Contextual advantage implies limited distinguishability in any physical theory

Technical Summary: Contextual Advantage Implies Limited Distinguishability in Any Physical Theory

Problem Statement
A central question in quantum information theory is determining whether a specific set of states can provide an advantage in information-processing tasks. Two such fundamental questions are: (1) How well can a set of states be discriminated (State Discrimination, SD)? and (2) Can a set of states power tasks where the advantage stems from generalized contextuality? While it is known that perfect discrimination requires linear independence, the relationship between the degree of distinguishability and the capacity for nonclassical (contextual) advantage in general physical theories remains less explored. Specifically, it is unclear if imperfect distinguishability is merely a necessary condition for contextuality or if there are strict quantitative thresholds below which contextuality is impossible.

Methodology
The authors work within the framework of Generalized Probabilistic Theories (GPTs), a formalism that encompasses classical and quantum theories as special cases while allowing for broader classes of hypothetical physical models. The study focuses on prepare-and-measure scenarios, where systems are prepared via procedures $P$ and measured via $M$, yielding statistics $p(k|M, P)$.

The methodology proceeds in two main stages:

1. Geometric Analysis of State Discrimination: The authors analyze the performance of a finite set of states $\{s_k\}$ in minimum-error state discrimination tasks with prior probabilities $\{p_k\}$. They utilize the convex geometry of the state space $\bar{\Omega}$, specifically focusing on the property of linear dependence. They establish that if a set of states is linearly dependent, the average state $\bar{s} = \sum p_k s_k$ admits multiple linearly independent decompositions.
2. Connection to Generalized Contextuality: The authors link the geometric properties of the state set to generalized noncontextuality. They utilize the definition that a system is noncontextual if operationally equivalent procedures admit identical representations in an ontological model. They demonstrate that linearly independent sets of states always admit a noncontextual explanation, whereas contextuality requires linear dependence.

Key Contributions and Results

1. Universal Upper Bounds on State Discrimination (Theorem 1 & Corollary 1)
The paper establishes that linear dependence imposes strict, nontrivial upper bounds on the success probability ($P_{guess}$) of any state discrimination task using the full set of states.

• Theorem 1: For a linearly dependent set of states $\{s_k\}$ and any SD task with priors $\{p_k\}$, the optimal guessing probability is bounded by:
  $$P_{guess} \leq 1 - P_{\{q_k\}}$$
  where $P_{\{q_k\}}$ depends on the minimum prior $p_{min}$ and the total variation distance between the prior $\{p_k\}$ and any linearly independent decomposition $\{q_k\}$ of the average state $\bar{s}$.
• Corollary 1: This yields a universal constraint: $P_{guess} \leq 1 - \frac{1}{2}p_{min}$.
• Invariance (Corollary 2): These bounds are invariant under injective linear maps. This means the bounds apply to equivalence classes of state sets across different GPTs, provided the geometric relations (linear dependence/independence) are preserved. The bounds depend only on the prior distribution and the convex geometry of the states, not on the specific theory (quantum, classical, or hypothetical).

2. The Trade-off Between Contextuality and Distinguishability (Theorem 2)
The paper's second main result connects the bounds on SD to the capacity for contextual advantage.

• Theorem 2: If a set of states $\{s_k\}$ provides a nonclassical advantage in a contextuality-powered task, it must be linearly dependent. Consequently, for any SD task using the full set $\{s_k\}$, the success probability must obey the bounds derived in Theorem 1.
• Contrapositive: If a set of states allows for a discrimination success probability that violates these bounds (i.e., is "too high"), that set cannot provide an advantage in any contextuality-powered task. Any prepare-and-measure experiment using such a set admits a noncontextual ontological model.

3. Application to Generalized Parity-Oblivious Multiplexing (Example 1 & Corollary 3)
The authors apply these results to generalized parity-oblivious multiplexing (gOM) tasks.

• They show that any encoding satisfying the obliviousness constraints is necessarily linearly dependent.
• Corollary 3: If the overall success probability in a gOM task is "too high" (specifically exceeding $1 - \frac{1}{2d}$), it implies that while the full ensemble might be nonclassical, at least one sub-ensemble of the codebook (corresponding to a specific query) is sufficiently distinguishable to be classically explainable. This reveals a trade-off where high global success forces local classicality in sub-ensembles.

Significance and Claims
The paper claims to establish a universal limitation on which state sets can power contextual advantage. The significance lies in the following points:

• Beyond Idealized Limits: The results go beyond the known fact that perfect discrimination implies linear independence. The authors show that even imperfect distinguishability, if it exceeds specific geometric thresholds, precludes contextuality.
• Theory Independence: The bounds hold for any GPT, not just quantum mechanics. This clarifies the operational principles that constrain physical theories and distinguishes quantum theory from broader classes of models.
• Operational Trade-off: The work exposes a fundamental trade-off between two operational resources: distinguishability and contextuality. A set of states cannot be simultaneously "too distinguishable" (violating the derived bounds) and "contextually powerful."
• Practical Utility: The bounds provide a feasibility test for state sets in discrimination tasks without requiring the solution of complex optimization problems (like semidefinite programs). They allow researchers to discard candidate state sets that are too distinguishable to support contextuality or too dependent to support high-fidelity discrimination.

The authors emphasize that these results are derived from simple geometric properties (linear dependence) and do not rely on specific quantum features, thereby offering a robust, theory-independent perspective on the relationship between nonclassicality and information processing.