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 ρ_AB. 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, $ρ̃B0=TrA[(00⊗IB)ρ]$⁠). 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

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 states⁹ $ρAB(η)∣η∈[0,1]$ given by

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 ρ_A and ρ_B are maximally mixed) has been given.^10–12 (Note that the requirement on ρ_B 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 ρ_A 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.,

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 result^15,16 that Werner states are RP-steerable if and only if η > 2/π (see Theorem 17).

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 ρ_AB 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 f_θ—they need not be diametrically opposite.)

We show that ρ_AB 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 $ρ̃B(θ)=TrA((θθ⊗IB)ρAB)$⁠, this is equivalent to the requirement

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.,

where ∥⋅∥₁ is the trace norm, then the state ρ_AB 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:

where $|X|=X†X$ is the absolute value of the operator. Moreover, the state ρ_AB is steerable if and only if

is steerable for all positive definite Y (see Lemma 16), and by substituting in $ρAB′$ for ρ_AB 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

is a scalar multiple of ρ_B. This compels (8) to either be greater than or less than or equal to ρ_B, 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 ρ_AB.

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 $iXj∈R$ for all i, j, where ${i}$ is the standard basis). If $A,B∈P≥(H)$⁠, we write A ≥ B to mean that $A−B∈P≥(H)$ and A ≱ B for the complement of this. For an operator X on $H$⁠, we use $|X|≔X†X$ and ∥X∥₁ ≔ 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., $X≔X/Tr(X)$⁠. In addition, if Y is also an operator on $H$⁠, then we use ⟨X, Y⟩ ≔ Tr(X^†Y).

Throughout this paper, we take $HA=HB=C2$ to be qubit systems possessed by Alice and Bob and use $RP1⊆RD(H)$ to denote the set of one-dimensional real projectors on $C2$⁠.

Any operator $λ∈RA(C2)$ can be expressed uniquely in terms of real numbers n, r₁, r₃ as

where $σ1=01+10$ and $σ3=00−11$ are the usual Pauli operators. Note that $λ∈RP1$ 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

Let $ρAB∈RD(C2⊗C2)$⁠. Then, the steering ellipse of ρ_AB on B is the function $ρ̃B:RP1→P≥(C2)$ given by

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.)

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 $ρ̃Bq,aq∈Q,a∈A$ is a probability distribution μ on $D(C2)$ and set of functions $fq,a:D(C2)→[0,1]q,a$ with $∑a∈Afq,a(x)=1$ such that

(To connect with the earlier description, f_q,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, f_a,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, f_q,a have support $RP1$⁠. We are thus led to the following definition.

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

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 $S1⊆R2$ which is given by $12(I+r1σ1+r3σ3)↔(r1,r3)$ with $r12+r32=1$⁠. We say that a sequence $s1,s2,s3∈RP1$ 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 $t1,…,tn∈RP1$ is clockwise (respectively, counterclockwise) if every 3-term subsequence of t₁, t₂, …, t_n, t₁ is clockwise (respectively, counterclockwise).

For any $x,y∈RP1$⁠, let [x, y] denote the set of all $z∈RP1$ 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.

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:

It is easiest to work with cases in which (22) is a scalar multiple of ρ_B. Our goal in the current section is to show that for any 2-qubit state ρ_AB whose steering ellipse has tilt <1, there is a $Y∈RD>(C2)$ such that operator (22) for $ρAB′=(IA⊗Y)ρAB(IA⊗Y)$ 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.

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

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

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 $P∈RP1$⁠. Let $Y:R2→RP≥(C2)$ be given by

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

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:

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.

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.