5th International Conference on Geometry and Applications by Atanasov D.V.

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By Atanasov D.V.

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An orthogonal projector is a projector that also satisfies P† = P. Recall the following definition from basic linear algebra. 13) for some constant c. The constant c is called the eigenvalue of T corresponding to the eigenvector |ψ . The following result is relevant to measurements in quantum mechanics. 9 If T = T † and if T |ψ = λ|ψ then λ ∈ R. In other words, the eigenvalues of a Hermitean operator are real. In linear algebra one learns that the trace of a square matrix is obtained by adding the elements on the main diagonal.

13) SOME COMMENTS ON THE DIRAC NOTATION 37 In all the above examples, the bipartite space was one in which each subspace had dimension 2. The Schmidt decomposition theorem can be applied to more complicated bipartite vector space, even in cases where the two subspaces have different dimensions. 2. 8 Some Comments on the Dirac Notation Juggling bras and kets can be somewhat confusing if you are not familiar with working in the Dirac notation. Now we discuss a convention that potentially adds to this confusion.

7) Technically we could describe quantum states by equivalence classes of unit vectors, but we will simply specify a unit vector, with the understanding that any two vectors that are related by a global phase are equivalent. We will motivate the fact that |ψ and eiθ |ψ are equivalent after we have introduced the Measurement Postulate. 3, which will be discussed shortly). Consider the analogous situation for a deterministic classical bit. The state of a classical bit can be described by a single binary value ψ, which can be equal to either 0 or 1.

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