Projector

Let \; \mathcal{H}_1 be a closed subspace of the Hilbert space \; \mathcal{H} and \; \mathcal{H}_2 its orthogonal complement in \; \mathcal{H}. Then any vector \; \psi \in \mathcal{H} can be decomposed into its components belonging to \; \mathcal{H}_1, respectively \; \mathcal{H}_2 as follows: :\; \psi = \psi_1 + \psi_2, \quad \langle \psi_1,\psi_2\rangle =0 . '''Definition: The linear operator \; Q_1 on \; \mathcal{H}_1 such that ''' : \; Q_1\psi = \psi_1 , \quad \mathcal{D}(Q_1)= \mathcal{H} , '''is said to be a ''projector'' on \; \mathcal{H}_1. ''' Having defined the operator \; Q_2 = \mathbf{1}- Q_1 such that : \; Q_2\psi =\psi - \psi_1 = \psi_2 , \quad \mathcal{D}(Q_2)= \mathcal{H} , the latter is a projector on \; \mathcal{H}_2. The operators \; Q_1 and \; Q_2 have the following properties: # They are self-adjoint; # Their eigenspaces are \; \mathcal{H}_1 and \; \mathcal{H}_2 respectively corresponding to the eigenvalues \; (1,0) respectively \; (0,1); # They have a complete set of eigenvectors and their descrete spectrum is \; Sp(Q_a) =\{1,0\}, \; a=1,2; # They are idempotent, i.e. such that \; Q_a^2 = Q_a; # They are orthogonal to each other, i.e. such that \; Q_1Q_2 = Q_2Q_1=0. We then have the following '''Definition: The linear, bounded, self-adjoint, idempotent operators \; Q_1 and \; Q_2 on \; \mathcal{H} are projectors on the orthogonal subspaces \; \mathcal{H}_1 and \; \mathcal{H}_2 respectively.''' These results can be generalized to the case of more than two projectors, as follows. '''Definition: Let \; \{Q_j : j \in I\}, with \; I an arbitrary finite or countably infinite subset, be a set of projectors such that :\; Q_jQ_k = \delta_{jk} , \quad j,k \in I , :\; \sum_{j \in I}Q_j = \mathbf{1} ,''' '''then we will say that \; \{Q_j : j \in I\} is a complete set of orthogonal projectors.''' Each projector \; Q_j belongs to a closed subspace \; \mathcal{H}_j of the Hilbert space \; \mathcal{H}, and these subspaces give a complete orthogonal decomposition of \; \mathcal{H}: :\; \mathcal{H} = \oplus_{j \in I} \mathcal{H}_j , \quad \mathcal{H}_j = Q_j\mathcal{H} . From completeness it follows that \; \psi= \sum_{j \in I}Q_j\psi, with \; Q_j\psi \in \mathcal{H}_j , and therefore each component of \; \psi in the subspace \; \mathcal{H}_j is given by the projection of \; \psi in \; \mathcal{H}_j : \; \psi_j=Q_j\psi. Moreover, if \; \Delta is a subset of \; I, then the operator \; Q_{\Delta}= \sum_{j \in \Delta}Q_j is a projector. In fact it can be easily verified that the following properties are satisfied: # \; \mathcal{D}(Q_\Delta)=\mathcal{H}; # \; Q_\Delta^\dagger=Q_\Delta; # \; Q_\Delta^2=Q_\Delta. The closed subspace of \; \mathcal{H} on which \; Q_{\Delta} projects is then \; \mathcal{H}_{\Delta}= \oplus_{j \in \Delta}\mathcal{H}_j. Category:Linear Algebra category:Handbook of Quantum Information

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Monday, October 26, 2015 - 17:56