wk4 Flashcards
What is the projection measurement
An observable (result of observation) has a spectral decomposition. Where M = Sum_m( m P_m), where p_m = projector. This means that all possible outcomes of measurement correspond to to the eigenvalues
What is the probability of getting some measurement m, when measuring state |ψ>
p(m)= <ψ|P_m|ψ>
After a measurement, M, what is the space of the system
(1/ root(P_m) ) P_m |ψ>
What if all measurement projectors are orthogonal?
they are hermitian and M_m . M_n = kroneckerdelta_{m,n}
What is the average value of a projective measurement
The expected value, which is (see image).
This value is also equal to the standard deviation
When can quantum states be distinguished
If states are orthogonal then they can be reliably distinguished, non orthogonal ones overlap and are hard to distinguish
what is POVM
A positive operator-valued measurement. A measure whose values are semi definite operators on a Hilbert space.
when POVM is defined as a finite number of elements acting on a finite-dimensional Hilbert space, POVM is a set of positive semi-definite Hermitian Matrices such that they sum to. an identity matrix
How are measurements defined in POVM
E_m := M^TM
such that E_m is positive to allow for the sum of all E_m to = Identity. And p(m) = <ψ|E_m|ψ>
Also P_m P_n = KoneckerDelta_{m,n}
Postulate 3 of quantum mechanics
Physical variables are real and hence represented by hermitian operators, measurement can be described by projection operators
For non commuting observables, Delta(c) . Delta(d) >= 1/2 |<ψ|[C,D]|ψ>|
Postulate 4:
State space of composite physical system is the tensor product of the state spaces of the component physical systems. e.g. |ψ_1> tensor|ψ_2> tensor…|ψ_n>
Postulate 1
Quantum systems are described by state vectors or density matrices
Postulate 2
isolated quantum systems time-evolve via unitary operators (observables) and or Schrödinger equation
