Math of Qbits Flashcards
Measurement
Observing state -> makes the superposition collapse into a single state with the corresponding probability (Born’s rule)
This is irreversible ! -> no matter how many times we measure again: stays in the collapsed state
As long as same orientation, otherwise direction depend on orientation
if |phi> makes angle theta with |0> => |phi> = cos theta |0> + sin theta |1>
Measurement -> projection onto standard base with corresponding probas
S-G expirement rotating the magnet
90° go from up down to right left. Change of measurement base ! 50% of either direction: superposition.
-> superposition depends on direction of measurement
Rotating the apparatus by 90° this means having an angle theta of 45° between the bases -> the angles are not the same ! theta apparatus angle, theta/2 base angle
Superposition basic states
2 superposition can have the same probabilities of collapsing to a basic state but if 1 has different signs in amplitude then they are different!
Basic state 0,1
0 (up), 1 (down)
|phi> = a|0> + b|1>
Basic state can be represented by vector of amplitudes
2 dimensional vector in complex or real space!
Normalized -> length 1 (sqrt(a^2+b^2=1) -> the amplitudes squared are the proba that must sum to 1)
|0> = (1 0)^t |1>=(0 1)^t
Graphical representation:
0(up) is x axis, 1(down) is y axis
Ket
> column vector
quantum quantities are described by kets
|a+b>=|b+a> just sum the corresponding indices
inner product: <a> row then column
0 if orthogonal, >0 pointing in same direction, <0 pointing in opposite direction
Length ket = sqrt(</a><a>)</a>
Measuring in another basis
Base makes angle theta with the starting states.
new base |u> (x axis) |u’> (y axis) will be
|phi> = cos theta |u> + sin theta |u’>
|u> = 1/sqrt(2) |0> + 1/sqrt(2) |1>
|u’> = - 1/sqrt(2) |0> + 1/sqrt(2) |1>
This measure if the Qbit is in the superposition in u or u’ of the starting base
=> measure if u or u’ state
Bra
<b></b>
Evolution quantum system
states are vectors in a vector space and the transformations of the quantum systems are linear operators.
Matrix multiplication
row multiplied by all columns.
Orthonormal base
all kets orthogonal to each other: inner product = 0
all kets normal: inner product itself = 1
Basic state right, left
|right> = [1/sqrt(2) 1/sqrt(2)] |left>=[1/sqrt(2) -1/sqrt(2)]
Tensor product
To describe the evolution of the quantum system
A(X)B = [aiB]
Linear combination basis vector
|a> = [ a1 a2]^t = a1[1 0]^t + a2[0 1]^t in right left: (a1+a2)/sqrt(2) right + (a1-a2)/sqrt(2) left in general basis |b1> |b2> |v> = w1|b1> +w2|b2> wi =
Change of base
M = matrix of the new basis
|s> = state
change of base = M|s> = |s’>
Equivalent state
Distinguish 2 state with same proba but different amplitude
Just measuring will collapse with same proba so not distinguishable
even by changing base it is not possible ! will keep getting same probas.
|phi> and -|phi> are undistinguishable
but [a b] and [a -b] are distinguishable when changing state
How to check if equivalent
Check if same proba and/or amplitude
If same proba:
90° rotation
new states a1|b1>+a2|b2> and a1’|b1>+a2’|b2>
(a1+a1’)/sqrt(2) |b1> + (a2+a2’)/sqrt(2) |b2>
Photon polarization
electron wave and magnetic field
polaroid filter to change/measure the polarization-> certain orientations allowed to pass
0 is horizontal polarization
1 is vertical polarization
photon = 1/sqrt(2) |0> + 1/sqrt(2) |1>
OR
photon = cos theta |0> + sin theta 1/sqrt(2) |1> if some angle with vertical/horizontal orientation
use vertical filter
P=1 photon pass if |1>/vertical polarization
P=0 photon pass if |0>/horizontal polarization
P=1/2 photon pass if 45° polarization
if put 2 parallel filters everything pass
if put 2 crossed filters nothing pass
if put 1 diagonal filter between 2 crossed filter some pass ! this is because change state to diagonal and thus have 1/sqrt(2)
total prob to pass 1/2 x 1/2 = 1/4
Uncertainty principle
Cannot know position and velocity of particle with high accuracy. 1 is a tradeoff of the other.
If have 2 different base for our qbit, the close we are to 1 the further we are from the other.
if we measure in one base then are interested to measure in another we get some probability to collapse to a state
=> the more certain we are about 1 state the more uncertain we are of the other
