Qbits 1

We pointed out in the last lecture that the quantity 

| ψ ⟩ = (Subscript[c, 1])| 1 ⟩ + (Subscript[c, 0])| 0 ⟩

is called a Qubit or a ket state.  The coeffcients (Subscript[c, 1])(Subscript[c, 0]) can be any complex number that satifies
| (Subscript[c, 1])((|)^2) + | (Subscript[c, 0])((|)^2) =1. For the special case where either (Subscript[c, 1])=1,(Subscript[c, 0])=0, or visa versa, we revert to the notation used for classical Cbits. The individual Qbits | 1 ⟩, | 0 ⟩ correspond to the physical states where the light bulb is either on or off respectively. So we need to give a meaning to the quantity | ψ⟩ which involves a superposition of those two physical ``states’’.  The rules that give meaning to such a construction are part of what is called the “Copenhagen Interpretation” of the Quantum Theory”.

According to it, when the system is in “state | ψ ⟩” we can make a measurement to see if the
physical system is in the “on” (lightbulb on) or “off” (lightbulb off) state. For now, these are the only measurements we can make. Thus 
| 1 ⟩, | 0 ⟩ play a special role and are called the computational basis states.  According to the Copenhagen interpretation if the system is in state | ψ ⟩  then a measurement of the lightbulb will either give a result where it is on, or off. Nothing else!   However there is uncertainty in which of the two choices you will measure

Born rule

| (Subscript[c, 1])((|)^2)    gives us the probability that we observe the lightbulb on.

| (Subscript[c, 0])((|)^2)    gives us the probability that we observe the lightbulb off.

What do we mean by probability?  Make lots of measurements of the identical system
in state  | ψ ⟩ , e.g. N. Let (Subscript[N, on]) be the number of times that you find the lightbulb on.  
Evaluate the fraction

((Subscript[N, on])/N)  

it should approximatley be equal to | (Subscript[c, 1])((|)^2) .

In the previous lecture notes we also made the association

| 0 ⟩ =(({{1}, {0}}))

| 1 ⟩ =(({{0}, {1}}))

and so we then expect that we can also make the association


| ψ ⟩ = (({{(Subscript[c, 0])}, {(Subscript[c, 1])}}))

In addition to the basis basis | 0 ⟩, | 1 ⟩  it is convenient, to introduce an additional notation by
defining quantities described as. Whoa what’s this ? 

⟨ 0 |

⟨ 1 |

According to Dirac, they are called bra states.  For our purposes it is most easy to think
of them as row matrices, i.e

⟨0 | = (1  0)

⟨1 |=  (0  1 )

so if | ψ ⟩ = (({{(Subscript[c, 0])}, {(Subscript[c, 1])}})) what do you think ⟨ ψ | looks like ?

Answer:

 ( (SuperStar[(Subscript[c, 0])])  (SuperStar[(Subscript[c, 1])]))  that is the transpose and complex conjugate of  (({{(Subscript[c, 0])}, {(Subscript[c, 1])}}))

Dirac invented this type of notation because it is now easy to give an interpretation to something like

⟨1 |  | ψ ⟩  ≡ ⟨1 | ψ ⟩   ( Do you now see why we call it a bra-ket ?) what kind of object is this?  Suppose

we replace the kets and bras with their matrix representations 


 (0  1 )  (({{(Subscript[c, 0])}, {(Subscript[c, 1])}}))    Do you know how to multiply a row matrix with a column matrix ?
 
 the answe should be just a number  i.e. (Subscript[c, 1]).
 
 
 Or     (Subscript[c, 1])= ⟨1 | ψ ⟩  and the probability for observing the light bulb on is  |⟨1 | ψ ⟩((|)^2)
 
 
 Exercise: Evaluate  ⟨ψ | 0 ⟩

But that s not all. Lets put a  ket and a bra together this way

|0 ⟩⟨ 1|     what do you think this is?  Lets try again by putting it in matrix form

(({{1}, {0}})) (0 1)   what do you get when you multiply a column matrix by a row matrix ?

Answer: another matrix

(({{0, 1}, {0, 0}})) 

we can say   

|0 ⟩⟨ 1| =(({{0, 1}, {0, 0}}))

But we know this matrix is the same as a gate (Question: is this a reversible gate?).

With Dirac notation we can also represent gates in terms of bra and kets.

Example consider the Qubit gate:   |0 ⟩⟨ 0|  and let that act on the state  | ψ ⟩

or

 |0 ⟩⟨ 0|   | ψ ⟩   =
 
 |0 ⟩⟨ 0| ψ ⟩ =  |0 ⟩ (Subscript[c, 0])
 
 what is the probability of finding the system in the “on” and “off” position respectively?
 
 Try doing this using the matrix form of the operators, could |0 ⟩⟨ 0|  really represent a physical gate?
 
 The above structures define what is called a vector space, and various operations on it. For a more detailed discussion
 see

Hyperlink["http://www.physics.unlv.edu/%7Ebernard/phy721_03/lectures/lecture1.pdf"]
Hyperlink["http://www.physics.unlv.edu/%7Ebernard/phy721_03/lectures/lecture2.pdf"]

The collapse postulate of the Copenhagen interpretation

Suppose we make a measurement on our system | ψ ⟩ for “on” or “off” . If we obtain
 “on”  then immediately after that measurement, the state of the system is 
| 1 ⟩   (up to a phase factor i.e. -1, i, etc). Let’ define the operator M = ((|0⟩⟨0|)/(Sqrt[⟨ψ| 0⟩⟨0| ψ⟩])) what is the
result of its action on | ψ⟩ ?

Some physical examples of Qubits

We have already seen one example of a Qubit. David Wineland’s experiment demonstrated that a
single (SuperPlus[Hg]) ion can be turned on and off. If the ion is in the lower energy state it is in the on position
(since in that case, when he shines a UV laser light on it it instantly (in about (10^(-9)) seconds)  fluoresces (i.e. it shines, just like the light bulb). Alternatively, when its in an excited energy state the
UV laser does not cause a fluorescence, i.e. the bulb is off. In general he can make superpositions
of those two states.

Another important example involves a single spin-1/2 particle (e. g. electron ) that has a magnetic moment. Suppose a magnetic field in the  z direction is turned on. This results in two energy states
one in which the magnetic moment is parallel to the field and the other in which it is anti-parallel.
We then have a situation as above, the two state atom, in its lower energy state we can call the system in state |g ⟩ and the upper energy state is called |e ⟩. This example also allows for a nice
illustration for how can obtain mixtures of |g ⟩,   |e ⟩. 

We now need to learn a little about quantum dynamics. One of the main tenets of the Copenhagen
interpretation is the following:

Given that at time t=0 the Qubit system (i.e. our electron in a magnetic field ) is in an arbitrary state
state |(Subscript[ψ, 0])⟩ then at later time t the system “evolves” into a new state given by

|ψ(t) ⟩ = U(t) |(Subscript[ψ, 0])⟩

where U(t) is a time-dependent unitary operator (see definition below). For a Qubit system U(t)
is simply a 2 x2 matrix that has the following properties

U (SuperDagger[U])= (SuperDagger[U]) U = 1   here 1 is the 2x 2 identity operator.

The dagger operator means the following: take the transpose of matrix U, then the complex conjugate
of each element.  Note: from the above definition U is a gate, and is reversible. Thus a type of
gate is already built into our theory simply by the existence of this physical postulate. The exact form
for U depends on another type of matrix called the Hamiltonian matrix or H (do not confuse it with the
Hadamard gate! same symbol, different meaning). Roughly speaking H determines the energy values allowed in our system. The fact that quantum states evolve in the way described above is another postulate of the Copenhagen interpretation.

Summary of Copenhagen interpretation

I) Physical systems are described by quantities called states |ψ⟩ which can be represented
as a linear combination of so called computational basis states. The number of computational
basis states determine the dimension of a Hilbert space (a type of vector space). The dimension
of a Qbit Hilbert space is 2, since the basis states are |0⟩, |1⟩.

2) Associated with the measurement is an Hermitian operator. Typically, the eigenstates of
these operators form a possible basis set for a Hilbert space, and the corresponding eigenvalues are the possible
results of a measurement. 

2)  The coefficients for the basis state |ϕ ⟩ in an expansion of |ψ ⟩ gives the probability for the
system to have measurement value ϕ, i.e. for a Qubit  |ψ⟩ =  (Subscript[c, 1])| 1 ⟩ + (Subscript[c, 0])| 0 ⟩,  |(Subscript[c, 1])((|)^2) is the probability
that a measurement of the system will reveal an “on” value. In Dirac’s notation this quantity can be represented as  (Subscript[c, 1])=⟨1|ψ⟩.

3) If the system is in state |ψ⟩ and if a measurement gives the value ϕ, then immediately after that measurement the system “collapses” to state |ϕ⟩.

4) If the system is not disturbed (no one is doing measurements) then it will evolve 
according to |ψ(t) = U(t) |(Subscript[ψ, 0])⟩, where U(t) is a unitary operator (quantum gate).

Qbit dynamics

(II) Spin -1/2 particle in a magnetic field (e.g. an electron). Suppose we have a constant magnetic
field pointing along the positive z-direction. If you place a stationary electron in it, the energy of the electron is lower  if its ‘spin’ points along the same direction as B. If its spin is anti-parallel to B its energy is lower. 






Lets represent the state in which the the electron has energy (Subscript[E, 1])  |0⟩=(({{1}, {0}})) and the state
in which it has energy  (Subscript[E, 2])  |1⟩=(({{0}, {1}})).  However the electron can also be in the superposition

| ψ⟩ = (Subscript[c, 1])| 1 ⟩ + (Subscript[c, 0])| 0 ⟩. The time evolution operator is given by

U(t)=(({{exp(-i (Subscript[ω, 2]) t) , 0}, {0, exp(-i (Subscript[ω, 1]) t) }}))       (Subscript[ω, 2])=(Subscript[E, 2])/ℏ , (Subscript[ω, 1])=(Subscript[E, 2])/ℏ

How can we tell that our system is in a superposition?


We treat our z-direction as a preferred direction and imagine that there is a small “perturbation” in
another direction. Suppose the effect of this perturbation is to change U(t) into the following form (Rabi Oscillations)

U(t)=(Subscript[U, 0])(t) (({{Cos(ω t)-i Cos(θ)Sin(ω t), -i Sin(θ) Sin(ω t)}, {-i Sin(θ) Sin(ω t), Cos(ω t)+ i Cos(θ)Sin(ω t)}}))

ω=1/2 (Sqrt[((((Subscript[ω, 2])-(Subscript[ω, 1])) )^2)+ 4 (Δ^2) ]);    tan(2θ)=((2 Δ)/(((Subscript[ω, 2])-(Subscript[ω, 1]))));   (Subscript[U, 0])(t)=  exp(-i  ((Subscript[ω, 2])-(Subscript[ω, 1])) t/2  )    (({{1, 0}, {0, 1}}))

So lets see what happens when the system starts at t=0 in the state |0⟩. According to
postulate (4) it evolves as U(t)| 0 ⟩. Or,

|ψ(t) ⟩= (Subscript[U, 0])(t) (({{Cos(ω t)-i Cos(2θ)Sin(ω t)}, {-i Sin(2θ) Sin(ω t)}})) = exp(-i  ((Subscript[ω, 2])-(Subscript[ω, 1])) t/2  ) (({{Cos(ω t)-i Cos(2θ)Sin(ω t)}, {-i Sin(2θ) Sin(ω t)}})) 

but this is the same as


|ψ(t) ⟩ = (Subscript[c, 0])(t) |0⟩ + (Subscript[c, 1])(t) |1⟩

(Subscript[c, 0])(t)= exp(-i  ((Subscript[ω, 2])-(Subscript[ω, 1])) t/2  ) ( Cos(ω t)-i Cos(2θ)Sin(ω t))
(Subscript[c, 1])(t)=-i exp(-i  ((Subscript[ω, 2])-(Subscript[ω, 1])) t/2  )Sin(θ) Sin(ω t)

So the probability that the system stays in Qubit |0⟩ at later time t is according to postulate (2)

 |(Subscript[c, 0])((|)^2)=   (Cos^2)(ω t) + (Cos^2)(2θ) (Sin^2)(ω t) = 1 - (Sin^2)(2θ) (Sin^2)(ω t)

Plot[(1-Sin[2 θ]^2 Sin[ω t]^2 ) /.{ θ->Pi/12,ω->π},{t,0,2Pi},PlotRange->{0,1}]

Neutrino Oscillations

Its been know for some time that when a neutron turns into a proton via the weak interaction, e.g

n ⟶ p + e + (Subscript[ν, e])

a neutrino | (Subscript[ν, e])⟩   (Why am I writing a particle as a ket?) is released. The neutrino does not have charge,(i.e. its neutral) and is almost massless.  It took a long time for this particle to be discovered (1950’s) but then physicists
also found other types of neutrinos, e.g. the muon neutrino. The muon is like a very heavy electron and in 1962
it was discovered that in reactions in which muons appear a neutrino - like particle also appears.
We can call it  | (Subscript[ν, μ])⟩. These different type of neutrinos are called flavors (today we know that there exists yet another
flavor the tau neutrino).

The Solar Neutrino Problem. 

We know that the energy of the sun comes mainly from the following chain of reactions (figure due to NASA)

Notice that two electron neutrinos are released. Since they interact very weakly with matter they escape un-impended from the sun, and traveling nearly at the speed of light, reach the earth in about 8 minutes.
If we could build a detector on earth and measure the neutrino flux from the sun we could also figure out
how many reactions (or energy released) occur in the Sun.  A good theoretical model of how these reactions
occur in the Sun was developed and from that theory one can estimate how many neutrinos we should measure
on Earth. In the 1960’s such a detector was built. Sadly,  the number of neutrinos detected was only about 1/3
the value demanded by theory!  What’s the problem? Is something wrong with the experiment?, is our theoretical model of the Sun incorrect?  Furthermore look at this data, it is a measure of the solar neutrino flux as a function
of time (months).

There was one theory, originally proposed in the 1960’s by Bruno Pontocervo, that suggested that the two different neutrino flavors | (Subscript[ν, e])⟩,  | (Subscript[ν, μ])⟩  can mix. There is an analogy with the Qbit system we discussed above.  Instead
of just | (Subscript[ν, e])⟩,  or  | (Subscript[ν, μ])⟩    we actually have a superposition 

 (Subscript[c, 1])(t) | (Subscript[ν, e])⟩ + (Subscript[c, 2])(t)  | (Subscript[ν, μ])⟩
 
 So if at t=0, an electron neutrino starts its journey the probability |(Subscript[c, 1])(t)((|)^2)  might change in time. It looks like
 | (Subscript[ν, e])⟩ has disappeared by the time it reaches the earth, but in reality it has just turned into | (Subscript[ν, μ])⟩ which is not
 detected by the experiments which can only detect  | (Subscript[ν, e])⟩.  This is a nice theory, is it true? only in the past few
 years has the answer been shown to be in the affirmative.
 
 It is now know that the probability for an electron neutrino to stay as one is given by the expression
 
 1 - (Sin^2)2θ (Sin^2)( Δ(m^2)(c^4) (L/(4p ℏ (c^2))))

2015 Nobel Prize in Physics  Takaaki Kajita (Japan) and Arthur MacDonald  (Canada)

Hyperlink["http://www.nobelprize.org/mediaplayer/index.php?id=2579"]

Hyperlink["http://www.nobelprize.org/mediaplayer/index.php?id=2580"]