Uniqueness of Spectral Decomposition In one of the papers (related to Quantum Computing) I am reading, I came across this statement which says,

An elementary result is that sets of orthogonal rank-one eigenprojectors of Hermitian operators
are not unique when the spectrum includes degenerate eigenvalues, and that uniqueness is recovered when
rank-one eigenprojectors are combined into full-rank eigenprojectors, corresponding to maximal subsets of
rank-one eigenprojectors for distinct eigenvalues.


For any finite-dimensional Hermitian operator $\rho$, there is a "unique" set of full-rank projectors $\Pi_k$ such that, $\rho = \sum_k Tr(\rho\Pi_k)\Pi_k$, which also satisfy $\sum_k\Pi_k = I $  and $\Pi_k\rho = \rho\Pi_k = Tr(\rho\Pi_k)\Pi_k$

Here I understand that Spectral decomposition may not be unique if there is degeneracy involved.
What exactly are the "full rank projectors" mentioned here, and why do they make the decomposition unique.
 A: Consider the following matrix:
$$A=\pmatrix{1 &0&0\\0&2&0\\0&0&2}$$
I can decompose $A$ as the sum of projectors onto 1D subspaces as follows:
$$A = 1 \cdot\pmatrix{1&0&0\\0&0&0\\0&0&0} + 2\cdot\pmatrix{0&0&0\\0&1&0\\0&0&0}+2\cdot\pmatrix{0&0&0\\0&0&0\\0&0&1}$$
But this decomposition is not unique, because I could also use this one:
$$A = 1 \cdot\pmatrix{1&0&0\\0&0&0\\0&0&0} + 2\cdot\pmatrix{0&0&0\\0&\frac{1}{2}&\frac{1}{2}\\0&\frac{1}{2}&\frac{1}{2}}+2\cdot\pmatrix{0&0&0\\0&\frac{1}{2}&-\frac{1}{2}\\0&-\frac{1}{2}&\frac{1}{2}}$$
Each projection matrix written above is a projection onto a 1D subspace of $\mathbb R^3$.  However, since the eigenspace corresponding to the eigenvalue $2$ is 2D, the second and third projections are not unique.
What I can do, however, is express $A$ as
$$A = 1 \cdot\pmatrix{1&0&0\\0&0&0\\0&0&0} + 2\cdot\pmatrix{0&0&0\\0&1&0\\0&0&1}$$
Now the first projection is onto a 1D subspace, but the second projection is onto the full 2D subspace corresponding to the eigenvalue $2$.  This is what your paper meant - the projections should be onto the full eigenspaces, not just 1D subspaces, if you want the decomposition to be unique.
