Paradox

🔬 Setup:

• Different angle of spins do not commute eg $[S_x,S_z]\ne 0$ ie no simultaneous measurements

• We 2 particles in entangled states

  $$ |S_\text{tot}=0\rang = \frac{1}{\sqrt{2}} (\ket{\uparrow} \otimes \ket{\downarrow} - \ket{\uparrow} \otimes \ket{\uparrow}) $$

  such that the total spin between them equal $0$ ie $S_\text{tot}=0$

• We have 2 perpendicular detectors

  1. Detector 1 measures spin along the $x$-axis for one particle
  2. Detector 2 measures spin along the $z$-axis for the other particle

💥 Experimental results:

• Say our results are the following
  • Particle 1 measured on the $x$-axis (detector 1) has spin up ($+\frac 12$)
  • Particle 2 measured on the $z$-axis (detector 2) has spin up ($-\frac 12$)

📊 Analysis:

• Since the total spin of the system must be $0$ due to condition 2 then we expect
  • Particle $1$’s $z$-axis spin must be up ($+\frac 12$) to compensate for particle 2’s spin down ($-\frac 12$)
  • Particle $2$’s $x$-axis spin must be down ($-\frac 12$) to compensate for particle $1$’s spin up ($+\frac 12$)

🔀 Paradox:

• Our analysis leads us to believe that

  • Particle $1$ ha spin up ($+\frac 12$) along both $x$- and $z$-axes
  • Particle $2$ has spin down $(-\frac 12$) along both $x$- and $z$-axes

  Now this is impossible because that would mean we measured both $S_x$ and $S_z$ at the same time which would break our first condition ie it would break the uncertainty principle

Solution

🔬 Setup:

• We define the following detectors
  • $A$ is a detector set to the $z$-direction
  • $B$ is a detector set to a 45 degree angle between $z$- and $x$-directions
  • $C$ is a detector set to $x$-direction
• We use the following notation for our 2 electrons
  • E.g. $(A\uparrow,B,\downarrow)$ means particle one got measured at $A$ and got spin up etc.…

💥 Experimental solution:

• Bell’s inequality: if the way the electrons will go through any give orientation is set in advance,

  $$ N(A\uparrow,B\uparrow)+N(B\uparrow,C\uparrow )\ge N(A\uparrow,C\uparrow) $$

  where $N(A\uparrow ,B\uparrow)$ is the number of $(A\uparrow,B\uparrow)$ pairs etc

📊 Proof:

💼 Case: consider 3 distinct binary properties of an object, ie, properties a, b, c which can either be True or False, for this we use the notation $A$ for True and $\overline A$ for False

• Each object is described using the letters for example $A\overline{BC}$ which stands for a is true and b and c are false

  The Venn diagram represents every possible combination