"""
This module implements Pauli algebra by subclassing Symbol. Only algebraic
properties of Pauli matrices are used (we don't use the Matrix class).
See the documentation to the class Pauli for examples.
References
~~~~~~~~~~
.. [1] https://en.wikipedia.org/wiki/Pauli_matrices
"""
from __future__ import print_function, division
from sympy import Symbol, I, Mul, Pow, Add
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.
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 not i 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__(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(Pauli, self).__mul__(other)
def _eval_power(b, e):
if e.is_Integer and e.is_positive:
return super(Pauli, b).__pow__(int(e) % 2)
[docs]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,y
>>> 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)) | ((start == arg) & (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
