Quantum Mechanical Background

Quan­tum me­chan­ics de­scribes the phys­i­cal prop­er­ties of nature at the scale of atoms and sub­atom­ic par­ti­cles.

Spin

Spin is an in­trin­sic form of an­gu­lar mo­men­tum car­ried by el­e­men­tary par­ti­cles. It's a quan­tum me­chan­i­cal prop­er­ty with­out a clas­si­cal ana­logue. For ex­am­ple, an elec­tron has spin $1/2$. When mea­sured along a spe­cif­ic axis (e.g., z-axis), its spin com­po­nent can be found to be either "up" ($+\hbar/2$) or "down" ($-\hbar/2$), where $\hbar$ (h-bar) is the re­duced Planck con­stant. We often sim­pli­fy this to out­comes $+1$ and $-1$.

En­tan­gle­ment

Quan­tum en­tan­gle­ment is a phe­nom­e­non where two or more quan­tum par­ti­cles become linked in such a way that their fates are in­ter­twined, re­gard­less of the dis­tance sep­a­rat­ing them. Mea­sur­ing a prop­er­ty of one par­ti­cle in­stan­ta­neous­ly in­flu­ences the prop­er­ties of the other(s).

Con­sid­er a pair of en­tan­gled spin-$1/2$ par­ti­cles, like elec­trons, cre­at­ed in a sin­glet state. The total spin of the pair is zero. If one par­ti­cle is mea­sured to have spin-up along a cer­tain axis, the other par­ti­cle, when mea­sured along the same axis, will in­stant­ly be found to have spin-down, and vice-versa. This cor­re­la­tion is per­fect, no matter how far apart the par­ti­cles are.

The quan­tum state of such a pair (e.g., an elec­tron-positron pair from a pion decay) can be writ­ten as:

Where:

• $|\psi\rangle$ (psi) rep­re­sents the quan­tum state of the en­tan­gled pair.

• $|\uparrow_A \downarrow_B\rangle$ means par­ti­cle A has spin-up and par­ti­cle B has spin-down along a chosen axis.

• $|\downarrow_A \uparrow_B\rangle$ means par­ti­cle A has spin-down and par­ti­cle B has spin-up along that axis.

Mea­sure­ment in Quan­tum Me­chan­ics

In quan­tum me­chan­ics, mea­sure­ment is an active process that gen­er­al­ly dis­turbs the system being mea­sured. The out­come of a mea­sure­ment is prob­a­bilis­tic. For a par­ti­cle in a su­per­po­si­tion of states (like one of the en­tan­gled par­ti­cles before mea­sure­ment), the act of mea­sure­ment "col­laps­es" the wave­func­tion into a def­i­nite state.