Quantum nonlocality without entanglement and state discrimination measures

Technical Summary: Quantum Nonlocality Without Entanglement and State Discrimination Measures

Problem Statement
The paper addresses the phenomenon of "quantum nonlocality without entanglement," defined as the inability of an ensemble of product states to be optimally discriminated using Local Operations and Classical Communication (LOCC), despite the fact that global measurements can achieve a higher success probability. A central question in this field is whether this nonlocality is an intrinsic property of the state ensemble or if it depends on the specific measure of state discrimination employed. Specifically, the authors investigate whether there exist product ensembles that exhibit nonlocality under one discrimination measure (e.g., minimum-error) but not under another (e.g., unambiguous discrimination).

Methodology
The authors construct a specific family of bipartite product ensembles, denoted as $E_T$, consisting of six linearly independent, equally probable states in $\mathbb{C}^3 \otimes \mathbb{C}^3$. These states are formed by tensor products of three symmetric pure states $\{|\psi_i\rangle\}_{i=1}^3$ with equal pairwise inner products $s \in (0,1)$, where the indices of the tensor factors are distinct ($i \neq j$).

The study employs two distinct discrimination frameworks:

1. Minimum-Error Discrimination: The goal is to maximize the probability of correctly identifying the state, allowing for errors. The authors utilize a theorem by Chefles and Barnett (referenced as [17] in the text) which reduces the problem of optimal minimum-error discrimination for linearly independent pure states to the problem of perfect discrimination of a unique orthonormal basis derived from the Square-Root Measurement (SRM).
2. Unambiguous Discrimination (UD): The goal is to identify the state with zero error, accepting a probability of inconclusive outcomes. This requires the states to be linearly independent. The authors formulate the optimization of UD success probability as a Semidefinite Program (SDP) and exploit the permutation symmetry of the ensemble to simplify the solution.

To extend these findings, the authors generalize the construction to multipartite systems ($N$ parties, $k$ subsystems) and provide numerical evidence using SDPs to compare LOCC performance against Positive Partial Transpose (PPT) measurements and global SRM performance.

Key Contributions and Results

• Measure-Dependent Nonlocality: The primary result (Theorem 1) demonstrates that for the constructed ensemble $E_T$:

  • Minimum-Error: LOCC is strictly suboptimal. The global optimum is achieved by the SRM, which corresponds to a unique orthonormal basis $\tilde{B}$. The authors prove that the elements of $\tilde{B}$ are entangled and cannot be perfectly discriminated by LOCC (Lemma 8). Consequently, $p^{me}_L(E_T) < p^{me}_G(E_T)$.
  • Unambiguous Discrimination: LOCC is optimal. The authors explicitly construct a local protocol that achieves the global optimum success probability of $(1-s)^2$. This is proven by showing that the minimum eigenvalue of the Gram matrix of the ensemble equals $(1-s)^2$, and a local strategy attains this bound. Thus, $p^{ud}_L(E_T) = p^{ud}_G(E_T)$.

• Multipartite Extension: The authors extend the construction to ensembles $E_{N,k}$ involving $k$ parties and $N$ local states. Numerical investigations for cases such as $E_{3,3}$, $E_{4,2}$, and $E_{4,3}$ provide strong evidence that the separation persists: LOCC remains suboptimal for minimum-error discrimination (showing a gap between SRM and PPT/LOCC performance) but achieves the global optimum for unambiguous discrimination.

• Structural Insights: The paper identifies that the nonlocality in the minimum-error case arises because the optimal measurement basis (SRM) consists of entangled states that cannot be perfectly distinguished locally. Conversely, for unambiguous discrimination, the symmetry of the ensemble allows a local protocol to match the global bound without requiring entangled measurement operators or classical communication to improve the success rate beyond a specific local strategy.

Significance and Claims
The paper claims to provide a positive answer to the question of whether nonlocality without entanglement is measure-dependent. It establishes that a set of product states can exhibit nonlocality under the minimum-error criterion while being locally distinguishable under the unambiguous criterion.

The authors highlight that this result challenges the notion that nonlocality is an absolute property of a state ensemble. Instead, it is contingent upon the discrimination task. The paper notes that while the double trine ensemble exhibits nonlocality under both measures, the constructed ensemble $E_T$ exhibits a "separation" where LOCC fails for one measure but succeeds for the other.

Furthermore, the authors discuss that for the unambiguous case, the optimal success probability can be achieved even without classical communication between parties, as the local measurement outcomes do not require updating the prior probabilities of the other parties to maintain optimality.

The work concludes by posing open questions regarding the precise gap between LOCC and PPT measurements in the minimum-error setting and whether the specific features of the construction (equiangularity and label correlations) are necessary conditions for such measure-dependent separations. The authors suggest that their construction serves as a foundational instance for a broader framework of measure-dependent nonlocality.