Lecture 10 (missing) - Entanglement

We just look at the slides:

![[Physics CPE 345 Quantum Computing Lecture slides Week 4 Multi qubit systems 240424 2-1.pdf#page=8]]

Note that if we measure a $| 0 ⟩$ :

\frac{1}{\sqrt{2}} (| 00 ⟩ + | 01 ⟩) = \frac{1}{\sqrt{2}} | 0 ⟩ (| 0 ⟩ + | 1 ⟩)

while if we measure a $| 1 ⟩$ :

\frac{1}{\sqrt{2}} (| 10 ⟩ - | 01 ⟩) = \frac{1}{\sqrt{2}} | 1 ⟩ (| 0 ⟩ - | 1 ⟩)

Note we have different common bases. We have the computational basis or $z$ -basis:

| 0 ⟩ = | ↕ ⟩, | 1 ⟩ = | \leftrightarrow ⟩

while we have the $x$ -basis:

| + ⟩, | - ⟩

where:

| + ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + | 1 ⟩), | - ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ - | 1 ⟩)

notice that:

⟨ + | + ⟩ = ⟨ - | - ⟩ = 1, ⟨ + | - ⟩ = ⟨ - | + ⟩ = 0

thus it's a valid orthonormal basis.

For our $y$ -basis:

| ↷ ⟩, | ↶ ⟩

defined by:

| ↷ ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + i | 1 ⟩) = | R ⟩

| ↶ ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ - i | 1 ⟩) = | L ⟩

We use the alternate $| R ⟩, | L ⟩$ symbols to not have the weird arrows.