Creation of Bell States:
Two qubits are maximally entangled when they are in a quantum Bell state. This means that one qubit is |j^> and the other |jv> and so are orthogonal to each other. Two qubits are put into a Bell state by first entangling them using a Haadamard gate and then putting them through a Controlled Not, CNOT, gate as in the diagram below. The CNOT bit-flips the second qubit in a multipart qubit if the first qubit is jv.
|??> ———– AH2 ————
¦
|??> ——————–CNOT—–
| Initial State | AH2 | CNOT |
| |j^j^> | 1/√2(|j^j^> + |jvj^>) | 1/√2(|j^j^> + |jvjv>) |
| ‑|j^j^> | –1/√2(|j^j^> + |jvj^>) | –1/√2(|j^j^> + |jvjv>) |
| |j^jv> | 1/√2(|j^jv> + |jvjv>) | 1/√2(|j^jv> + |jvj^>) |
| –|j^jv> | –1/√2(|j^jv> + |jvjv>) | –1/√2(|j^jv>+ |jvj^>) |
| |jvj^> | 1/√2(|j^j^> + –|jvj^>) | 1/√2(|j^j^> + –|jvjv>) |
| –|jvj^> | –1/√2(|j^j^> +|jvj^>) | 1/√2(–|j^j^> + |jvjv>) |
| |jvjv> | 1/√2(|j^jv> + –|jvjv>) | 1/√2(|j^jv> + –|jvj^>) |
| –|jvjv> | –1/√2(|j^jv> + |jvjv>) | 1/√2(–|j^jv> + |jvj^>) |
Opposite Direction: Bell Measurement
Note that all gates are unitary and as such the creation of quantum Bell states is reversible. Execute the CNOT and AH2 gates in reverse order to the creation of Bell states.
| Initial State | CNOT | AH2(Result of CNOT) |
| | | |
| 1/√2(|j^j^> + |jvjv>) from quantum Bell state created from |j^j^> | 1/√2(|j^j^> + |jvj^>) | 1/√2*1/√2 (|j^j^> + |jvj^> + |j^j^> + –|jvj^>) = 1/2(|j^j^> + |j^j^>) = |j^j^> |
| 1/√2(|j^jv> + –|jvj^>) from quantum Bell state created from |jvjv> | 1/√2(|j^jv> + –|jvjv>) | 1/√2*1/√2 (|j^jv> + |jvjv> + –|j^jv> + – –|jvjv>) = 1/2(|jvjv> + |jvjv>) = |jvjv> |
Quantum entanglement is very subtle. Even with two entangled qubits, the state of one of the qubits can still change as shown in the Bellagio Circuit. In this case q0 and q1 are entangled and then an X (Not) gate is applied to q1. This reverses later.
The effect of entanglement only occurs during measurement. Wikipedia describes this as:
‘For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise‘.
Next: Bellagio Circuit
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