# weylchamber.perfect_entanglers module¶

## Summary¶

Functions:

 F_PE Evaluate the Perfect-Entangler Functional concurrence Calculate the concurrence directly from the Weyl Chamber coordinates make_PE_krotov_chi_constructor Return a constructor for the χ’s in a PE optimization. project_to_PE Project onto the boundary surface of the perfect entanglers

## Reference¶

weylchamber.perfect_entanglers.project_to_PE(c1, c2, c3, check_weyl=True)[source]

Project onto the boundary surface of the perfect entanglers

Return new tuple (c1’, c2’, c3’) obtained by projecting the given input point (c1, c2, c3) onto the closest boundary of the perfect entanglers polyhedron. If the input point already is a perfect entangler, it will be returned unchanged

Example

>>> from weylchamber.visualize import WeylChamber
>>> print("%.2f, %.2f, %.2f" % tuple(project_to_PE(*WeylChamber.A3)))
0.50, 0.25, 0.25
>>> print("%.3f, %.3f, %.3f" % tuple(project_to_PE(0.5, 0.5, 0.25)))
0.500, 0.375, 0.125
>>> print("%.3f, %.3f, %.3f" % tuple(project_to_PE(0.25, 0, 0)))
0.375, 0.125, 0.000
>>> print("%.3f, %.3f, %.3f" % tuple(project_to_PE(0.75, 0, 0)))
0.625, 0.125, 0.000
>>> print("%.3f, %.3f, %.3f" % tuple(project_to_PE(0.3125, 0.0625, 0.01)))
0.375, 0.125, 0.010
>>> print("%.1f, %.1f, %.1f" % tuple(project_to_PE(0.5, 0, 0)))
0.5, 0.0, 0.0
>>> print("%.1f, %.1f, %.1f" % tuple(project_to_PE(0.5, 0.2, 0.2)))
0.5, 0.2, 0.2
>>> try:
...     project_to_PE(1.0, 0.5, 0)
... except ValueError as e:
...     print(e)
(1, 0.5, 0) is not in the Weyl chamber

Return type: Tuple[float, float, float]
weylchamber.perfect_entanglers.concurrence(c1, c2, c3)[source]

Calculate the concurrence directly from the Weyl Chamber coordinates

Example

>>> import qutip
>>> from weylchamber.coordinates import c1c2c3
>>> '%.1f' % concurrence(*c1c2c3(qutip.gates.swap()))
'0.0'
>>> '%.1f' % concurrence(*c1c2c3(qutip.gates.cnot()))
'1.0'
>>> '%.1f' % concurrence(*c1c2c3(qutip.gates.identity([2, 2])))
'0.0'

Return type: float
weylchamber.perfect_entanglers.F_PE(g1, g2, g3)[source]

Evaluate the Perfect-Entangler Functional

Example

>>> import qutip
>>> from weylchamber.local_invariants import g1g2g3
>>> "%.1f" % F_PE(*g1g2g3(qutip.gates.cnot()))
'0.0'
>>> "%.1f" % F_PE(*g1g2g3(qutip.gates.identity([2, 2])))
'2.0'

Return type: float
weylchamber.perfect_entanglers.make_PE_krotov_chi_constructor(canonical_basis, unitarity_weight=0)[source]

Return a constructor for the χ’s in a PE optimization.

Return a chi_constructor that determines the boundary condition of the backwards propagation in an optimization towards a perfect entangler in Krotov’s method, based on the foward-propagtion of the Bell states. In detail, the function returns a callable function that calculates

$\ket{\chi_{i}} = \frac{\partial F_{PE}}{\partial \bra{\phi_i}} \Bigg|_{\ket{\phi_{i}(T)}}$

for all $$i$$ with $$\ket{\phi_{0}(T)}, ..., \ket{\phi_{3}(T)}$$ the forward propagated Bell states at final time $$T$$, cf. Eq. (33b) in Ref. [1]. $$F_{PE}$$ is the perfect-entangler functional F_PE(). For the details of the derivative see Appendix G in Ref. [2].

References:

Parameters: canonical_basis (list[qutip.Qobj]) – A list of four basis states that define the canonical basis $$\ket{00}$$, $$\ket{01}$$, $$\ket{10}$$, and $$\ket{11}$$ of the logical subspace. unitarity_weight (float) – A weight in [0, 1] that determines how much emphasis is placed on maintaining population in the logical subspace. a function chi_constructor(fw_states_T, *args) that receive the result of a foward propagation of the Bell states (obtained from canonical_basis via weylchamber.gates.bell_basis()), and returns a list of statex $$\ket{\chi_{i}}$$ that are the boundary condition for the backward propagation in Krotov’s method. Positional arguments beyond fw_states_T are ignored. callable