Is the optimal probability of error in discriminating $n$ states a function of the $n^2$ pairwise inner products?

I am interested in proving/understanding some relationship between state discrimination and inner products. Given a set of (pure) quantum states $$s_1, s_2, \ldots, s_n$$:

1. Is the optimal probability of error (PoE) in discriminating the given states -- a function of the set of $$n^2$$ pairwise inner products? I.e., if I know the $$n^2$$ pairwise inner products (without knowing the actual states), does that give me the optimal probability of error (PoE) too? This seems obvious, but I don't have a formal proof. Informally, I believe it's true because a set of $$n^2$$ pairwise inner products uniquely determines the equivalence class of the set of states (where two sets of states are in the same equivalence class if they can be transformed from one to another via rotation and/or renumbering which keeps the PoE equal).

2. Perhaps related to the above: If the sum of inner products is constant, then does it seem obvious/intuitive that the PoE would be minimized if the inner products are all equal? This is likely very non-trivial to prove -- but I would appreciate any intuition and/or insights. I haven't tried to prove it empirically yet; I'm more interested in theoretical insights/proofs.

Thanks.

• Are you talking about pure states specifically, or general quantum (potentially mixed) states? Commented Aug 1 at 8:47
• I'm primarily interested in pure states. Commented Aug 1 at 14:18

It's unclear what exactly the state discrimination task you have in mind is. Is $$S$$ known? An instructive example case to consider is $$|0\rangle, |+\rangle, |+i\rangle$$. The squared norm of the inner products between pairs is equal, but you can still discriminate.
• I think the OP is rather observing that the Gram matrix $G$ (containing pairwise inner products) should be related to the optimal success probability $p^*$ for a measurement to output $i$ when provided with state $\rho_i$. In your example, $G$ is diagonal $\Rightarrow p^*=1$. But is there a way to determine $p^*$ in general given $G$? Commented Aug 5 at 15:21
• @forky40 What do you mean the optimal success probability for a measurement to output $i$ when given $\rho_i$? Also, $G$ is not diagonal in the example I provided, and $p^* < 1$. Commented Aug 5 at 17:31
• I meant that if $\{h_1, \dots, h_n\} := H$ is a set of positive operators, the definition I'm used to is $p^* = \max_{H \text{ is POVM}} \sum_i \text{Pr}(\rho_i) \text{Tr}(\rho_i h_i)$, which is average error. Maybe the OP meant worst-case error though, I'm not sure. And yea, forget I mentioned your example its not actually relevant to what I was saying Commented Aug 5 at 18:04