Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem ever since the concept of steerability was first defined. In this paper, using our recent geometrical approach to steerability, we suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. As an example, we calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently suggested local hidden state model.
Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem ever since the concept of steerability was first defined. In this paper, using our recent geometrical approach to steerability, we suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. As an example, we calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently suggested local hidden state model.
According to the geometric characterization of measurement assemblages and local hidden state (LHS) models, we propose a steering criterion which is both necessary and sufficient for two-qubit states under arbitrary measurement sets. A quantity is introduced to describe the required local resources to reconstruct a measurement assemblage for two-qubit states. We show that the quantity can be regarded as a quantification of steerability and be used to find out optimal LHS models. Finally we propose a method to generate unsteerable states, and construct some two-qubit states which are entangled but unsteerable under all projective measurements.
When two qubits, A and B, are in an appropriate state, Alice can remotely steer Bob's system B into different ensembles by making different measurements on A. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on A (made by Alice) and an ensemble of B (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of B corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the four-dimensional geometry of Alice's steering outcomes completely determines both the nonseparability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into nonseparable and steerable classes is equivalent to geometrically classifying certain four-dimensional skewed double cones.
ABSTRACT Quantum steering refers to the phenomenon where one quantum system can instantaneously influence another via local measurements. The critical radius, proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)], is a sufficient‐and‐necessary steering criterion (SNSC). However, its broad applicability has been restricted by the absence of analytical solutions for general two‐qubit states. To address this, we develop the first comprehensive analytical framework for the critical radius, unifying theory and practice. We first derive an exact SNSC for all two‐qubit T‐states under infinite measurements and verify its equivalence to the critical radius. Leveraging the critical radius's symmetry and concavity, we generalize the framework to arbitrary two‐qubit states, aiming to transform the upper and lower bounds of the critical radius into an equal pattern via optimized local operations, thereby deriving an accurate formula for the critical radius. Critically, we discover the first exact analytical SNSCs for two distinct classes of asymmetric states, precisely mapping the boundary between steerable and unsteerable regions. These breakthroughs eliminate reliance on the numerical semidefinite programming methods, provide an operationally efficient method for discovering the hidden steerability.
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.
Quantum steering can be detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Here we discuss the converse problem, namely, ensuring that an entangled state is unsteerable and hence Bell local. We present a simple criterion, applicable to any two-qubit state, that guarantees that the state admits a local hidden state model for arbitrary projective measurements. Specifically, we construct local hidden state models for a large class of entangled states, which thus cannot violate any steering or Bell inequality. In turn, this leads to sufficient conditions for a state to be only one-way steerable and provides the simplest possible example of one-way steering. Finally, by exploiting the connection between steering and measurement incompatibility, we give a sufficient criterion for a continuous set of qubit measurements to be jointly measurable.
Quantum steering is the ability that one system affects another one without delay by performing local measurements. During the past few years, the research on quantum steering has been verified in a number of experiments by the choices of measurements. But this research cannot detect all steerable states. Recently, the definition of critical radius was proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)]. The critical radius can detect all steerable states for two-qubit states. Therefore the critical radius can be regarded as the best method for detecting the steerability of two-qubit states. However, relevant experimental research is still lacking. In this work, for comparison, we experimentally investigate three steering criteria based on the critical radius, the geometric Bell-like inequality, and the three-setting measurements steering inequality and find that the critical radius can detect the most steerable states. Furthermore, we demonstrate that the critical radius satisfies the scaling property. The results show that we only need to calculate the critical radii of some quantum states to obtain the critical radii of more scaling quantum states. Thus this property can provide the convenience for detecting the steerability of many quantum states.