Abstract
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.
Highlights
In 1935, Einstein, Podolsky and Rosen (EPR) questioned the completeness of quantum mechanics under the assumption of locality and reality[1] that underlie the classical world view
It is easy to verify that the GHZ state is the common eigenstate of the following four mutually commutative operators: σ1xσ2xσ3x, σ1xσ2yσ3y, σ1yσ2xσ3y, and σ1yσ2yσ3x, with the eigenvalues being +1, −1, −1, −1, respectively
We denote the supposedly definite values of σ1x, σ2y, ...as v1x, v2y, ...(with v’s being 1 or −1), a product of the last three operators, according to local hidden variables (LHV) models, yields v12y v22y v32y v1xv2xv3x = −1, in sharp contradiction to the first operator v1xv2xv3x =+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
Summary
In 1935, Einstein, Podolsky and Rosen (EPR) questioned the completeness of quantum mechanics under the assumption of locality and reality[1] that underlie the classical world view. By considering continuous-variable entangled state, EPR proposed a famous thought experiment that involves a dilemma concerning local realism against quantum mechanics. This dilemma is nowadays well-known as the EPR paradox. It is easy to verify that the GHZ state is the common eigenstate of the following four mutually commutative operators: σ1xσ2xσ3x, σ1xσ2yσ3y, σ1yσ2xσ3y, and σ1yσ2yσ3x (here σ1x denotes the Pauli matrix σx measured on the 1st qubit, for the others), with the eigenvalues being +1, −1, −1, −1, respectively. In the original formulation of the EPR paradox[1], a bipartite entangled state is considered which is a common eigenstate of the relative position x1 − x2 and the total linear momentum p1 + p 2 and can be expressed as
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