ai-content-maker/.venv/Lib/site-packages/sympy/physics/paulialgebra.py

"""
This module implements Pauli algebra by subclassing Symbol. Only algebraic
properties of Pauli matrices are used (we do not use the Matrix class).

See the documentation to the class Pauli for examples.

References
==========

.. [1] https://en.wikipedia.org/wiki/Pauli_matrices
"""

from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.numbers import I
from sympy.core.power import Pow
from sympy.core.symbol import Symbol
from sympy.physics.quantum import TensorProduct

__all__ = ['evaluate_pauli_product']


def delta(i, j):
    """
    Returns 1 if ``i == j``, else 0.

    This is used in the multiplication of Pauli matrices.

    Examples
    ========

    >>> from sympy.physics.paulialgebra import delta
    >>> delta(1, 1)
    1
    >>> delta(2, 3)
    0
    """
    if i == j:
        return 1
    else:
        return 0


def epsilon(i, j, k):
    """
    Return 1 if i,j,k is equal to (1,2,3), (2,3,1), or (3,1,2);
    -1 if ``i``,``j``,``k`` is equal to (1,3,2), (3,2,1), or (2,1,3);
    else return 0.

    This is used in the multiplication of Pauli matrices.

    Examples
    ========

    >>> from sympy.physics.paulialgebra import epsilon
    >>> epsilon(1, 2, 3)
    1
    >>> epsilon(1, 3, 2)
    -1
    """
    if (i, j, k) in ((1, 2, 3), (2, 3, 1), (3, 1, 2)):
        return 1
    elif (i, j, k) in ((1, 3, 2), (3, 2, 1), (2, 1, 3)):
        return -1
    else:
        return 0


class Pauli(Symbol):
    """
    The class representing algebraic properties of Pauli matrices.

    Explanation
    ===========

    The symbol used to display the Pauli matrices can be changed with an
    optional parameter ``label="sigma"``. Pauli matrices with different
    ``label`` attributes cannot multiply together.

    If the left multiplication of symbol or number with Pauli matrix is needed,
    please use parentheses  to separate Pauli and symbolic multiplication
    (for example: 2*I*(Pauli(3)*Pauli(2))).

    Another variant is to use evaluate_pauli_product function to evaluate
    the product of Pauli matrices and other symbols (with commutative
    multiply rules).

    See Also
    ========

    evaluate_pauli_product

    Examples
    ========

    >>> from sympy.physics.paulialgebra import Pauli
    >>> Pauli(1)
    sigma1
    >>> Pauli(1)*Pauli(2)
    I*sigma3
    >>> Pauli(1)*Pauli(1)
    1
    >>> Pauli(3)**4
    1
    >>> Pauli(1)*Pauli(2)*Pauli(3)
    I

    >>> from sympy.physics.paulialgebra import Pauli
    >>> Pauli(1, label="tau")
    tau1
    >>> Pauli(1)*Pauli(2, label="tau")
    sigma1*tau2
    >>> Pauli(1, label="tau")*Pauli(2, label="tau")
    I*tau3

    >>> from sympy import I
    >>> I*(Pauli(2)*Pauli(3))
    -sigma1

    >>> from sympy.physics.paulialgebra import evaluate_pauli_product
    >>> f = I*Pauli(2)*Pauli(3)
    >>> f
    I*sigma2*sigma3
    >>> evaluate_pauli_product(f)
    -sigma1
    """

    __slots__ = ("i", "label")

    def __new__(cls, i, label="sigma"):
        if i not in [1, 2, 3]:
            raise IndexError("Invalid Pauli index")
        obj = Symbol.__new__(cls, "%s%d" %(label,i), commutative=False, hermitian=True)
        obj.i = i
        obj.label = label
        return obj

    def __getnewargs_ex__(self):
        return (self.i, self.label), {}

    def _hashable_content(self):
        return (self.i, self.label)

    # FIXME don't work for -I*Pauli(2)*Pauli(3)
    def __mul__(self, other):
        if isinstance(other, Pauli):
            j = self.i
            k = other.i
            jlab = self.label
            klab = other.label

            if jlab == klab:
                return delta(j, k) \
                    + I*epsilon(j, k, 1)*Pauli(1,jlab) \
                    + I*epsilon(j, k, 2)*Pauli(2,jlab) \
                    + I*epsilon(j, k, 3)*Pauli(3,jlab)
        return super().__mul__(other)

    def _eval_power(b, e):
        if e.is_Integer and e.is_positive:
            return super().__pow__(int(e) % 2)


def evaluate_pauli_product(arg):
    '''Help function to evaluate Pauli matrices product
    with symbolic objects.

    Parameters
    ==========

    arg: symbolic expression that contains Paulimatrices

    Examples
    ========

    >>> from sympy.physics.paulialgebra import Pauli, evaluate_pauli_product
    >>> from sympy import I
    >>> evaluate_pauli_product(I*Pauli(1)*Pauli(2))
    -sigma3

    >>> from sympy.abc import x
    >>> evaluate_pauli_product(x**2*Pauli(2)*Pauli(1))
    -I*x**2*sigma3
    '''
    start = arg
    end = arg

    if isinstance(arg, Pow) and isinstance(arg.args[0], Pauli):
        if arg.args[1].is_odd:
            return arg.args[0]
        else:
            return 1

    if isinstance(arg, Add):
        return Add(*[evaluate_pauli_product(part) for part in arg.args])

    if isinstance(arg, TensorProduct):
        return TensorProduct(*[evaluate_pauli_product(part) for part in arg.args])

    elif not(isinstance(arg, Mul)):
        return arg

    while not start == end or start == arg and end == arg:
        start = end

        tmp = start.as_coeff_mul()
        sigma_product = 1
        com_product = 1
        keeper = 1

        for el in tmp[1]:
            if isinstance(el, Pauli):
                sigma_product *= el
            elif not el.is_commutative:
                if isinstance(el, Pow) and isinstance(el.args[0], Pauli):
                    if el.args[1].is_odd:
                        sigma_product *= el.args[0]
                elif isinstance(el, TensorProduct):
                    keeper = keeper*sigma_product*\
                        TensorProduct(
                            *[evaluate_pauli_product(part) for part in el.args]
                        )
                    sigma_product = 1
                else:
                    keeper = keeper*sigma_product*el
                    sigma_product = 1
            else:
                com_product *= el
        end = tmp[0]*keeper*sigma_product*com_product
        if end == arg: break
    return end