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An operator is a mathematical object that transforms a function into another function. Applying this to linear algebra, an operator is something that transforms a vector into another vector.
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The outer product of a bra and a ket is an operator.
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The Pauli X operator is the bit flip or NOT operator
Don’t forget to flip the kets
The projection operator is hermitian.
- An Eigen value of an measurment operator represents a measurment result
Projection coefficient: The value ⟨a|b⟩ tells us the coefficient of |a⟩ when expanding |b⟩ in terms of the basis that includes |a⟩.
Physically: It “projects” the state described by ρ \rho ρ onto the subspace spanned by ∣ϕ⟩ |\phi\rangle ∣ϕ⟩, keeping only the component of ρ \rho ρ that aligns with ∣ϕ⟩ |\phi\rangle ∣ϕ⟩. The result is a new density matrix, but it’s no longer a full description of the original state—it’s a restricted version.
After doing the fisrt measurment the 2nd measurement will also give the same result, the first measurment alters the wave function giving 100% at the place of collapse
TODO:
watch the density matrix lecture again, there is more into it
A new intution I am starting to build is that the dansity matrix is a vector inside the vector and the projector is projecting this into a subspace inside the bloch sphere
