QC 4 QuantumBrain

Below is a photo of the recent experiment I was involved with at the quantum technology lab, UQ. The setup is an implementation of a quantum walk for a single photon. The laser light shows the path a single photon takes through the setup.

Copyright © University of Queensland

The quantum walk is the quantum analogue of its classical counterpart the random walk. A walker, in this case a quantum bit (qubit), experiences a coin flip and depending on the outcome of the coin flip either moves left or right one step on a one-dimensional lattice. Unlike the classical case where the outcome is either heads or tails, the qubit can be both heads and tails. This is a property of quantum systems known as quantum superposition. As the walker spreads out across the lattice it interferes with itself at various points along the walk. Due to the quantumnness of the walk the probability of finding the walker at a given lattice site after some time is very different to the classical case.

Check out the work here.

The Bunching Effect

I know I talked about beam splitters in a previous post, but I want to touch on a really cool effect that I have recently learned about.

Let us start from the quantum operation of a beam splitter again. I am going to use the bra-ket notation here as it is the most widely used in the field of quantum optics and in particular quantum information.

Schematic diagram of a beam splitter

We can represent the prepared input and output modes of the beam splitter in the following way

 \left|10\right>

The first and second elements in the bra-ket unit represent photons in spatial modes traced out by ‘a’ and ‘b’ respectively (see diagram above). The above state represents one photon in the ‘a’ spatial mode and no photons in the ‘b’ spatial mode. This is effectively a two qubit system where the qubit state is represented by either the presence of a photon in the mode, or not, this is known as the spatial mode degree of freedom. There are two spatial modes thus making it a two qubit system.

By defining the reflectivity of the beam splitter as R and the transmitivity as 1-R it is easy to convince yourself that the following equations represent zero, single and two photon events at the beam splitter.

 \hat{U}\left|00\right>=\left|00\right>

 \hat{U}\left|10\right>=i\sqrt{R}\left|01\right> + \sqrt{1-R}\left|10\right>

 \hat{U}\left|01\right>=i\sqrt{R}\left|10\right> + \sqrt{1-R}\left|01\right>

 \hat{U}\left|11\right>=i(2R-1)\left|11\right> + \sqrt{2R(1-R)}(\left|20\right>+\left|02\right>)

The first equation is the most trivial and of least interest, it represents no photons entering either mode of the beam splitter. The next two equations map the process of a single photon in one mode and no photon in the other. Both equations are equivalent and there is a phase difference of 180 degrees between the reflected and transmitted photon, equivalent to multiplying the reflected mode by i.

The last equation shows the consequence of having a 50/50 beam splitter. That is to say, R = 0.5 will always result in both photons leaving via the same mode (you do the math!). This effect is known as the bunching effect.

It is interesting to note that the two qubits (represented by the spatial modes) are now quantum mechanically entangled. Knowing information about one spatial mode will give you information about the other instantly. Entanglement lies at the heart of quantum information and quantum computation architecture and whilst this example is nice it by no means introduces entanglement in its full glory.

There, the bunching effect. If you find flaws in the math please tell me. I’m not entirely sure it is correct from my working. However, it gives the idea of the bunching effect. The phases are chosen arbitrarily, here I chose a phase change upon reflection, which is standard practice in linear optics.

Something just occured to me…

I’m working intensly with bosonic creation and annihilation operators in the field of quantum optics and whilst doing some simple calculations I realised the brilliance that are these two operators, it can be summed up in the expression:

[\hat{a}^\dagger,\hat{b}^\dagger] = \hat{a}^\dagger\hat{b}^\dagger - \hat{b}^\dagger\hat{a}^\dagger = 0

They commute! Of course I already knew this, but it took me a while to realize how nice this is and how thankful I am that I’m not working with stupid fermionic systems.

I apologize for all the geek speak in this post, but I had to tell someone. Now I have it off my chest I can live the rest of my days.

Optical components II

Beam splitters - A quantum mechanical approach

Schematic diagram of a beam splitter

A beam splitter can be represented by a quantum operator. From the diagram, the beam splitter maps the input states to the output states as;

 \hat{a}^\dagger_{out} \rightarrow r \hat{b}^\dagger_{in} + t \hat{a}^\dagger_{in}

 \hat{b}^\dagger_{out} \rightarrow r' \hat{a}^\dagger_{in} + t' \hat{b}^\dagger_{in}

For a 50/50 beam splitter, i.e. one that transmits 50% and reflects 50%, |r|=|t| and r=r’ and t=t’. The operation of the beam splitter is thus written as,

\begin{bmatrix}
\hat{a}^\dagger_{out} \\ \hat{b}^\dagger_{out}
\end{bmatrix} =
\hat{U}
\begin{bmatrix}
\hat{a}^\dagger_{in} \\ \hat{b}^\dagger_{in}
\end{bmatrix} =
\begin{bmatrix}
t & r\\
r & t
\end{bmatrix}
\begin{bmatrix}
\hat{a}^\dagger_{in} \\ \hat{b}^\dagger_{in}
\end{bmatrix}

An electromagnetic wave experiences a phase change upon reflection and transmission and so the operator looks like,

 \hat{U} = \frac{1}{\sqrt{2}}\begin{bmatrix}
e^{i\phi_{t}} & e^{i\phi_{r}}\\
e^{i\phi_{r}} & e^{i\phi_{t}}
\end{bmatrix}

By absorbing a global phase and defining,

\phi = \phi_{r} - \phi_{t}

 \hat{U} = \frac{1}{\sqrt{2}}\begin{bmatrix}
e^{i\phi} & 1\\
1 & e^{i\phi_}
\end{bmatrix}

For the beam splitter to conserve photon number its operator must be unitary, that is

 \hat{U}^\dagger \hat{U} = I

Therefore

 \frac{1}{2}\begin{bmatrix}
e^{-i\phi} & 1\\
1 & e^{-i\phi_}
\end{bmatrix}
\begin{bmatrix}
e^{i\phi} & 1\\
1 & e^{i\phi_}
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow \frac{1}{2}
\begin{bmatrix}
2 & (e^{i\phi}+e^{-i\phi})\\
(e^{i\phi}+e^{-i\phi}) & 2
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow
\begin{bmatrix}
1 & \cos \phi\\
\cos \phi & 1
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow \cos\phi = 0

\Rightarrow \phi = \frac{\pi}{2}+n\pi

In order to conserve photon number there must be a phase difference of 90 degrees between the transmitted and reflected waves.

The first in what may be the ‘What I’ve leared archive‘…

Optical components

A mathematical description of some of the basic building blocks of linear optics.

Single-mode phase shift

A phase shifter has an index of refraction that is different to that of free space.

The operation of the single mode phase shifter can be represented by the creation and annihilation operators defined as

 \hat{a}^\dagger\left|n\right>=\sqrt{n+1}\left|n+1\right>

 \hat{a}\left|n\right>=\sqrt{n}\left|n-1\right>

respectively. Where the operation of the single-mode phase shift is given by

 \hat{a}^\dagger_{out}=e^{i\phi\hat{a}^\dagger_{in}\hat{a}_{in}}a^\dagger_{in}e^{-i\phi\hat{a}^\dagger_{in}\hat{a}_{in}}=e^{i\phi}\hat{a}^\dagger_{in}

The interaction Hamiltonian is

 H_{\phi}=\phi\hat{a}^\dagger_{in}\hat{a}_{in}

Where

\hbar=1

Since the Hamiltonian is proportional to the number operator

\hat{n}=\hat{a}^\dagger\hat{a}

this means the number of photons is conserved.

Beam splitters

Beam splitters are fun, and far more complex. I shall return to them in the near future!

A demonstration of the link between the no cloning theorem and Heisenberg’s uncertainty principle.

In quantum information theory the no cloning theorem states that it is not possible to copy an unknown quantum state or quantum bit known as a qubit. The simplest proof I have come across to date goes a little something like this.

Assume there exists an unitary operator U (or a quantum gate) that can clone the unknown state

\left|\phi\right>=\alpha\left|0\right>+\beta\left|1\right>

We will show that acting on the tensor product of the unknown state \left|\phi\right> and a prepared state \left|0\right> with U, in two equivalent ways, gives an inconsistent result.

1. Acting on the unexpanded form of our unknown state

U\left|\phi\right>\left|0\right>=\left|\phi\right>\left|\phi\right>

which can be written as

U \left|\phi\right>\left|0\right>=(\alpha\left|0\right>+\beta\left|1\right>)(\alpha\left|0\right>+\beta\left|1\right>)

U \left|\phi\right>\left|0\right>=\alpha^2\left|00\right>+\alpha\beta\left|01\right>+\beta\alpha\left|10\right>+\beta^2\left|11\right>

2. Acting on the expanded form of our unknown state

U(\alpha\left|0\right>+\beta\left|1\right>)\left|0\right>=\alpha\left|00\right>+\beta\left|11\right>

From this we see that

U(\alpha \left| 0 \right>+\beta\left| 1 \right>)\left| 0 \right>\neq U\left|\phi\right>\left|0\right>

An inconsistency. This proves that there cannot exist a unitary operator that clones an unknown quantum state.

This is good.

Corollary

When a measurement is made on a quantum state, the state itself changes and information contained within the state is lost forever. It is therefore not possible to know simultaneously certain properties of a quantum state; such pairs of properties are known as conjugate variables. This premise of quantum mechanics is known as Heisenberg’s uncertainty principle (HUP). If we could clone an unknown quantum state we could simultaneously measure these conjugate variables and hence would be in complete violation of HUP.

Another way of looking at this is if we could measure all properties of an unknown quantum state simultaneously we could indeed recreate that state i.e. clone it.

Therefore we see that the no cloning theorem and Heisenberg’s uncertainty principle are two just two sides of the same coin.

Nice.

__________________________________

Unitary operators

A unitary operator has the property

U^\dagger U = I = identity \: matrix

In quantum computation we talk of a quantum gate acting on a quantum bit or qubit. A quantum gate can be thought of as an operator for the purposes of this discussion.

Since

\left|\varphi\right>=\alpha\left|0\right>+\beta\left|1\right>

has normalization condition

|\alpha|^2+|\beta|^2=1

It only makes sense that the post-operated state

\left|\varphi '\right>=\alpha '\left|0\right>+\beta '\left|1\right>

has the normalization condition

|\alpha '|^2+|\beta '|^2=1

This is accomplished by demanding the operator U is unitary. Amazingly it turns out that the unitary constraint is the only constraint placed on quantum gates, that is, any unitary matrix qualifies as a quantum gate.

Quantum Gates

We can represent a qubit in matrix form

\left|\varphi\right> =
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}
=\alpha\left|0\right>+\beta\left|1\right>

The quantum NOT gate, X, for example, which acts to change 0 –> 1 and 1 –> 0, takes the matrix form

X =
\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}

It operates on a single qubit as

X
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}
 =
\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}
=
\begin{bmatrix}
\beta\\
\alpha
\end{bmatrix}

And it is unitary

X^\dagger X =
\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}
=
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
=I