We study superspace concentration as a quantum resource, formalized through the focus measure F(\r{ho}) = λ_max(\r{ho}_super) - the largest eigenvalue of the reduced superspace state - which quantifies the capacity of a quantum system to concentrate informational weight into a preferred subspace of an extended degree-of-freedom space. We develop a complete resource-theoretic framework around this measure and validate its properties through GPU-accelerated numerical simulation. Analytic decoherence predictions are confirmed to machine precision (1.11 x 10^{-16}) for superspace dimensions dS in {2,4,8,16,32}. Focus monotonicity holds across 10,000 random states with zero violations under four focus-non-generating channels across six system configurations. Focused quantum states resist coherent unitary attacks with significantly greater resilience than standard fidelity predicts, with focus remaining above 0.9 at attack strength ε = 0.302 versus ε = 0.174 for fidelity. We further demonstrate that the focus measure and the U(dS)-asymmetry measure are operationally distinct: asymmetry remains near zero and provides no robustness signal under coherent and targeted attacks while focus tracks spectral concentration and remains robust until ε > 0.3. The connection between Grover's algorithm and superspace concentration is made explicit via the identity F(|ψ_k><ψ_k|) = P(marked), providing a resource-theoretic interpretation of oracle query complexity. Finally, we provide the first numerical characterization of the focus capacity gap ΔF, identifying a log_2(dS) scaling law confirmed for both product and correlated noise channels.

翻译：我们研究超空间浓度作为一种量子资源，通过聚焦度量 \( F(\rho) = \lambda_{\max}(\rho_{\text{super}}) \)（约化超空间态的最大特征值）加以形式化，该度量量化了量子系统将信息权重集中到扩展自由度空间中某一特定子空间的能力。我们围绕此度量发展了一个完整的资源理论框架，并通过GPU加速数值模拟验证其性质。解析退相干预言在超空间维度 \( d_S \in \{2,4,8,16,32\} \) 下被确认至机器精度（\(1.11 \times 10^{-16}\)）。聚焦单调性在10,000个随机态中成立，在六种系统配置下四种非聚焦生成通道中均无违规。聚焦量子态抵抗相干幺正攻击的韧性显著高于标准保真度预测：攻击强度 \( \varepsilon = 0.302 \) 时聚焦仍高于0.9，而保真度在 \( \varepsilon = 0.174 \) 时已下降。我们进一步证明聚焦度量与 \( U(d_S) \)-非对称性度量在操作上具有区别：在相干和定向攻击下，非对称性保持接近零且不提供鲁棒性信号，而聚焦追踪谱浓度并保持鲁棒直至 \( \varepsilon > 0.3 \)。通过恒等式 \( F(|\psi_k\rangle\langle\psi_k|) = P_{\text{marked}} \)，明确了Grover算法与超空间浓度的联系，为预言查询复杂度提供了资源理论解释。最后，我们首次对聚焦容量缺口 \( \Delta F \) 进行数值表征，确认了对于乘积与关联噪声通道均满足 \( \log_2(d_S) \) 标度律。