coordinate transformation

I got confused over time of coordinate transformation for a point in space v.s. transformation of a point location with fixed coordinates. It turns out that the former is the exact reverse of the latter. I don’t include a proof here, but it should be self-evident. Also it is much easier to understand coordinate transformation…

Free energy

When we model probability of a variable $latex x$ by $latex p(x) = {e^{-\frac{F(x)}{T}}}$, $latex F(x)$ is often referred to as the free energy. The name is coming from historical reason. The Gibbs-Boltzmann distribution for a configuration is proportional to $latex e^{-\frac{H}{k_B T}}$. And the closest reason I found is from here $latex p(H,T) =…

Convex conjugate

I wasn’t aware that convex conjugate is just Legendre transformation. That is, $latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde {x}}\rangle -f({\tilde {x}})\}\geq \langle p,x\rangle -f(x)$   Note that it is also known as the Legendre-Fenchel transformation. Btw, we have the Fenchel inequality $latex \langle p,x\rangle \le f(x)+f^*(p)$ directly from the definition since $latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde…

Bernstein-Vazirani Algorithm

Here is some personal notes on Bernstein-Vazirani algorithm. Assume that we have a black box that does nothing but compute $latex u \cdot x$, where $latex x$ is an input binary vector and $latex u$ is another binary vector that we don’t know its value. Note that all operation is bit-wise and so the output…

Hadamard gates on multiple qubits

The effect of applying Hadamard gates to multiple qubits is rather subtle and is used in many quantum algorithms. Consider $latex n$ input qubits, $latex |u_1 u_2 u_3 \cdots u_n\rangle$. What will be the output if we apply the Hadamard gate to each of the qubit? Note that a Hadamard gate maps $latex |0\rangle$ to…

Quantum Teleportation

Let say Alice wants to pass her qubit $latex |\psi\rangle=a|0\rangle+b|1\rangle$ to Bob. By no cloning theorem, we know that Alice cannot duplicate her qubit to Bob. However, she can “teleport” her qubit to Bob if they share a Bell state $latex \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. First, Alice will apply CNOT gate from $latex |\psi\rangle$ to her qubit in…

Proof of no cloning theorem

I’m still studying the online course on quantum computing where Prof. Vazirani went through the no cloning theorem. It is very interesting to see that something that can lead to some very subtle consequence can also have a very simple proof. Below is a summary. Basically, the theorem states that no quantum state can be…

Bell’s Experiment

I am studying the online class on quantum computing (coursera). I did a quantum mechanic course more than a decade ago but I didn’t go into the detail of the Bell’s experiment then. The explanation from Prof. Vazirani is great and I feel that I finally understand the setup after all these years! Here, I…