12.3 Entangled and separable states

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §12.3. Rewritten in original wording — no textbook Entangle.py dump. Continues 12.2b. Next: 12.3b.

A state built as a tensor product $|\psi_A\rangle\otimes|\psi_B\rangle$ is separable. If two qubits interact so the joint state cannot be written as such a product, they are entangled. Distant spins in a total-spin-zero state remain correlated — measuring one constrains the other (Bell / Nobel 2022 context).

Separability test and Bell states

For $|\Psi\rangle=[w,x,y,z]^T$ in the computational basis, the state is separable iff a simple amplitude condition holds. Bell states fail that test. Pick $wz=xy$ and $|\beta_{00}\rangle$ among lookalikes.

Product states satisfy the test identically. The four Bell states are the usual maximally entangled basis (sign and pairing variants of $|00\rangle\pm|11\rangle$ and $|01\rangle\pm|10\rangle$).

Density matrix (preview)

$$\rho=\sum_i p_i\,|\psi_i\rangle\langle\psi_i|$$

encodes ensembles of pure states (useful later for mixed / noisy QC).

Pauli gates (QC names)

$$X=\begin{bmatrix}0&1\\1&0\end{bmatrix},\ Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix},\ Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.$$

Actions: $X|0\rangle=|1\rangle$, $X|1\rangle=|0\rangle$, $Z|0\rangle=|0\rangle$, $Z|1\rangle=-|1\rangle$, etc.

Physics: two dipoles

Dipole–dipole $H\propto \boldsymbol\sigma_A\cdot\boldsymbol\sigma_B-3(\boldsymbol\sigma_A\cdot\hat r)(\boldsymbol\sigma_B\cdot\hat r)$. For $\hat r=\hat z$,

$$H=\frac{\mu^2}{r^3}\bigl(X\otimes X+Y\otimes Y+Z\otimes Z-3\,Z\otimes Z\bigr) =\frac{\mu^2}{r^3} \begin{bmatrix}-2&0&0&0\\0&2&2&0\\0&2&2&0\\0&0&0&-2\end{bmatrix}.$$

Eigenvalues of $H/(\mu^2/r^3)$: $4,0,-2,-2$. Eigenvectors include Bell-like

$$\frac{|01\rangle+|10\rangle}{\sqrt2},\quad |11\rangle,\quad |00\rangle,\quad \frac{|01\rangle-|10\rangle}{\sqrt2}$$

— some separable, some entangled. Labs lock matrix entries and the separability measure.

Takeaway

Idea	Test / fact
Separable	$wz=xy$ for two-qubit amplitudes
Bell	Maximally entangled examples
$X,Y,Z$	Single-qubit Pauli gates
Dipoles	$H$ in $\otimes$ space; mixed eigenstates

Quiz

Entanglement

1 / 4

Two-qubit [w,x,y,z] is separable iff…

Pick an answer to see if you are right.

Next: 12.3b Entanglement labs.