Let  H1 be a closed subspace of the Hilbert space  H and  H2 its orthogonal complement in  H. Then any vector ψ ∈ H can be decomposed into its components belonging to  H1, respectively  H2 as follows:

ψ = ψ1 + ψ2,  ⟨ψ1, ψ2⟩ = 0.

'''Definition: The linear operator Q1 on  H1 such that '''

Q1ψ = ψ1,  D(Q1) = H, 

'''is said to be a projector on  H1. '''

Having defined the operator Q2 = 1 − Q1 such that

Q2ψ = ψ − ψ1 = ψ2,  D(Q2) = H, 

the latter is a projector on  H2.

The operators Q1 and Q2 have the following properties:

  1. They are self-adjoint;
  2. Their eigenspaces are  H1 and  H2 respectively corresponding to the eigenvalues  (1, 0) respectively  (0, 1);
  3. They have a complete set of eigenvectors and their descrete spectrum is Sp(Qa) = {1, 0},  a = 1, 2;
  4. They are idempotent, i.e. such that Qa2 = Qa;
  5. They are orthogonal to each other, i.e. such that Q1Q2 = Q2Q1 = 0.

We then have the following

Definition: The linear, bounded, self-adjoint, idempotent operators Q1 and Q2 on  H are projectors on the orthogonal subspaces  H1 and  H2 respectively.

These results can be generalized to the case of more than two projectors, as follows.

'''Definition: Let  {Qj : j ∈ I}, with I an arbitrary finite or countably infinite subset, be a set of projectors such that

QjQk = δjk,  j, k ∈ I, 

 ∑j ∈ IQj = 1, 
''' then we will say that  {Qj : j ∈ I} is a complete set of orthogonal projectors.

Each projector Qj belongs to a closed subspace  Hj of the Hilbert space  H, and these subspaces give a complete orthogonal decomposition of  H:

 H =  ⊕ j ∈ IHj,  Hj = QjH.

From completeness it follows that ψ = ∑j ∈ IQjψ, with Qjψ ∈ Hj, and therefore each component of ψ in the subspace  Hj is given by the projection of ψ in  Hj: ψj = Qjψ.

Moreover, if  Δ is a subset of I, then the operator QΔ = ∑j ∈ ΔQj is a projector. In fact it can be easily verified that the following properties are satisfied:

  1.  D(QΔ) = H;
  2. QΔ †  = QΔ;
  3. QΔ2 = QΔ.

The closed subspace of  H on which QΔ projects is then  HΔ =  ⊕ j ∈ ΔHj.

Category:Linear Algebra category:Handbook of Quantum Information

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Monday, October 26, 2015 - 17:56