Quantum Computing

Quantum Entanglement

When two or more quantum particles interact in such a way that their properties become strongly correlated, they can enter a remarkable condition known as quantum entanglement. In this state, the particles no longer have independent descriptions; instead, they share a single, unified quantum state. As a result, measuring one particle instantly provides information about the other, even if they are separated by large distances.

Example: Two Entangled Electrons

Consider two electrons that are initially close to each other. According to the Pauli exclusion principle, no two electrons in the same atom can have identical sets of four quantum numbers (n, l, m, s). Therefore, even if both electrons share the same values for the first three quantum numbers (n, l, m), they must differ in their spin quantum number.

If the two-electron system is prepared in a special entangled configuration called the singlet state, their combined spin wavefunction is expressed as:

\( |\psi\rangle = \frac{1}{\sqrt{2}} \left( | \uparrow \rangle_1 | \downarrow \rangle_2 \;-\; | \downarrow \rangle_1 | \uparrow \rangle_2 \right) \)
  • \(| \uparrow \rangle\) represents spin-up
  • \(| \downarrow \rangle\) represents spin-down
  • Subscripts 1 and 2 identify electron 1 and electron 2

Now imagine physically separating the two electrons: one stays on Earth while the other is transported to a distant location in space. Despite the vast separation, they still share the same entangled state. Their measurement outcomes remain perfectly correlated:

  • If the electron on Earth is measured to have spin-up, then the distant electron will immediately be known to have spin-down.
  • If the Earth electron is measured to have spin-down, the distant one will collapse to the spin-up state.
Quantum Entanglement

Figure 1: Example of Quantum entanglement