T1: 1. Quantum Mechanics Background Flashcards
Define classical uncertainty
Uncertainty characterised by a lack of knowledge.
Define quantum uncertainty
The uncertainty of outcome given full knowledge of a wavefunction.
Give an example of classical uncertainty in a quantum system
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.
Define a Hilbert space
A complex vector space with a Hermitian inner product
Define Hermitian (in an orthonormal basis)
An object which equals its conjugate transpose.
Define Unitary
An object whose inverse equals its conjugate transpose.
How do we define an adjoint operator (using inner products)
Swap the outer states, remove the dagger from the operator and take the complex conjugate of the whole object.
Define the expectation value of an operator (words)
The average outcome after taking an infinite number of measurements of an operator on a given state.
Under what condition is the time-evolution operator unitary?
If the Hamiltonian is Hermitian (self-adjoint)
State the time evolution operator U(t,t_0)
U(t,t_0) = exp(-i/ℏ H(t-t_0))
What conservation is imparted on a state if time-evolution is unitary?
The norm of a state is conserved for all t.
How does time-evolution change if the Hamiltonian is time-dependent?
We still use the unitary time evolution operator, this time with a time-ordered exponential
Give two properties of the density operator
Linear and Hermitian
How to find the (m,n) element of the matrix representing the density operator
Sandwich the operator between states ⟨m|and|n⟩
Give the spectral decomposition of the identity on an orthogonal set of basis states
Sum over n of |n⟩⟨n|= identity in n dimensions
Give the trace of an operator (typical for density)
Sum over n of ⟨n|A|n⟩ for operator A
Give the spectral decomposition of a Hermitian operator L
Sum over n of λ_n|n⟩⟨n| where λ_n is the eigenvalue corresponding to eigenvector |n⟩.
How do we evolve an operator in time?
Sandwich it between the time-evolution operator and its hermitian conjugate
Define a pure state
A state with no classical uncertainty; it is described by the density matrix ϱ =|ψ⟩⟨ψ|
Define a mixed state
A state with classical uncertainty; it is described as the sum of density matrices representing pure states, weighted with a probability of occuring.
What is a condition on projectors and what does this tell us about pure states?
The square of a projector equals the projector and hence Tr(ϱ^2)=Tr(ϱ)=1 for a pure state.
Are mixed-state density matrices unique to an ensemble?
No, many ensembles can produce the same density matrix.
Are there any requirements on the states forming an ensemble?
The do not need to be orthogonal, however they are normalised.
What are the three conditions on the density matrix by construction?
They are normalised such that Tr(ϱ)= 1, Hermitian ϱ^†=ϱ and semi-positive definite.
Define semi-positive definite for an operator
The expectation value of said operator is geq zero.
In the matrix representation of the density matrix, what do r and θ denote?
r is a bloch vector which denotes a position on the unit sphere, while θ is a vector of pauli matrices.
Defining r as a point on the Bloch sphere, what is a condition on r? (and corresponding condition on pure states?)
|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.
State the Cauchy-Schwarz inequality for a dot product
|a⋅b|<=|a||b|
Can mixing two pure states produce a pure state?
No!
Give a fun fact about the distance of a mixed state from the origin of the Bloch sphere.
A mixed state can never be farther from the origin than any of the states mixed to produce it.
Given a mixed state on the Bloch sphere, how many pairs of pure states can be used to produce this mixed state and how?
An infinite number. Any pair of states on the radius of the sphere whose connecting line passes through the mixed state.
How does the ‘mixed-ness’ of a state relate to position on the Bloch sphere.
The more mixed a state is, the closer to the origin (|r| closer to 0).
What does the trace distance tell us about two states?
How distinguishable two states are.
Give three features of the trace distance
Non-negative, symmetric and satisfies the triangle inequality.
What density matrix corresponds to the most mixed state?
ϱ = I/2
What does the most mixed state look like on the Bloch sphere?
Two antipodal points mixed with probability 50% each.
Give a shortcut for the expectation value of an operator A
<a> = Tr(ϱA)</a>
Give the magnitude of some operator |A|
|A| = sqrt(A† A)
State the trace distance D(ϱ1, ϱ2)
D(ϱ1, ϱ2) = 1/2 Tr|ϱ1 - ϱ2| = 1/2 |r1 - r2|
