tools/entanglement.py

Perp

Return the orthogonal projector onto the complement of a basis span.

Given a list/array of vectors $\{|b_i⟩\}$, forms $P=\sum _i|b_i⟩⟨b_i|$ and returns $I-P$.

def Perp(basis: 'np.ndarray | list') -> np.ndarray

Implementation

def Perp(basis: 'np.ndarray | list') -> np.ndarray:
    if not isinstance(basis, np.ndarray):
        basis = np.array(basis)
    projector = sum(np.array([np.outer(basis[i], basis[i].conj().T) for i in range(len(basis))]))
    return np.eye(len(basis[0])) - projector

Loss

Quadratic form $⟨F|\mathrm{\Pi }|F⟩$ used as an objective.

Computes $F^{†}\mathrm{\Pi }F$ (with $\mathrm{\Pi }$ typically a projector).

NOTE

for np.ndarray, .H may not exist; callers typically pass an np.matrix.

def Loss(F: 'np.matrix | np.ndarray', projector: np.ndarray) -> np.ndarray

Implementation

def Loss(F: 'np.matrix | np.ndarray', projector: np.ndarray) -> np.ndarray:
    prod = F.H.dot(projector).dot(F)
    return np.real_if_close(prod, tol=ROUNDOFF_TOL)

rank

Multi-start minimization over a rank-parameterized family.

Runs scipy.optimize.minimize from several random initial points and returns the smallest objective value found.

def rank(f: object, D: int, r: int) -> float

Implementation

def rank(f: object, D: int, r: int, **kwargs: object) -> float:
    kwargs_any = dict(kwargs)
    if 'method' not in kwargs_any:
        kwargs_any['method'] = 'Powell'
    if 'tol' not in kwargs_any:
        kwargs_any['tol'] = ROUNDOFF_TOL
    tries = int(kwargs_any.get('tries', 1))
    rmax = float(kwargs_any.get('r_max', 2 ** 7))
    size = int((2 * D + 1) * (r - 1))
    if 'tries' in kwargs_any:
        del kwargs_any['tries']
    minimas = np.ones(tries)
    for i in range(tries):
        try:
            minimas[i] = minimize(f, x0=np.random.uniform(-rmax, rmax, size), **kwargs_any).fun
        except Exception:
            pass
    return float(np.min(minimas))