Postulates of Quantum Mechanics

Question 1

Quantum Mechanics

Answer 1

Framework for the development of physical theories aside of gravity.

Question 2

Postulates of Quantum Mecanics

Answer 2

4 postulates

1. State space: how can a quantum state be described
2. Evolution: how a quantum state is allowed to change over time
3. Measurement: effect on a quantum state of interaction with a classical system that yields to classical information
4. Composition: how quantum systems can be composed ?

Question 3

State space

Answer 3

State is an inner product of vectors in a vector space. System completely described by a unit vector that is its state vector.

Question 4

Evolution

Answer 4

The evolution over time of quantum system from on step to the next is described by U|s>
U is a unitary transformation and s the current state.
Unitary makes sure it is a valid superposition -> sum of the amplitudes squared are 1 -> we preserve length

Question 5

Unitary Matrix

Answer 5

invM=M transposed
They have orthogonal columns and are diagonalizable => eigenvalues 0
inner products are preserved
if U done on a state |s> with an amplitude with opposite sign then there will be a destructive interference: that outcome is cancelled
2x2 to transform qbit of 2
4x4 to transform 2qbits

Question 6

Quantum interference

Answer 6

probas can be negative in quantum and thus have destructive interferance : an outcome is cancelled

Question 7

Total probability

Answer 7

length -> inner product |>

Question 8

Pauli matrices

Answer 8

operators X,Y,Z

Question 9

X

Answer 9

pauli matrix
Flip operator
effect on standard base
X|0> = |1>
X|1> = |0>
X = [0 1;1 0]

Question 10

Y

Answer 10

pauli matrix 
effect on standard base
Y|0> = i|1>
Y|1> = -i|0>
Y = i [0 -1;1 0]
flips, change sign and put in complex space -> phase change

Question 11

Z

Answer 11

pauli matrix 
effect on standard base
Z=[1 0;0 -1]
Z|0> = |0>
Z|1> = -|1>
sign change -> phase
bit is unchanged

Question 12

H

Answer 12

Hadarmard matrix
H = 1/sqrt(2) [ 1 1; 1 -1]
effect on standard base
H|0> = 1/sqrt(2)(|0> + |1>) = |+> = right
H|1> = 1/sqrt(2)(|0>-|1>) = |-> = left
Totally random states !!
It is reversible
Hright = |0>
Hleft = |1>

Question 13

Mutliple qbits

Answer 13

eg: |00>
vector of 4 amplitudes
tensor of the 2 bits

Question 14

Composition

Answer 14

state space of composite is the tensor product of all the components
if system made of |q1> and |q2> then |phi>=|q1>|q2>