1 About the videos to watch last time

Any questions?

2 Today

It's time to review some math fundamentals, from Quantum Computing for Computer Scientists. This will help with entanglement.

1. Read chapter 1 on your own; you should know it.

2. Chapter 2:

   1. Complex vector space, page 34.

      1. n-dim vector $\begin{vmatrix} c_0 \\ \cdots \\ c_{n-1}\end{vmatrix}$

      2. Add 2 vectors.

      3. Multiply vector by scalar.

      4. etc.

   2. Set of $m\times n$ complex matrices, $\mathbb{C}^{m\times n}$, is also a complex vector space.

      1. Transpose, conjugate, adjoint.

      2. Matrix mult is $\star$ in book.

   3. $\mathbb{C}^{m\times n}$ is a complex algebra.

   4. Set of polynomials in one variable of degree $le n$ is a complex vector space.

   5. State of a quantum system is a complex vector.

   6. You can make new vector spaces from combos of old ones.

      1. Cartesian product or direct sum.

      2. Just an ordered pair.

      3. $(v_1, v_2)$.

   7. Set of basis vectors for the vector space.

      1. Every vector $v$ in the space is a linear combo of the basis vectors.

      2. Represent $v$ as the list of weights.

      3. There are many possible basis sets.

      4. Each different basis set causes a different representation for the vectors.

      5. Convert: change of basis.

      6. Cartography example: NAD27, WGS84.

         A bridge between Switzerland and Germany across the Rhine River at Laufenberg had its two ends at different elevations because of a conversion error between two different basis systems.

      https://www.science20.com/news_articles/what_happens_bridge_when_one_side_uses_mediterranean_sea_level_and_another_north_sea-121600

      1. Hadamard matrix is an example.

   8. Section 2.4 Inner product, etc, p 53 will be covered later.

   9. Section 2.7 Tensor product of vector spaces, p 66.

      1. $\mathbb{V} \otimes \mathbb{V'}$.

      2. Let $dim(\mathbb{V})=p$ and $dim(\mathbb{V'})=q$ . Then $dim(\mathbb{V} \otimes \mathbb{V'}) = pq$.

      3. This is how quantum systems combine.

      4. Example 2.7.2 p 70.

3. Chapter 3 through 3.2, p 74-88.

3 Entanglement of 2-qbit system

State is a 4-vector of length 1.

It was originally created as the exterior product of two 2-vectors, the states of two separate 1-qbit systems.

Originally the separate 1-qbit systems didn't affect each other. Either could be transformed and measured.

Then 2-qbit system was rotated with a transformation matrix.

Now, perhaps it can be decomposed into the exterior product of two 2-vectors. Perhaps not.

1. Case 1: The 4-vector representing the new state can be decomposed.

   Then it's still really two separate 1-qbit systems.

   They can still either be transformed and measured.

   Measuring one qbit does not affect the other qbit.

2. Case 2: The 4-vector representing the new state cannot be decomposed.

   So the 2 qbits are now entangled.

   That means that measuring one qbit affects what you will see when you measure the other.

   It might just bias the probabilities of measuring the other qbit as 0 or 1.

   Or, it might totally control what you will see.

4 Paper asked about

Last time, there was a question about Asher Peres, Quantum Disentanglement and Computation, https://cds.cern.ch/record/330754/files/9707047.pdf

It's a matrix multiplication.