Quantum Entanglement
Quantum Entanglement
Entangled QuBits consist of a superposition of single-particle states. A Bell pair describes the state of maximal entanglement for two entangled QuBits QA, QB:
\Ket{Q_{A},Q_{B}}=\frac{1}{\sqrt{2}}(\Ket{00}\pm\Ket{11}); \Ket{Q_{A},Q_{B}}=\frac{1}{\sqrt{2}}(\Ket{01}\pm\Ket{10})Excursus: Representation of two QuBits via tensor product
\Ket{a}=a_{00}\Ket{00}+a_{01}\Ket{01} +a_{10}\Ket{10} +a_{11}\Ket{11}=\begin{bmatrix} a_{00}\\ a_{01}\\ a_{10}\\ a_{11} \end{bmatrix}
For the probabilities, again:
|a_{00}|^2+|a_{01}|^2+|a_{10}|^2+|a_{11}|^2=1
In general, the state or states of two (or more QuBits) can be described by the tensor product (dyadic product):
\Ket{a}=\begin{bmatrix} a_{0} \\ a_{1} \end{bmatrix}; \Ket{b}=\begin{bmatrix} b_{0} \\ b_{1} \end{bmatrix}; \Ket{ab}=\Ket{a} \otimes \Ket{b}=\begin{bmatrix} a_{0} \times \begin{bmatrix} b_{0}\\ b_{1} \end{bmatrix} & \\ a_{1} \times \begin{bmatrix} b_{0}\\ b_{1} \end{bmatrix} & \end{bmatrix}=\begin{bmatrix} a_{0}b_{0}\\ a_{0}b_{1}\\ a_{1}b_{0} \\ a_{1}b_{1} \end{bmatrix}
In a quantum computer, entanglement can be created by using the Hadamard and CNOT gates:
The Hadamard gate
The Hadamard gate has the task of transferring the (absolute) state of a QuBit (|0〉or |1〉) into the superposition state [1]:
\Ket{1}
This corresponds on the Bloch Sphere to a projection in the equatorial plane. The Hadamard gate transforms the state |0〉to |+〉or |1〉to |-〉. If the QuBit would be measured in the state |+〉or |-〉, it would transition to the |1〉or |0〉state with a probability of 50%.
Excursus: Mathematical representation of a Hadamard gate
Mathematically, the Hadamard gate (Hadmard operator) is a 2×2 matrix that transforms the vectors (states) |0〉or |1〉into the vector (state) |+〉or |-〉:
H=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 &1 \\ 1 & -1 \end{bmatrix}\\ H\Ket{0}=\Ket{+}=\frac{1}{\sqrt{2}}(\Ket{0}+\Ket{1})\\ H\Ket{1}=\Ket{-}=\frac{1} {\sqrt{2}}(\Ket{0}-\Ket{1})
The CNOT gate
Two QuBits are required for this gate QA and QB. The QuBit QA is called the control bit, because the state of QB depends on its state [1]:
The CNOT gate inverts the value of QuBit QB only if QA has the value 1. Otherwise the value of QB remains unchanged.
Excursus: Mathematical representation of a CNOT gate
CNOT=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}; \Ket{a}=\begin{bmatrix} a_{00}\\ a_{01}\\ a_{10}\\ a_{11} \end{bmatrix}; CNOT\Ket{a}=\begin{bmatrix} a_{00}\\ a_{11}\\ a_{10}\\ a_{01} \end{bmatrix}
The CNOT gate thus swaps the amplitudes of a01 and a11.
Now a Bell pair or entanglement is to be generated with these two gates. This circuit entangles the two qubits QA and QB with the initial values 0 [1]:
Depending on the measurement result of the control QuBit QA, the QuBit QB is changed or not. After a measurement, the state of both qubits is |0,0〉or |1,1〉with a probability of 50%.
Sources
[1] M. Ellerhoff. Mit Quanten Rechnen. ISBN 978-3-658-31221-3