On estimating Schatten norm and power distances between quantum states

We study the computational complexity of estimating the quantum Schatten $\alpha$-norm distance ${\rm T}_\alpha(\rho_0,\rho_1)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $\rho_0$ and $\rho_1$. This quantity serves as a lower bound on the trace distance and, for $\alpha > 1$, is interchangeable with its powered version $\Lambda_\alpha(\rho_0,\rho_1)$. For any constant $\alpha > 1$, we develop an efficient rank-independent quantum estimator for ${\rm T}_\alpha(\rho_0,\rho_1)$ with time complexity ${\rm poly}(n)$, achieving an exponential speedup over the prior best results of $\exp(n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). When $0<\alpha<1$ is a constant, the quantum Schatten $\alpha$-power distance $\Lambda_\alpha(\rho_0,\rho_1)$ becomes a distance metric. Accordingly, we provide a rank-efficient quantum estimator for this quantity. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $\alpha$-norm (QSD $_\alpha$), which involves deciding whether ${\rm T}_\alpha(\rho_0,\rho_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of $\alpha > 1$ and $0 < \alpha\leq 1$: 1. For any constant $\alpha>1$, QSD$_{\alpha}$ is $\sf BQP$-complete. 2. For any $1 \leq \alpha(n) \leq 1+{\rm negl}(n)$, QSD$_\alpha$ is $\sf QSZK$-complete, implying that no efficient quantum estimator for ${\rm T}_\alpha(\rho_0,\rho_1)$ exists unless ${\sf BQP}={\sf QSZK}$. This $\sf QSZK$-hardness result also extends to the promise problem defined by $\Lambda_\alpha(\rho_0,\rho_1)$ for constant $0<\alpha<1$. The hardness results follow from reductions based on new rank-dependent inequalities for ${\rm T}_\alpha(\rho_0,\rho_1)$ when $1\leq \alpha \leq \infty$ and for $\Lambda_\alpha(\rho_0,\rho_1)$ when $0<\alpha<1$, which are of independent interest.