## Convert str(array) back to numpy array

If we print a numpy array, which actually use str(), we will find it almost irreversible.

In [5]:
l=arange(16).reshape(4,4)
print('l is printed as:\n', l)

l is printed as:
[[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
[12 13 14 15]]


Use print() will fallback to str(), so str() is not the correct way.

• repr()
• .tolist()

## # Reduced density matrix

Suppose we have two quantum systems $a, b$, with dimension $N_a, N_b$ respectively. Then the Hilbert space of $a+b$ is of dimension $N=N_aN_b$. Suppose we have a density matrix $$\hat\rho=\sum_{i,j}\rho_{ij}\lvert i\rangle \langle j\rvert=\sum_{i,j,k,l}\rho_{ijkl}\lvert i\rangle_a\lvert j\rangle_b \langle k\rvert_a\langle l\rvert_b$$

Then the reduced density matrix of $a$ is defined as $$\hat\rho_a=\mathrm{tr}_b\hat\rho=\sum_i \langle i\rvert_b\hat\rho\lvert i\rangle_b$$

i.e. reduced density matrix problem is equivalent to partial trace problem.

### # Tensor

In fact, if we take $\hat\rho$ as a 4-tensor $\rho_{ijkl}$, then the reduced density matrix is $$\rho^{(a)}_{ij}=\delta^{\mu\nu}\rho_{i\mu k\nu}$$ For simple density matrix $\rho=\lvert \psi\rangle \langle \psi\rvert$, the reduced matrix is $$\rho^{(a)}_{ik}=\delta^{jl}\rho_{ijkl}=\delta^{jl}\psi_{ij}\psi^+_{lk}=\sum_i |\langle i_b\lvert \psi\rangle|^2=[\psi\psi^+]_{ik}$$ Here we are taking $\psi$ as an $N_a\times N_b$ matrix.

For general case, if we find decomposition $$\rho=\sum_c \lambda_c\lvert \psi_c\rangle \langle \psi_c\rvert,\quad \sum_c \lambda_c=1$$ then we have $$\rho^{(a)}_{ik}=\left[\sum_c\lambda_c\psi_c\psi^+_c\right]_{ik}$$

## One way quantum computer

The one-way or measurement based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.

## Cluster States

In quantum information and quantum computing, a cluster state is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d-dimensional lattice, and a cluster state is a pure state of the qubits located on C. Another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d-dimensional lattice. Cluster states are especially useful in the context of the one-way quantum computer.

## Renyi entropy of the wormholes

In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The Rényi entropy is named after Alfréd Rényi.

## Fermat’s Principle

Fermat’s Principle is

$$\delta s=\delta\int n\dd \ell=0$$

If we choose the path …

## Invariance of Euler-Lagrange Equations

E-L is deduced from the Hamilton’s principle

## Nowtonian

Assume $$\bm F$$ is active force and $$\bm R$$ is constraint force. Then for …