wk9/10

Question 1

How is density matrix generated from ensemble

Answer 1

sum of all (p|𝛙><𝛙|). Where p is probability (scalar) factor and 𝛙 is the normalised pure state in the ensemble

Question 2

What is the difference between probabilistic mixtures and density matrices

Answer 2

density matricies display quantum inteference whereas probabilistic mixture does not

Question 3

What are the properties of density matricies

Answer 3

1) they are hermitian
2) it is positie semi-definitie
3) unitary trace, trace( density matrix) = 1

Question 4

what property does a density matrix of a pure state satisfy

Answer 4

tr( density matrix^2 ) == 1

Question 5

How do you apply unitary map onto density matrix

Answer 5

U rho U^dagger

Question 6

in terms of teleportation, what does a density matrix of a mixed state imply

Answer 6

the receiver of any teleported message that is a mixed state will not be able to distinguish the mixed state without any classically transferred information from the sender

Question 7

what are the properties of pure states

Answer 7

• density matrix = |𝛙><𝛙|
• <𝛙|𝛙>=1
• trace ( density matrix ^ 2) = 1

Question 8

which arbitrary quantum states are physically indistinguishable

Answer 8

( e^i(theta) |𝛙> ) for all theta’s are indistinguishable from each other. This is because their scalar factors are not taken into account in density matrices

Question 9

what is the significance of two ensembles giving rise to same density matrix

Answer 9

the two ensembles are indistinguishable in terms of their physical behaviour as quantum behaviour is exhaustively defined by the density matrix

Question 10

What are general quantum operations described by

Answer 10

general quantum operation ε( density matrix ) is defined by the set of linear operators {E_k}
ε( density matrix ) = (probability)_k E_k (density matrix) E_k^dagger

Note we only care about trace-preserving operators E_k

Question 11

Which gates correspond to bit flip and which to phase flip

Answer 11

X = bit flip
Z = phase flip