Critical properties of EPR steering

Technical Summary: Critical Properties of EPR Steering

Problem Statement
The paper investigates the conditions under which entangled two-qubit states, which are unsteerable under single-copy projective measurements (PVMs), become steerable when multiple copies are measured jointly. Specifically, it addresses the "activation" of EPR steering at the critical boundary of unsteerability. The study seeks to determine:

1. Whether a uniform copy number exists for the entire class of PVM-critical entangled states.
2. The role of entanglement in the joint measurements required for activation.
3. The relationship between the number of measurement settings, the number of hidden states in a Local Hidden State (LHS) model, and the ability to detect steering near the critical boundary.
4. Whether steering activation can occur while Bell nonlocality remains absent and whether the strength of steering violations can be unbounded with only two copies.

Methodology
The authors employ a combination of geometric characterizations of steering, convex duality, and tensor product analysis:

• Canonical Form and LHS Models: The analysis utilizes the canonical form of two-qubit states derived via invertible filtering. The existence of an LHS model is characterized by the inclusion relation between two convex sets: the set of conditional states reproducible by a hidden-state ensemble ($K_\mu$) and the set of states required by the quantum state ($N$). A state is PVM-unsteerable if and only if the minimum total weight of a positive measure satisfying specific moment constraints ($c_P$) is less than or equal to 1.
• Criticality Definition: A state is defined as PVM-critical if $c_P = 1$. At this threshold, the LHS model has no "freedom" to accommodate changes in the experiment.
• Two-Copy Activation Theorem: The authors prove that for any two PVM-critical states $\rho_1$ and $\rho_2$, the tensor product $\rho_1 \otimes \rho_2$ is steerable. The proof relies on showing that any assumed LHS model for the two-copy system leads to a contradiction regarding the first-order response of conditional states to entangling rotations.
• Entanglement Thresholds: The study analyzes the concurrence of the joint measurement basis vectors to determine the minimum entanglement required to break the LHS model.
• Finite Resource Constraints: The paper constructs specific states and uses symmetric extensions to demonstrate limitations imposed by finite measurement settings and finite copy numbers.
• Isotropic States and Amplification: High-dimensional isotropic states are used to calculate the ratio of quantum scores to LHS bounds, testing the amplification of steering violations.

Key Contributions and Results

1. Uniform Two-Copy Activation:
   The central theorem proves that every PVM-critical entangled two-qubit state becomes steerable with exactly two copies. This holds for both identical copies ($\rho \otimes \rho$) and distinct critical inputs ($\rho_1 \otimes \rho_2$). This establishes a uniform activation number of two for the entire critical boundary, contrasting with general entangled states where the required copy number may vary or be arbitrarily large.

2. Zero Entanglement Threshold for Measurements:
   The paper demonstrates that at the critical boundary, the threshold of entanglement required in the joint measurement to reveal steering is zero. Arbitrarily weakly entangled joint measurements (with arbitrarily small concurrence) suffice to activate steering. Crucially, this activation does not require the joint measurement to create entanglement between Bob's two systems; all conditional states on Bob's side can remain separable.

3. Infinite Hidden States and Setting Divergence:

   • Hidden States: A PVM-critical state requires an LHS model with infinitely many hidden states. Any finite ensemble fails to satisfy the steering constraints for all PVM directions simultaneously.
   • Measurement Settings: For any fixed number of measurement settings $m$, there exists a neighborhood of a critical input that remains unsteerable under all PVM families with at most $m$ settings. Consequently, the number of settings required to detect steering diverges as the input approaches the critical boundary.
   • Copy Number vs. Settings: Increasing the number of copies does not universally compensate for limited measurement choices. The authors construct critical states that remain unsteerable under any prescribed finite setting bound, even for arbitrary tensor powers, provided the settings are restricted.

4. Activated Bell-Local States:
   The authors construct fixed-dimensional states (specifically $2 \times 3$ systems) that are:

   • Unsteerable on one copy.
   • Steerable on two copies.
   • Bell local for all joint POVMs across any prescribed finite range of copies ($n_0$).
     This is achieved using an "erasure" mechanism that preserves forward steering while suppressing reverse steering and Bell nonlocality.

5. Unbounded Steering Violations:
   Addressing a question by Hsieh, Liang, and Lee, the paper provides an affirmative solution: two copies are sufficient to produce unbounded violations of steering inequalities. Using high-dimensional isotropic states, the ratio of the quantum score to the LHS bound scales as $\frac{1}{2} \log d$, which diverges as the dimension $d \to \infty$.

Significance
The paper identifies a distinct class of mixed entangled states where the transition from unsteerability to steerability is sharp and uniform: the critical boundary. The results clarify the distinct roles of three resources in revealing quantum correlations:

• Measurement Entanglement: At criticality, the threshold for measurement entanglement vanishes, yet the measurement must still be non-product to break the LHS model.
• Measurement Number: Finite measurement settings are insufficient to resolve steering arbitrarily close to the critical boundary; the setting requirement is unbounded.
• Collective Access: While two copies are sufficient for activation at the critical boundary, they do not automatically overcome restrictions on the number of settings.

The work establishes that the "critical" nature of these states is defined by the vanishing of the measurement-entanglement threshold and the necessity of infinite hidden states. It further demonstrates that steering activation can occur independently of Bell nonlocality and that the strength of steering violations can be arbitrarily amplified with just two copies in high dimensions. These findings refine the hierarchy of quantum resources (entanglement, steering, nonlocality) and the operational requirements for detecting them.