Source code for netket.experimental.observable.renyi2.S2_operator

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from textwrap import dedent

import jax.numpy as jnp
import numpy as np

from netket.hilbert import HomogeneousHilbert

from netket.operator._abstract_observable import AbstractObservable

[docs]
class Renyi2EntanglementEntropy(AbstractObservable):
r"""
RÃ©nyi2 bipartite entanglement entropy of a state :math:| \Psi \rangle
between partitions A and B.

"""

[docs]
def __init__(
self,
hilbert: None,
partition: jnp.array,
):
r"""
Constructs the operator computing the RÃ©nyi2 entanglement entropy of
a state :math:| \Psi \rangle for a partition with partition A:

.. math::

S_2 = -\log_2 \text{Tr}_A [\rho^2]

where :math:\rho = | \Psi \rangle \langle \Psi | is the density
matrix of the system and :math:\text{Tr}_A indicates the partial
trace over the partition A.

The Monte Carlo estimator of S_2 [Hastings et al., PRL 104, 157201 (2010)] is:

.. math::

S_2 = - \log \langle \frac{\Psi(\sigma,\eta^{\prime}) \Psi(\sigma^{\prime},\eta)}{\Psi(\sigma,\eta) \Psi(\sigma^{\prime},\eta^{\prime})} \rangle

where the mean is taken over the distribution
:math:\Pi(Ïƒ,Î·) \Pi(Ïƒ',Î·'), :math:\sigma \in A,
:math:\eta \in \bar{A} and
:math:\Pi(\sigma, \eta) = |\Psi(\sigma,\eta)|^2 / \langle \Psi | \Psi \rangle.

Args:
hilbert: hilbert space of the system.
partition: list of the indices identifying the degrees of
freedom in one partition of the full system. All
indices should be integers between 0 and hilbert.size

Returns:
RÃ©nyi2 operator for which computing the expected value.
"""

# Homogeneos, not discrete... because we don't support
# tensorhilbert and such. We could generalize easily by
# setting psi(x not in hilbert) = 0
if not isinstance(hilbert, HomogeneousHilbert):
raise TypeError(
dedent(
"""
Entanglement Entropy estimation is only implemented for
Homogeneous discrete Hilbert spaces.

It can be easily generalised to continuous spaces, so if
you want this feature get in touch with us!"
"""
)
)
else:
if hilbert.constrained:
raise ValueError(
dedent(
"""
Entanglement entropy estimation is not implemented
for constrained Hilbert spaces.

It can be generalised, so get in touch with us if you
need this feature.
"""
)
)

super().__init__(hilbert)

self._partition = np.array(list(set(partition)))

if (
np.where(self._partition < 0)[0].size > 0
or np.where(self._partition > hilbert.size - 1)[0].size > 0
):
raise ValueError(
"Invalid partition: possible negative indices or indices outside the system size."
)

@property
def partition(self):
r"""
list of indices for the degrees of freedom in the partition
"""
return self._partition

def __repr__(self):
return f"Renyi2EntanglementEntropy(hilbert={self.hilbert}, partition={self.partition})"