Bell State Preparation

Bell states are the simplest entangled states and the foundation of quantum information. They were proposed by physicist John Bell in 1964 to test whether quantum mechanics could be explained by hidden local variables. The answer was no — and Bell states became the cornerstone of quantum teleportation, superdense coding, quantum key distribution, and error correction. If you understand Bell states, you understand the core resource that makes quantum computing different from classical computing.

Measuring 00 or 11 with equal probability (never 01 or 10) confirms the Bell state. The perfect correlation cannot be explained by each qubit independently deciding its value.

The Bell state $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ is a maximally entangled state of two qubits. It cannot be written as a product of individual qubit states ($|{\psi }_A⟩$ times $|{\psi }_B⟩$, which is the mathematical definition of entanglement. If you mathematically remove one qubit from the description (a process called tracing out), the remaining qubit is in a completely mixed state -- meaning it carries zero information about the outcome. All the information lives in the correlations between the two qubits. There are four Bell states, and they form a complete basis for the two-qubit state space: |$\Phi$±⟩ = ($|00⟩$±$|11⟩/\sqrt{2}$ and |$\Psi$±⟩ = ($|01⟩$±$|10⟩/\sqrt{2}$.