EPR Steering inequalities with Communication Assistance

In this paper, we investigate the communication cost of reproducing Einstein-Podolsky-Rosen (EPR) steering correlations arising from bipartite quantum systems. We characterize the set of bipartite quantum states which admits a local hidden state model augmented with c bits of classical communication from an untrusted party (Alice) to a trusted party (Bob). In case of one bit of information (c = 1), we show that this set has a nontrivial intersection with the sets admitting a local hidden state and a local hidden variables model for projective measurements. On the other hand, we find that an infinite amount of classical communication is required from an untrusted Alice to a trusted Bob to simulate the EPR steering correlations produced by a two-qubit maximally entangled state. It is conjectured that a state-of-the-art quantum experiment would be able to falsify two bits of communication this way.

Quantum entanglement is a remarkable phenomenon that has no counterpart in classical physics 1,2 . Beyond its fundamental importance, it is a crucial resource in quantum information and quantum computing 3 . Entanglement gives rise to the phenomenon of Bell nonlocality 4,5 , which lies at the heart of device-independent quantum information processing 6 . Such device-independent protocols are greatly immune against errors which are due to deviations of the ideal description of the setup from the actual physical implementation.
Quantum correlations can be phrased in terms of an information task wherein a referee, say Charlie, wants to verify that two parties, called Alice and Bob, share an entangled state (see Fig. 1 displaying the setup). In the preparation stage of the protocol, Alice and Bob share a number of copies of a bipartite state ρ, and for each of those states Charlie asks them to perform one of a number of measurements chosen by Charlie at random. Alice's and Bob's measurements are denoted by M a x and M b y , respectively, where x and y denote the choice of measurements and a and b their corresponding outputs. By repeating the procedure many times, they form the joint probability distribution ( ) P ab xy Q , which is given by That is, the object of our study is the probability distribution of the outputs of the two parties dependent on each party's input (i.e. choice of measurement settings). Throughout we will assume that measurements are projective ones, that is, . Note that for two-outcome settings (which is our main concern) this is not a limitation 32 .
Basically, there are three options to certify entanglement depending on the number of trusted parties participating in the protocol. Charlie trusts both Alice and Bob (and their apparatuses). Charlie trusts (say) Bob, but not Alice. Finally, Charlie trusts neither Alice nor Bob.
In the latter case of no trust at all (i.e. the Bell nonlocality scenario), we say that a quantum state ρ is Bell local or equivalently admits a local hidden variables (LHV) model (for projective measurements), when the statistics On the other hand, in case of partial trust (i.e. an EPR steering scenario), we obtain the data ( ) P ab xy Q with an additional knowledge that Bob's system is well-characterized. That is, Charlie trusts Bob's measurements {M b|y } b,y . In that case, the state ρ shared by Alice and Bob is said to be unsteerable or equivalently to admit a local hidden state (LHS) model, when the statistics ( ) P ab xy Q can be reproduced by a distribution of the form (2), where now In that case, the distribution ( ) P ab xy Q cannot violate the so-called steering inequalities. Conversely, if the distribution ( ) P ab xy Q cannot be written in the form (2) with the above restriction (3) on ( ) λ P b y , it violates a steering inequality. Again, this means that some communication has to be taken place between Alice and Bob to reproduce the obtained statistics ( ) P ab xy Q . Finally, in the case that Charlie trusts both Alice's and Bob's measurement devices, the LHS model above becomes a quantum separable (QS) model, where there exist local density operators σ λ A and σ λ B such that the response functions of Alice and Bob in the formula (2) are given respectively by σ Failure of satisfying this model implies entanglement. This again can be detected through the violation of certain inequalities which are conventionally called entanglement witnesses.
However, observing either a violation of a LHV model in the Bell nonlocality scenario, or violation of a LHS model in the EPR steering scenario, or violation of a QS model in the entanglement scenario will not quantify the amount of communication beyond the fact that some communication was indeed required. In case of Bell nonlocality, where no trust is assumed in either devices, Bacon and Toner provided a general framework for a measure of nonlocality by allowing the parties to communicate some bits of information after selecting the measurement settings 33 . They in particular proved that correlations produced by projective measurements on the two-qubit singlet state can be simulated with a local hidden variables model (LHV) augmented by a single bit of communication 34 .
In this paper, we pose an analogous question in the EPR steering scenario. Our aim is to quantify the correlations arising from quantum (projective) measurements conducted by Alice on her share of an entangled particle. In (a) the simulation protocol is as follows. The two parties distribute shared randomness λ. Charlie sends settings ( , ) x y to the two parties. After obtaining the settings, Alice is allowed to communicate to Bob a classical message consisting of c bits. Finally, Alice and Bob give outputs a and b as a function of available information for each party. The (b) protocol is similar to (a) with the difference that Charlie fully trusts Bob, hence, we can assume that Bob performs a given set of quantum measurements {M b|y } b,y on σ λ . To do so, we allow some amount of classical communication from the untrusted party Alice to a trusted party Bob.
The structure of the paper is as follows. We first present the Bell nonlocality setup by Bacon and Toner 33 . Then we translate this setup to the EPR-steering scenario. In particular, we develop a computational framework to decide if an EPR correlation produced by quantum theory can be simulated by a LHS model plus exchanging a number of bits of communication. We provide an efficient code based on semidefinite programming (SDP) 35 to solve such a membership problem. This allows us to explore the shape of the set of two-qubit states admitting a LHS model augmented with one bit of communication, a set we denote by ( ) LHS 1 . Specifically, we prove that the set ( ) LHS 1 is strictly larger than the set of states admitting a LHS model (for projective measurements). On the other hand, we conduct an extensive numerical search which indicates that there exist two-qubit quantum states admitting a LHV model (assuming projective measurements), which nevertheless cannot be described by a LHS model assisted with 1 bit of classical communication. Finally, we show that an infinite amount of classical communication is required from an untrusted Alice to a trusted Bob to simulate the statistics of any bipartite pure entangled state in this scenario.

Results
Bell nonlocality with communication. Bacon  In the simulation protocol, we also would like to take into account the possibility that Alice and Bob's devices are correlated due to a common random variable λ, which was prepared and distributed between the parties before receiving inputs x and y from Charlie. In that case, the set of admissible distributions is formed by all convex combination of strategies labeled by λ: where each λ ( ) ≥ P 0 and λ ∑ ( )= λ P 1. Bacon and Toner 33 quantify the amount of resource by the number of bits of communication c to match ( ) P ab xy Q with the distribution ( ) P ab xy in Eq. (5). They prove that c = 1 bit of communication assistance (i.e. sending a message with = d 2 levels) is enough for Alice and Bob to reproduce any correlations ( ) P ab xy Q by measuring arbitrary projective measurements on a maximally entangled two-qubit state 34 . Figure 1(a) displays this setup.
EPR steering with communication. We ask the analogous question what happens if (unlike in case of the Bell nonlocality scenario) Charlie completely trusts Bob's measurement device. This is the framework of EPR steering. In this case, we obtain the data ( ) P ab xy Q with an additional knowledge that Bob's equipment is well-characterized. Hence our model is the same as (5), but with the constraint that where Bob's measurements {M b|y } b,y are trusted by Charlie and the states σ λ,c are of unit trace and positive. Note, however, that in this case Bob's measurements will not depend on the communication c, since they can be considered as supplied by Charlie. Please see Fig. 1.(b), which shows this setup. Hence we get λ D a c x are some deterministic strategies labeled by λ taking values 0 or 1. Note that we used the fact that unshared randomness of ( , ) λ P a c x can always be considered as part of shared randomness (represented by λ) and hence can be absorbed into it. For instance, if Alice performs m measurements ( = ,…, where the data ( ) P ab xy Q is arising from formula (1) and M b y are fixed measurements of Bob. We can in fact simplify a bit the above code by eliminating Bob's measurements M b y . To this end, let us define the conditional (unnormalized) states prepared by Alice on Bob's subsystem by This set of states is called assemblage 18,19 and captures the whole physics of an EPR steering scenario. With this assemblage, we have σ ( )= ( ) P ab xy M tr Q a x b y . Let us assume that Bob's measurements M b y form a complete basis of Bob's Hilbert space. In case of a two-dimensional Hilbert space, this can be the three Pauli matrices, It is worth noting that the above code simplifies to the one derived in ref. 18

Exploring the shape of the LHS(1) set of states.
In this subsection, we explore the shape of the ( ) LHS c set, where we set = c 1 bit. Specifically, we ask how it fits into the set of bipartite states which admit a LHS or LHV model. Let us note that the new set ( ) LHS 1 is also convex by construction. We find a nontrivial structure of this new set. To this end, we investigate special one-parameter slices of the full two-qubit state space. In particular, we choose two special one-parameter families, the Werner states of two-qubits and another family, which coincides with the two-qubit reduced state of the n-qubit W n state 37 for parameters = / p n 2 . The obtained results suggest that the ( ) LHS 1 set of states has a nontrivial shape as depicted in the schematic picture of Fig. 2. We use the SDP code (10) to test one-parameter families of two-qubit quantum states. Namely, let us write the state as a mixture of a pure entangled state and a noisy part parameterized by the weight v: where ρ noise is some fixed separable state and ψ is any two-qubit entangled pure state. A small variation of the semi-definite program (10) (please see Methods section for the actual code) gives us an efficient method to place an upper bound on v crit , where v crit denotes the boundary of states admitting a LHS(1) model. Such numerical computations, as well as all subsequent ones presented in this paper, were carried out using the Matlab packages YALMIP 38 and the SDP solver SeDuMi 39 . Then, using a heuristic search (e.g., we used an Amoeba routine 40 ), we lower the value of this upper bound on v crit by varying the set of measurements {M a|x } a,x . This way, we get better and better upper bounds to the true value of v crit by minimizing the parameters entering Alice's set of measurements {M a|x } a,x . We remark that due to the heuristic nature of the search, the program may not provide us a global minimum for v crit , however, for reasonable number of settings (say, ≤ m 6 A ) and a fair number of independent iterations, the obtained bound to our experience is quite reliable.
We first consider the Werner state of two qubits 41 , which is given by Concerning the LHS(1) model, we find the following results. First, in the Methods section a LHS(1) model is provided up to visibility = / .  v 1 2 0 7071 of the Werner states. On the other hand, Amoeba optimization provides us with a steering inequality for = m 4 settings which is violated above the parameter = .
v 0 7842. Also, by setting Alice's 12 measurements to point toward the vertices of an icosahedron on the Bloch sphere, we get a more powerful steering inequality which is violated above = .
v 0 7423 (note that due to reflection symmetry of the icosahedron, it is enough to consider = m 6 vertices in the actual code). Therefore, there is no LHS(1) model below = .
v 0 7423, as depicted in Fig. 3. We state it as an open problem what the exact value of the critical v above which no LHS(1) model exists if m goes to infinity. On the other hand, there is a 465 setting Bell inequality 42,43 , which is violated by a Werner state above = .
v 0 7056, which implies that there exists no LHV model for the Werner state for > .
v 0 7056. Again, this bound is shown in Fig. 3. The above bounds entail that the ( ) LHS 1 set has portions outside the LHV (i.e. Bell local) set of states. This is the shaded region depicted in Fig. 3 and proves in turn the existence of point A in the schematic Fig. 2 (i.e., a state which is nonlocal and admits a LHS(1) model).
Since the set of states admitting a LHV model is a strict superset of the states admitting a LHS model 44,45 , it follows that the ( ) LHS 1 set is strictly different from the LHS set. We provide an alternative proof of this fact in the Methods section.
Note that the hierarchy Notice that this state is the two-qubit reduced state of the n-qubit W n state 37 for = / p n 2 . We note that for this particular = / p n 2 , the state is ( − ) n 1 -symmetric extendable 46 , hence there is a LHV model (and therefore also a LHS model) for − n 1 settings (with arbitrary number of outcomes). The LHS bound seems to be tight, as we could recover the bound of = / v n 2 up to numerical precision for ≤ n 6 settings using the SDP method developed in ref. 18 (please see second column of Table 1). This correspondence suggests that there is no LHS model for any finite p > 0 if the number of settings is large enough. Using our numerical search described in the Methods section, we find the threshold values p regarding the LHS(1) model in the third column of Table 1. On the other hand, we conjecture that the local bound p LHV is / .  1 2 0 7071. Our conjecture is based on a linear programming approach combined with a heuristic search over the measurement angles 47 of Alice and Bob to get an upper bound on p LHV for a given number of measurement settings. For two settings per party ( = m 2), we have the (analytical) upper bound of / 1 2 on p LHV . However, by moving up to = m 8 settings per party, using numerical computations, this upper bound value did not become lower. Note that due to the heuristic nature of the search we cannot guarantee that / 1 2 is the optimal value. Though, at this level of complexity we are fairly confident about the validity of this threshold value. Moreover, we conjecture that this bound cannot be beaten beyond = m 8 settings as well. Similar conclusion was drawn by Amirtham 48 . The above results (modulo our conjecture) indicate a point B in Fig. 2, displaying portions of the LHV set lying outside the ( ) LHS 1 set.

Steering-like inequalities with any finite number of communication.
In this subsection, we go beyond the case of one bit of communication (i.e., = c 1). To this end, we construct a steering inequality with = ( ) c d log 2 number of bits of communication, which can be violated by a 2-qubit maximally entangled state for any finite d if the number of settings m for Alice is large enough. Violation implies that there is no LHS(c) model for any finite number of c bits for a 2-qubit maximally entangled state. Combing this result with a recent work of ref. 20 entails that the same applies to any pure bipartite entangled state. More details about equivalence of states with respect to LHS(c) models are found in the Methods section.
Let us also remark that there is an interesting nested feature of the sets ( ) LHS c , namely they satisfy    (7), in case of communicating = c d log 2 bits (i.e., a d level classical message is sent from Alice to Bob), the L c value corresponding to the LHS(c) limit is defined by maximizing the following expression from Alice to Bob. Note, however, that as m goes to infinity the L c value becomes close to 1, resulting in a very poor noise resistance. We pose it as an intriguing problem to construct more powerful steering-like inequalities exhibiting better noise tolerance.
As an experimentally relevant case, let us choose = m 5, in which case the number of communicated bits is In that case, the LHS(c = 2) bound in formula (20) becomes = . L 0 9804 2 . Due to our result, a Werner state with visibility larger than L 2 along with well-chosen measurements violates this two-bit bound L 2 . In light of recent experimental progress demonstrating EPR steering [24][25][26] , we believe this bound should be overcome in state-of-the-art photonic experiments.

Discussion
In this paper, we extended the notion of Bell inequalities with auxiliary communication to the EPR steering scenario. To do so, we introduced a general framework based on an efficient SDP method. With this tool, we characterized the set of bipartite states which admits a local hidden state model augmented with 1 bit of classical communication (the so-called LHS(1) model) from untrusted Alice to trusted Bob. This ( ) LHS 1 set of states was proven to be strictly larger than the set of states admitting an LHS model (for projective measurements). Moreover, this ( ) LHS 1 set turns out to have portions outside the LHV set. On the other hand, we conducted an extensive numerical search which indicates that there exist local two-qubit quantum states, which nevertheless cannot be described by an LHS(1) model (assuming projective measurements). We also showed that an infinite amount of classical communication is required from Alice to trusted Bob to simulate the EPR-steering statistics arising from any bipartite pure entangled state.
There is a number of open questions which deserves further investigations. bound). Would it be possible to close this gap either by improving the lower bound or by improving the upper bound value? • Based on extensive numerical search we conjectured that the LHV set has portions outside the ( ) LHS 1 set of states. Is there a formal proof of this conjecture? • We quantified quantum steering with the amount of classical communication between the two parties. What happens if we consider other resources such as certain no-signalling resources? • Another question concerns one-way steerability of quantum states 49 . As an extension of one-way steerable states, we ask whether there exists a bipartite quantum state, such that Alice can steer Bob's state, however, it is impossible for Bob to steer Alice's state even allowing 1 bit of classical communication between them. • It would be also interesting to see how our results relate to LHV models allowing classical communication.
We know that 2 bits of communication suffice to simulate projective measurements on any two-qubit entangled state 34 . However, in the EPR steering scenario due to our results any finite number of bits is not enough. Does the same result hold true if we add some noise to the singlet state? • In case of = c 1 we have shown that the nested relation holds true. It would be interesting to see if this strict hierarchical relation generalizes to any finite number of c bits.
• Finally, it is also interesting to consider the extension of the steering task with communication to the multipartite realm (see, e.g., refs 23,29,50).

Methods
Semidefinite program to compute critical weights. Here we provide an SDP program to compute an upper bound on v crit in the formula (11). Assuming the form of the state (11), the assemblage σ a x defined by Eq. (9) in function of parameter v is given by

LHS(1) model for a Werner state.
Here we present a simulation protocol which gives an LHS model augmented with 1 bit of classical communication for the 2-qubit Werner state up to the visibility / 1 2 . We proceed in two steps. Our first protocol will work for visibility up to / 2 3, whereas the second one, building on the first protocol, works up to the higher visibility of / 1 2 . Our first one bit protocol is as follows. Alice and Bob share two independently and uniformly distributed random variables λ 1 and λ 2 over the unit sphere. The protocol proceeds as follows: The goal of this protocol is to reproduce the assemblage (9) originating from a two-qubit Werner state (12). The assemblage of a Werner state is given by  Comparing this formula with (25), the critical visibility v is given by the closed form expression where we used the fact that because of spherical symmetry we can take for uniformly distributed u 1 , u 2 in the interval , [0 1]. We now improve the above one bit protocol up to visibility = / v 1 2. To this end, we use the same protocol as before, but this time λ 1 and λ 2 are correlated variables. We choose them as λ = Ue z 1 , λ = Ue x 2 , such that the × 2 2 matrix U is distributed according to the Haar measure on ( ) SU 2 . In that case, the protocol gives where integration was performed over the unit sphere.
The LHS(1) set is strictly larger than the LHS set. Here we prove the title. For the two-qubit Werner states the LHS set of states is bounded by = / v 1 2 8,9,28 . Hence any LHS(1) model giving a threshold value higher than = / v 1 2 does the job. Hence, the LHS(1) model with threshold = / v 1 2 presented in Methods section previously provides us with the desired proof. We give here a LHS(1) model with a smaller threshold = .
v 0 5899. Though, this value is worse than our previous threshold = / v 1 2, the present proof is completely different and maybe of independent interest. In fact, the proof below for an LHS(1) model is a special instance of the algorithmic procedure to construct LHS models appeared in refs 52,53.
Let us pick the icosahedron, a platonic solid which has 12 vertices and 20 faces. Using the SDP defined in Methods A, we compute = . v 0 7423 crit for the measurements pointing toward the 12 vertices of the icosahedron. Note that the icosahedron has a reflection symmetry through the center, and it is enough to take only 6 of its vertices: where ϕ is the golden ratio ϕ = ( + )/ 1 5 2. Following refs 50-53, any vector  u which is within the (largest) inscribed sphere of this icosahedron, can be expressed as the convex combination of the 12 vertices (the ones in (31) and its inverted versions). The computation takes roughly 1 min on a normal desktop PC. If we normalize the vertices (31) such that all of them have unit length from the origin, the radius of the inscribed sphere is = ( + )/ ∼ .  No QS(∞) model for Werner states for v > 1/2. Here we provide a sketch of the proof for the title. We first give an inequality which proves the (known) result that the Werner state is entangled above = / v 1 3. The same inequality will be used to prove that the Werner state does not admit a QS(∞) model for > / v 1 2. The inequality is as follows. where  u is distributed uniformly on the unit sphere and u z denotes ⋅   e u z . Note that the maximum of the right-hand-side of inequality (32) is 1, attainable with a maximally entangled two-qubit state (i.e. Werner state with v = 1). Then we obtain the result that Werner states with v > 1/3 violate the quantum separability inequality (32), hence they are entangled in this range.
Next we deal with = ∞ c . Since the amount of communication is unbounded, Alice is able to communicate her measurement settings x to Bob, which permits Bob to adjust his hidden state σ λ B according to x. This in turn implies the maximum , a n d σ ψ θ ψ θ = ( ) ( ) tr A , w h e r e ψ θ θ θ ( ) = + sin 01 cos 10 . This result implies that ρ ( ) θ v has a LHS (1) model for any θ > 0 below = / v 1 2. However, this threshold may not be tight, that is, it does not rule out the possibility of a higher θ ( ) v crit for θ π ≤ /4.