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: 'pt.Tensor | list') -> pt.Tensor

Implementation

def Perp(basis: 'pt.Tensor | list') -> pt.Tensor:
    if not isinstance(basis, pt.Tensor):
        basis = pt.as_tensor(np.asarray(basis))
    projector = sum(pt.stack([pt.outer(basis[i], basis[i].conj()) for i in range(len(basis))]))
    return pt.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: pt.Tensor, projector: pt.Tensor) -> pt.Tensor

Implementation

def Loss(F: pt.Tensor, projector: pt.Tensor) -> pt.Tensor:
    prod = F.H.dot(projector).dot(F)
    return prod.real

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 = pt.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(pt.min(minimas).item())