T1: 1. Quantum Mechanics Background

Question 1

Define classical uncertainty

Answer 1

Uncertainty characterised by a lack of knowledge.

Question 2

Define quantum uncertainty

Answer 2

The uncertainty of outcome given full knowledge of a wavefunction.

Question 3

Give an example of classical uncertainty in a quantum system

Answer 3

Relevant example.

I.e. Consider a system which produces one wavefunction with a prob of 50% and another with prob 50%. We can improve the system to reduce this uncertainty; it is independent of the uncertainty in the wavefunction.

Question 4

Define a Hilbert space

Answer 4

A complex vector space with a Hermitian inner product

Question 5

Define Hermitian (in an orthonormal basis)

Answer 5

An object which equals its conjugate transpose.

Question 6

Define Unitary

Answer 6

An object whose inverse equals its conjugate transpose.

Question 7

How do we define an adjoint operator (using inner products)

Answer 7

Swap the outer states, remove the dagger from the operator and take the complex conjugate of the whole object.

Question 8

Define the expectation value of an operator (words)

Answer 8

The average outcome after taking an infinite number of measurements of an operator on a given state.

Question 9

Under what condition is the time-evolution operator unitary?

Answer 9

If the Hamiltonian is Hermitian (self-adjoint)

Question 10

State the time evolution operator U(t,t_0)

Answer 10

U(t,t_0) = exp(-i/ℏ H(t-t_0))

Question 11

What conservation is imparted on a state if time-evolution is unitary?

Answer 11

The norm of a state is conserved for all t.

Question 12

How does time-evolution change if the Hamiltonian is time-dependent?

Answer 12

We still use the unitary time evolution operator, this time with a time-ordered exponential

Question 13

Give two properties of the density operator

Answer 13

Linear and Hermitian

Question 14

How to find the (m,n) element of the matrix representing the density operator

Answer 14

Sandwich the operator between states ⟨m|and|n⟩

Question 15

Give the spectral decomposition of the identity on an orthogonal set of basis states

Answer 15

Sum over n of |n⟩⟨n|= identity in n dimensions

Question 16

Give the trace of an operator (typical for density)

Answer 16

Sum over n of ⟨n|A|n⟩ for operator A

Question 17

Give the spectral decomposition of a Hermitian operator L

Answer 17

Sum over n of λ_n|n⟩⟨n| where λ_n is the eigenvalue corresponding to eigenvector |n⟩.

Question 18

How do we evolve an operator in time?

Answer 18

Sandwich it between the time-evolution operator and its hermitian conjugate

Question 19

Define a pure state

Answer 19

A state with no classical uncertainty; it is described by the density matrix ϱ =|ψ⟩⟨ψ|

Question 20

Define a mixed state

Answer 20

A state with classical uncertainty; it is described as the sum of density matrices representing pure states, weighted with a probability of occuring.

Question 21

What is a condition on projectors and what does this tell us about pure states?

Answer 21

The square of a projector equals the projector and hence Tr(ϱ^2)=Tr(ϱ)=1 for a pure state.

Question 22

Are mixed-state density matrices unique to an ensemble?

Answer 22

No, many ensembles can produce the same density matrix.

Question 23

Are there any requirements on the states forming an ensemble?

Answer 23

The do not need to be orthogonal, however they are normalised.

Question 24

What are the three conditions on the density matrix by construction?

Answer 24

They are normalised such that Tr(ϱ)= 1, Hermitian ϱ^†=ϱ and semi-positive definite.

Question 25

Define semi-positive definite for an operator

Answer 25

The expectation value of said operator is geq zero.

Question 26

In the matrix representation of the density matrix, what do r and θ denote?

Answer 26

r is a bloch vector which denotes a position on the unit sphere, while θ is a vector of pauli matrices.

Question 27

Defining r as a point on the Bloch sphere, what is a condition on r? (and corresponding condition on pure states?)

Answer 27

|r|= 1 since it is a point on the unit sphere. This indicates that a pure state also has |r|=1, while a mixed state has |r|>1.

Question 28

State the Cauchy-Schwarz inequality for a dot product

Answer 28

|a⋅b|<=|a||b|

Question 29

Can mixing two pure states produce a pure state?

Answer 29

No!

Question 30

Give a fun fact about the distance of a mixed state from the origin of the Bloch sphere.

Answer 30

A mixed state can never be farther from the origin than any of the states mixed to produce it.

Question 31

Given a mixed state on the Bloch sphere, how many pairs of pure states can be used to produce this mixed state and how?

Answer 31

An infinite number. Any pair of states on the radius of the sphere whose connecting line passes through the mixed state.

Question 32

How does the ‘mixed-ness’ of a state relate to position on the Bloch sphere.

Answer 32

The more mixed a state is, the closer to the origin (|r| closer to 0).

Question 33

What does the trace distance tell us about two states?

Answer 33

How distinguishable two states are.

Question 34

Give three features of the trace distance

Answer 34

Non-negative, symmetric and satisfies the triangle inequality.

Question 35

What density matrix corresponds to the most mixed state?

Answer 35

ϱ = I/2

Question 36

What does the most mixed state look like on the Bloch sphere?

Answer 36

Two antipodal points mixed with probability 50% each.

Question 37

Give a shortcut for the expectation value of an operator A

Answer 37

<a> = Tr(ϱA)</a>

Question 38

Give the magnitude of some operator |A|

Answer 38

|A| = sqrt(A† A)

Question 39

State the trace distance D(ϱ1, ϱ2)

Answer 39

D(ϱ1, ϱ2) = 1/2 Tr|ϱ1 - ϱ2| = 1/2 |r1 - r2|