Trading Quantum Ensembles

We consider an example scenario where we require several copies of a pure quantum state $|\psi\rangle$ for some quantum information processing task. Due to practical limitations, we only have access to $N = 10^3$ depolarized copies of $|\psi\rangle$ such that the fidelity $F$ of each copy with $|\psi\rangle$ is $0.75$. We denote this quantum asset with the ensemble $\mathcal{A}: (10^3, 0.75)_{|\psi\rangle}$. A genie appears and offers to trade $\mathcal{A}$ with either $\mathcal{B}: (10^4, 0.65)_{|\psi\rangle}$ or with $\mathcal{C}: (10^2, 0.90)_{|\psi\rangle}$. Should we accept the trade with either of these two ensembles? In this article, we attempt to answer this question with arbitrary $N$ and $F$. More specifically, we derive resource equivalence curves from quantum resource theory of purity, quantum state distinguishability, quantum state purification, and quantum state tomography. These curves enable ranking of these ensembles according to their operational usefulness for these tasks and allow us to answer the question of trading the aforementioned ensembles.