In Nielsen and Chuang exercise 2.64, the following problem is given:
Suppose Bob is given a quantum state chosen from a set $\{ \lvert \psi_1 \rangle, \ldots , \lvert \psi_m \rangle \}$ of linearly independent states [in some Hilbert space]. Construct a POVM $\{E_1, E_2, \ldots, E_{m+1} \}$ such that if outcome $E_i$ occurs, $1 \le i \le m$, then Bob knows with certainty that he was given the state $\lvert \psi_i \rangle$. (The POVM must be such that $\langle \psi_i | E_i | \psi_i\rangle > 0$ for each i.)
Here a POVM is a positive operator-valued measurement, or a set $\{E_i\}$ of positive (semidefinite) operators satisfying the completeness condition $$ \sum_i E_i = I. $$ Note that in this problem $\lvert \psi_i \rangle$ are not assumed to be orthogonal.
My approach to this problem is to construct each $E_i$ for $i \le m$ as a scaled projector: $$ E_i = \alpha_i P_i, $$ where $P_i$ is the projector onto the orthogonal complement of $\mathrm{Span}(\lvert \psi_1 \rangle, \ldots, \lvert \hat{\psi_i} \rangle, \ldots, \lvert \psi_m \rangle)$ in $\mathrm{Span}(\lvert \psi_1 \rangle, \ldots, \lvert \psi_m \rangle)$ satisfying $\langle \psi_i | P_i | \psi_i \rangle > 0$, and $\alpha_i$ is some positive real number.
Then we have the desired properties that each $E_i$ is positive, and $\langle \psi_i | E_j | \psi_i \rangle = \delta_{ij} \beta_i$ for some real positive $\beta_i$. Indeed, $E_i$ has eigenvalues $\alpha_i$ and $0$; $E_j | \psi_i \rangle = 0$ if $i \ne j$ as $P_j$ is a projector onto a space orthogonal to $\lvert \psi_i \rangle$; and $$\langle \psi_i | E_i | \psi_i \rangle = \alpha_i \cos \theta_i \langle \psi_i | \psi_i \rangle = \beta_i > 0, $$ where $\theta_i$ is the angle between $\lvert \psi_i \rangle$ and $P_i \lvert \psi_i \rangle$.
Then for any choice of $\{\alpha_i\}$ we have constructed positive $E_i$ for $i \le m$, but this set of positive operators is not necessarily complete. To complete the set, let $E_{m+1} = I - \sum_{i \le m} E_i$. It seems that as long as we can ensure $E_{m+1}$ is positive, then we have constructed the desired POVM. However, I am not sure how to assign $\{\alpha_i\}$ in order to do so.
Naively, I can't see why we don't have a great deal of freedom over $\alpha_i$. As a sum of Hermitian operators, $E_{m+1}$ is automatically Hermitian. And $\langle \psi_i | E_{m+1} | \psi_i \rangle = \langle \psi_i | \psi_i \rangle - \beta_i \ge 0$ so long as $\alpha_i \cos \theta_i \le 1$. However, this so-called POVM would be able to distinguish the states $\lvert \psi_i \rangle$ perfectly if we were allowed to choose $\alpha_i = 1/\cos \theta_i$, in violation of the no-cloning theorem (as shown in Nielsen–Chuang, exercise 1.2, with stronger bounds implicit in Box 2.3).
I feel I must be mistaken about the criterion for $E_{m+1}$ to be positive—it does not seem sufficient to ensure that $\langle \psi_i | E_{m+1} | \psi_i \rangle \ge 0$ for all $i$, even though they are linearly independent.
Is this a fruitful approach for this problem? If so, what conditions on $\{\alpha_i\}$ would ensure $E_{m+1}$ is positive? What have I missed?