Sharp Contradiction for Local-Hidden-State Model in Quantum Steering

In quantum theory, no-go theorems are important as they rule out the existence of a particular physical model under consideration. For instance, the Greenberger-Horne-Zeilinger (GHZ) theorem serves as a no-go theorem for the nonexistence of local hidden variable models by presenting a full contradiction for the multipartite GHZ states. However, the elegant GHZ argument for Bell’s nonlocality does not go through for bipartite Einstein-Podolsky-Rosen (EPR) state. Recent study on quantum nonlocality has shown that the more precise description of EPR’s original scenario is “steering”, i.e., the nonexistence of local hidden state models. Here, we present a simple GHZ-like contradiction for any bipartite pure entangled state, thus proving a no-go theorem for the nonexistence of local hidden state models in the EPR paradox. This also indicates that the very simple steering paradox presented here is indeed the closest form to the original spirit of the EPR paradox.

where |0〉 and |1〉 are the eigenstates of the Pauli matrix σ z with the eigenvalues + 1 and − 1. respectively. It is easy to verify that the GHZ state is the common eigenstate of the following four mutually commutative operators: σ 1x σ 2x σ 3x , σ 1x σ 2y σ 3y , σ 1y σ 2x σ 3y , and σ 1y σ 2y σ 3x (here σ 1x denotes the Pauli matrix σ x measured on the 1st qubit, similarly for the others), with the eigenvalues being + 1, − 1, − 1, − 1, respectively. However, a contradiction arises if one tries to interpret the quantum result with LHV models. Specifically, we denote the supposedly definite values of σ 1x , σ 2y , … as v 1x , v 2y , … (with v's being 1 or − 1), then a product of the last three operators, according to LHV models, yields , in sharp contradiction to the first operator v 1x v 2x v 3x = + 1. Such a full contradiction "1 = − 1" indicates that the GHZ theorem is a no-go theorem for quantum nonlocality, i.e., there is no room for the LHV model to completely describe quantum predictions of the GHZ state. The GHZ theorem has Scientific RepoRts | 6:32075 | DOI: 10.1038/srep32075 already been verified by photon-based experiment 6 , and recently a fault-tolerant test of the GHZ theorem has also been proposed based on nonabelian anyons 7 .
In the original formulation of the EPR paradox 1 , a bipartite entangled state is considered which is a common eigenstate of the relative position −x x 1 2 and the total linear momentum +p p 1 2 and can be expressed as  where r > 0 is the squeezing parameter, Since the discovery of the EPR paradox, the question of whether the original EPR state possesses the LHV models has pushed many researchers to achieve intriguing and thought provoking results [9][10][11][12][13][14] . Bell first showed that the Wigner function of the EPR state, due to its positive definiteness, can directly be used to construct the LHV models 9 . However, attempt has also been made to reveal its nonlocality in phase space by considering displaced parity operators upon the NOPA state in the large r limit 10 . Moreover, maximal violations of the EPR state by multicomponent Bell's inequalities have also been investigated in refs 15,16. Very recently the notion of "steering" 17,18 has stimulated people to reconsider the exact implication of the EPR argument. For instance, Werner has remarked on why Einstein did not go all the way to discover Bell's inequality 19 Steering is indeed a quite old concept. In response to the EPR paper 20 , Schrödinger, who believed the validity of quantum mechanical descriptions of Nature, introduced in the same year of EPR's paper a term "steering" to depict the "spooky action at a distance" which was mentioned in the EPR paper. Specifically, steering in a bipartite scenario describes an ability of one party, say Alice, to prepare the other party's (say Bob's) particle with different quantum states by simply measuring her own particle with different settings. This is also at the heart of remote state preparation protocol using EPR state 21 . However, steering lacked operational meanings, until in the year 2007 Wiseman et al. 17,18 gave a rigorous definition of it through the quantum information task. It then turns out that the EPR paradox concerns more precisely the existence of local hidden state (LHS) models, rather than that of LHV models leading to Bell's inequality.
That is, the exact type of quantum nonlocality in the EPR paradox is EPR steering, rather than Bell nonlocality. After that, there has been rapid development in EPR steering both theoretically and experimentally [22][23][24][25][26] , such as in the test of steering inequalities [27][28][29][30] and the experimental observation of one-way EPR steering 31 .
Thus, a natural question arises: since there exist a simple GHZ paradox, i.e., "1 = − 1", which rule out the LHV models more uncompromisingly than Bell inequalities, one may ask whether a similar contradiction can be found so as to completely rule out the LHS models, especially for the EPR state. The merits of confirmatively answering this question include not only finding out the aforementioned missing piece of proofs of steering in analogy to proofs of Bell nonlocality, but also accomplishing the demonstration of the EPR paradox in its most original sense.
The aim of this paper is to present a very simple steering paradox, i.e., "2 = 1", which intuitively demonstrates the steerability for the EPR state, directly confirming that EPR steering is exactly the type of quantum nonlocality inherited in the EPR paradox, henceforth proving a no-go theorem for nonexistence of LHS models in EPR's original sense.

Results
Simple steering paradox in two qubits. We shall show that in the original EPR's scenario, there exists a simple steering paradox that leads to "2 = 1". A two-setting EPR steering scenario together with a bipartite entangled state are sufficient to demonstrate this full contradiction.
To illustrate the central idea, let us first consider the two-qubit case. In a two-setting steering protocol of ˆn n { , } 1 2 (with ≠n n 1 2 ), Alice prepares a two-qubit state ρ AB , she keeps one and sends the other to Bob. Bob asks Alice to perform his choice of either one of two possible projective measurements (i.e. two-setting) â n 1  and â n 2  on her qubit and tell him the measurement results of a. Here is the projector, with = n n n n ( , , ) x y z the measurement direction, a (with a = 0, 1) the Alice's measurement result, the 2 × 2 identity matrix, and σ σ σ σ =  ( , , ) where ℘ ξ (with ℘ > ξ 0) and ξ ℘â n ( , ) are probabilities satisfying ∑ ℘ = ξ ξ 1, and ξ ∑ ℘ = a n ( , ) 1 a for a fixed ξ, and ρ B = tr A (ρ AB ) is Bob's reduced density matrix (or Bob's unconditioned state) 17,18 .
Then, Bob will check the following set of four equations: If these four equations have a contradiction (or say they cannot have a common solution of ρ ℘ ξ ξ { } and ξ ℘â n ( , )), then Bob is convinced that a LHS model does not exist and Alice can steer the state of his qubit. Now, let the state ρ AB be an arbitrary two-qubit pure entangled state, which is given in its Schmidt form as where θ ∈ (0, π/2). The pure entangled state ρ AB = |Ψ (θ)〉 〈 Ψ (θ)| has a remarkable property: Bob's normalized conditional states ρ ρ ρ =  ˆ/tr( ) a n a n a n are always pure, and ρ ρ ρ ρ ≠ˆˆ{ , } { , }  , this leads to a full contradiction of "2 = 1". The above simple paradox "2 = 1" offers a transparent argument of nonexistence of LHS models (or existence of EPR steering) for a two-qubit pure entangled state. The subtlety of the paradox lies in the fact the wavefunction |Ψ (θ)〉 can have different decompositions, such as with ± = ± ( 0 1 ) 1 2 and χ θ θ = ± ± cos 0 sin 1 . In practice, the two-setting protocol can be chosen as ˆẑ x { , }. Namely, Bob asks Alice to measure her qubit along the ẑ-direction and the x-direction, respectively. Suppose Alice performs her measurement in the ẑ-direction (or the x-direction), for convenient, one may denote the set of her projectors as  = Generalization to bipartite high-dimensional systems. Suppose in the steering scenario, the quantum state that Alice prepares is a pure entangled state of two d-dimensional systems (two-qudit), then one can have the same simple paradox "2 = 1".
Let us consider the two-qudit pure entangled state in its Schmidt form In principle, the choice of  and  is rather arbitrary, as long as any element in  does not fully overlap with that in . For simplicity  and  here can be taken as two of the mutually unbiased bases for a d-dimensional system, such that |〈 m|m′ 〉 | 2 = 1/d for any pair of m and m′ . After Alice's measurements, Bob obtains 2d conditional states Similarly, Bob can check the following set of 2d equations: . Due to the fact that a density matrix of pure state can only be expanded by itself, therefore, from equation (13)  From (15), one sees that the left-hand side gives 2trρ B = 2 and the right-hand side gives trρ B = 1, leading to a full contradiction of "2 = 1". The above analysis is also valid when d tends to infinity. By chosing λ = m hr hr (tan ) cos m and let d → ∞ , then one can have a similar paradox "2 = 1" for the continuous-variable state |NOPA〉 , which includes the original EPR state by taking the infinite squeezing limit. Thus, we complete the demonstration of the simple steering paradox for the original EPR scenario, which is a no-go theorem for nonexistence of LHS models in the EPR paradox. In other words, the sharp contradiction "2 = 1" indicates that there is no room for the LHS description of any bipartite pure entangled state, including the original EPR state.   on |Ψ 〉 EPR that does not change the state |Ψ 〉 EPR ,  is a real number, and Thus the two-setting steering protocol can be chosen as

Discussions
The EPR paradox has resulted in search for local hidden variable models with locality and reality as starting points, but Bell's inequaliy rules out such mdels as the predictions of LHV models do not match quantum theory. The GHZ paradox demonstrates sharp contradiction between the predictions of local hidden variable theory and quantum mechanics without using any inequality. However, the GHZ paradox is not applicable to bipartite systems. Hardy did attempt to extend the all-versus-nothing argument to a two-qubit system to reveal Bell's nonlocality 32,33 , and this proof is usually considered as "the best version of Bell's theorem" 34 . However, Hardy's proof works for only 9% of the runs of a specially constructed experiment, and moreover, it is not valid for two-qubit maximally entangled state. Thus, in this sense, Hardy's proof may not be considered appropriately as the closest form to the spirit of EPR's original scenario. In summary, we have presented a simple steering paradox that shows the incompatibility of the local hidden state model with quantum theory for any bipartite pure entangled state, including the original EPR state. The full contradiction that results in "2 = 1"; not only intuitively demonstrates the steerability for the EPR state, directly confirming that EPR steering is exactly the type of quantum nonlocality inherited in the EPR paradox, but also indicates that the very simple steering paradox is the closest in its form to the spirit of the EPR paradox. Furthermore, if one considers the EPR steering scenario in k-setting, then following the similar derivation one can arrive at a full contradiction, i.e., "k = 1". We expect that the simple steering paradox can be demonstrated in both two-qubit system and continuous-variable system by photon entangled based experiments in the near future.

Methods
Detail derivation of the steering paradox for two qubits. It can be directly verified that, if the state ρ AB = |Ψ (θ)〉 〈 Ψ (θ)| is a pure entangled state, then ρ ρ ρ ρˆˆ, , , n n n n 0 1 0 1 1 1 2 2 are four different pure states. For example and for convenient, let us take Then in the two-setting steering protocol of ˆẑ x { , }, Bob asks Alice to perform his choice of either one of two possible projective measurements along the z-direction (with the projector â z  ) and the x-direction (with the projector â x ) on her qubit and tell him the measurement results of a (with a = 0, 1). More precisely, one has the projectors as  which is just the set of equations given in (9). It can be verified that It then turns out that there exists a local hidden state model, with Alice's strategy based on a single hidden state, that could simulate the above Bob's four conditional states: Thus, local hidden state model is possible for pure separable states.