Direct sum decomposition into orthogonal subspaces

[sindhuja]

Hello All, I am trying to understand quantum information processing. I am reading the book "Quantum Computing A Gentle Introduction" by Eleanor Rieffel and Wolfgang Polak. I want to understand the following better:

" Let V be the N = 2^n dimensional vector space associated with an n-qubit system. Any device that measures this system has an associated direct sum decomposition into orthogonal subspaces V = S1 ⊕ · · · ⊕ Sk for some k ≤ N. The number k corresponds to the maximum number of possible measurement outcomesfor a state measured with that particular device."

Could anyone explain the intuition behind this statement. I think it is a quiet simple beginner level concept which I have not been getting a satisfactory explanation for. Thank you!

 

[vanhees71]

I don't know this book, but I guess what's meant is the following: If you measure some observable (in this case on a system $n$ qubits), this observable is described by some self-adjoint operator on the $2^n$-dimensional Hilbert space, describing the $n$-qubit system. You can think of it as a matrix $\hat{A}$ operating on $\mathbb{C}^{2^n}$-column vectors, which are the components of a vector wrt. an aribtrary orthonormal basis (e.g., the product basis of the $n$ qubits). The possible outcomes of measurements are the eigenvalues of this operator/matrix. To each eigenvalue $a$ there is at least one eigenvector. There's always a basis of eigenvectors, and you can always choose this basis to be an orthonormal set. The eigenvectors for each eigenvalue $a$ span a subspace $S_i=\mathrm{Eig}(a_i)$. The vectors in eigenspaces of different eigenvalues are always orthogonal to each other (again, because the matrix is self-adjoint). Thus the entire vector space is decomposed into the orthogonal sum of these eigenspaces, $V=S_1 \oplus S_2 \oplus \cdots \oplus S_k$, where the $a_i$ with $i \in \{1,\ldots,k \}$ are the different eigenvectors. Of course the dimensions of these subspaces are such that
$$\sum_{i=1}^k \mathrm{dim} \text{Eig}(a_i)=\mathrm{dim} V=2^n.$$

 

[Haborix]

You can also think of it as saying that when there are degenerate eigenvalues, a measuring device capable of measuring only the associated observable cannot give complete state information. The measuring device is incapable of resolving the decomposition of the state within the degenerate subspace, $S_i$.