QC 4 QuantumBrain

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.

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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