Math of Qbits

Question 1

Measurement

Answer 1

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

Question 2

S-G expirement rotating the magnet

Answer 2

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

Question 3

Superposition basic states

Answer 3

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!

Question 4

Basic state 0,1

Answer 4

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

Question 5

Ket

Answer 5

> 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>

Question 6

Measuring in another basis

Answer 6

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

Question 7

Bra

Answer 7

<b></b>

Question 8

Evolution quantum system

Answer 8

states are vectors in a vector space and the transformations of the quantum systems are linear operators.

Question 9

Matrix multiplication

Answer 9

row multiplied by all columns.

Question 10

Orthonormal base

Answer 10

all kets orthogonal to each other: inner product = 0

all kets normal: inner product itself = 1

Question 11

Basic state right, left

Answer 11

|right> = [1/sqrt(2) 1/sqrt(2)] |left>=[1/sqrt(2) -1/sqrt(2)]

Question 12

Tensor product

Answer 12

To describe the evolution of the quantum system

A(X)B = [aiB]

Question 13

Linear combination basis vector

Answer 13

|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 =

Question 14

Change of base

Answer 14

M = matrix of the new basis
|s> = state
change of base = M|s> = |s’>

Question 15

Equivalent state

Answer 15

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

Question 16

How to check if equivalent

Answer 16

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>

Question 17

Photon polarization

Answer 17

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

Question 18

Uncertainty principle

Answer 18

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