def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', None)
exp = tensor_product_simp(self)
if indices is None or len(indices) == 0:
return Mul(*[Tr(arg).doit() for arg in exp.args])
else:
return Mul(*[
Tr(value).doit() if idx in indices else value
for idx, value in enumerate(exp.args)
])
def fidelity(state1, state2):
""" Computes the fidelity [1]_ between two quantum states
The arguments provided to this function should be a square matrix or a
Density object. If it is a square matrix, it is assumed to be diagonalizable.
Parameters
==========
state1, state2 : a density matrix or Matrix
Examples
========
>>> from sympy import S, sqrt
>>> from sympy.physics.quantum.dagger import Dagger
>>> from sympy.physics.quantum.spin import JzKet
>>> from sympy.physics.quantum.density import fidelity
>>> from sympy.physics.quantum.represent import represent
>>>
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> amp = 1/sqrt(2)
>>> updown = (amp*up) + (amp*down)
>>>
>>> # represent turns Kets into matrices
>>> up_dm = represent(up*Dagger(up))
>>> down_dm = represent(down*Dagger(down))
>>> updown_dm = represent(updown*Dagger(updown))
>>>
>>> fidelity(up_dm, up_dm)
1
>>> fidelity(up_dm, down_dm) #orthogonal states
0
>>> fidelity(up_dm, updown_dm).evalf().round(3)
0.707
References
==========
.. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states
"""
state1 = represent(state1) if isinstance(state1, Density) else state1
state2 = represent(state2) if isinstance(state2, Density) else state2
if not isinstance(state1, Matrix) or not isinstance(state2, Matrix):
raise ValueError("state1 and state2 must be of type Density or Matrix "
"received type=%s for state1 and type=%s for state2" %
(type(state1), type(state2)))
if state1.shape != state2.shape and state1.is_square:
raise ValueError("The dimensions of both args should be equal and the "
"matrix obtained should be a square matrix")
sqrt_state1 = state1**S.Half
return Tr((sqrt_state1 * state2 * sqrt_state1)**S.Half).doit()
def test_outer_product():
k = Ket('k')
b = Bra('b')
op = OuterProduct(k, b)
assert isinstance(op, OuterProduct)
assert isinstance(op, Operator)
assert op.ket == k
assert op.bra == b
assert op.label == (k, b)
assert op.is_commutative is False
op = k * b
assert isinstance(op, OuterProduct)
assert isinstance(op, Operator)
assert op.ket == k
assert op.bra == b
assert op.label == (k, b)
assert op.is_commutative is False
op = 2 * k * b
assert op == Mul(Integer(2), k, b)
op = 2 * (k * b)
assert op == Mul(Integer(2), OuterProduct(k, b))
assert Dagger(k * b) == OuterProduct(Dagger(b), Dagger(k))
assert Dagger(k * b).is_commutative is False
#test the _eval_trace
assert Tr(OuterProduct(JzKet(1, 1), JzBra(1, 1))).doit() == 1
# test scaled kets and bras
assert OuterProduct(2 * k, b) == 2 * OuterProduct(k, b)
assert OuterProduct(k, 2 * b) == 2 * OuterProduct(k, b)
# test sums of kets and bras
k1, k2 = Ket('k1'), Ket('k2')
b1, b2 = Bra('b1'), Bra('b2')
assert (OuterProduct(k1 + k2,
b1) == OuterProduct(k1, b1) + OuterProduct(k2, b1))
assert (OuterProduct(k1, b1 + b2) == OuterProduct(k1, b1) +
OuterProduct(k1, b2))
assert (OuterProduct(1 * k1 + 2 * k2,
3 * b1 + 4 * b2) == 3 * OuterProduct(k1, b1) +
4 * OuterProduct(k1, b2) + 6 * OuterProduct(k2, b1) +
8 * OuterProduct(k2, b2))
def test_eval_trace():
up = JzKet(S.Half, S.Half)
down = JzKet(S.Half, Rational(-1, 2))
d = Density((up, 0.5), (down, 0.5))
t = Tr(d)
assert t.doit() == 1
#test dummy time dependent states
class TestTimeDepKet(TimeDepKet):
def _eval_trace(self, bra, **options):
return 1
x, t = symbols('x t')
k1 = TestTimeDepKet(0, 0.5)
k2 = TestTimeDepKet(0, 1)
d = Density([k1, 0.5], [k2, 0.5])
assert d.doit() == (0.5 * OuterProduct(k1, k1.dual) +
0.5 * OuterProduct(k2, k2.dual))
t = Tr(d)
assert t.doit() == 1
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert str(t) == 'Tr(A*B)'
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', [])
return Tr(self.doit(), indices).doit()
def test_permute():
A, B, C, D, E, F, G = symbols('A B C D E F G', commutative=False)
t = Tr(A * B * C * D * E * F * G)
assert t.permute(0).args[0].args == (A, B, C, D, E, F, G)
assert t.permute(2).args[0].args == (F, G, A, B, C, D, E)
assert t.permute(4).args[0].args == (D, E, F, G, A, B, C)
assert t.permute(6).args[0].args == (B, C, D, E, F, G, A)
assert t.permute(8).args[0].args == t.permute(1).args[0].args
assert t.permute(-1).args[0].args == (B, C, D, E, F, G, A)
assert t.permute(-3).args[0].args == (D, E, F, G, A, B, C)
assert t.permute(-5).args[0].args == (F, G, A, B, C, D, E)
assert t.permute(-8).args[0].args == t.permute(-1).args[0].args
t = Tr((A + B) * (B * B) * C * D)
assert t.permute(2).args[0].args == (C, D, (A + B), (B**2))
t1 = Tr(A * B)
t2 = t1.permute(1)
assert id(t1) != id(t2) and t1 == t2
def test_trace_new():
a, b, c, d, Y = symbols('a b c d Y')
A, B, C, D = symbols('A B C D', commutative=False)
assert Tr(a + b) == a + b
assert Tr(A + B) == Tr(A) + Tr(B)
#check trace args not implicitly permuted
assert Tr(C * D * A * B).args[0].args == (C, D, A, B)
# check for mul and adds
assert Tr((a * b) + (c * d)) == (a * b) + (c * d)
# Tr(scalar*A) = scalar*Tr(A)
assert Tr(a * A) == a * Tr(A)
assert Tr(a * A * B * b) == a * b * Tr(A * B)
# since A is symbol and not commutative
assert isinstance(Tr(A), Tr)
#POW
assert Tr(pow(a, b)) == a**b
assert isinstance(Tr(pow(A, a)), Tr)
#Matrix
M = Matrix([[1, 1], [2, 2]])
assert Tr(M) == 3
##test indices in different forms
#no index
t = Tr(A)
assert t.args[1] == Tuple()
#single index
t = Tr(A, 0)
assert t.args[1] == Tuple(0)
#index in a list
t = Tr(A, [0])
assert t.args[1] == Tuple(0)
t = Tr(A, [0, 1, 2])
assert t.args[1] == Tuple(0, 1, 2)
#index is tuple
t = Tr(A, (0))
assert t.args[1] == Tuple(0)
t = Tr(A, (1, 2))
assert t.args[1] == Tuple(1, 2)
#trace indices test
t = Tr((A + B), [2])
assert t.args[0].args[1] == Tuple(2) and t.args[1].args[1] == Tuple(2)
t = Tr(a * A, [2, 3])
assert t.args[1].args[1] == Tuple(2, 3)
#class with trace method defined
#to simulate numpy objects
class Foo:
def trace(self):
return 1
assert Tr(Foo()) == 1
#argument test
# check for value error, when either/both arguments are not provided
raises(ValueError, lambda: Tr())
raises(ValueError, lambda: Tr(A, 1, 2))
def __new__(cls, *args):
from sympy.physics.quantum.trace import Tr
return Tr(*args)
def test_doit():
x, y = symbols('x y')
A, B, C, D, E, F = symbols('A B C D E F', commutative=False)
d = Density([XKet(), 0.5], [PxKet(), 0.5])
assert (0.5*(PxKet()*Dagger(PxKet())) +
0.5*(XKet()*Dagger(XKet()))) == d.doit()
# check for kets with expr in them
d_with_sym = Density([XKet(x*y), 0.5], [PxKet(x*y), 0.5])
assert (0.5*(PxKet(x*y)*Dagger(PxKet(x*y))) +
0.5*(XKet(x*y)*Dagger(XKet(x*y)))) == d_with_sym.doit()
d = Density([(A + B)*C, 1.0])
assert d.doit() == (1.0*A*C*Dagger(C)*Dagger(A) +
1.0*A*C*Dagger(C)*Dagger(B) +
1.0*B*C*Dagger(C)*Dagger(A) +
1.0*B*C*Dagger(C)*Dagger(B))
# With TensorProducts as args
# Density with simple tensor products as args
t = TensorProduct(A, B, C)
d = Density([t, 1.0])
assert d.doit() == \
1.0 * TensorProduct(A*Dagger(A), B*Dagger(B), C*Dagger(C))
# Density with multiple Tensorproducts as states
t2 = TensorProduct(A, B)
t3 = TensorProduct(C, D)
d = Density([t2, 0.5], [t3, 0.5])
assert d.doit() == (0.5 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
0.5 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density with mixed states
d = Density([t2 + t3, 1.0])
assert d.doit() == (1.0 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
1.0 * TensorProduct(A*Dagger(C), B*Dagger(D)) +
1.0 * TensorProduct(C*Dagger(A), D*Dagger(B)) +
1.0 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density operators with spin states
tp1 = TensorProduct(JzKet(1, 1), JzKet(1, -1))
d = Density([tp1, 1])
# full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(1, -1) * Dagger(JzKet(1, -1))
t = Tr(d, [1])
assert t.doit() == JzKet(1, 1) * Dagger(JzKet(1, 1))
# with another spin state
tp2 = TensorProduct(JzKet(S.Half, S.Half), JzKet(S.Half, Rational(-1, 2)))
d = Density([tp2, 1])
#full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(S.Half, Rational(-1, 2)) * Dagger(JzKet(S.Half, Rational(-1, 2)))
t = Tr(d, [1])
assert t.doit() == JzKet(S.Half, S.Half) * Dagger(JzKet(S.Half, S.Half))
def test_eval_trace():
# This test includes tests with dependencies between TensorProducts
#and density operators. Since, the test is more to test the behavior of
#TensorProducts it remains here
A, B, C, D, E, F = symbols('A B C D E F', commutative=False)
# Density with simple tensor products as args
t = TensorProduct(A, B)
d = Density([t, 1.0])
tr = Tr(d)
assert tr.doit() == 1.0 * Tr(A * Dagger(A)) * Tr(B * Dagger(B))
## partial trace with simple tensor products as args
t = TensorProduct(A, B, C)
d = Density([t, 1.0])
tr = Tr(d, [1])
assert tr.doit() == 1.0 * A * Dagger(A) * Tr(B * Dagger(B)) * C * Dagger(C)
tr = Tr(d, [0, 2])
assert tr.doit() == 1.0 * Tr(A * Dagger(A)) * B * Dagger(B) * Tr(
C * Dagger(C))
# Density with multiple Tensorproducts as states
t2 = TensorProduct(A, B)
t3 = TensorProduct(C, D)
d = Density([t2, 0.5], [t3, 0.5])
t = Tr(d)
assert t.doit() == (0.5 * Tr(A * Dagger(A)) * Tr(B * Dagger(B)) +
0.5 * Tr(C * Dagger(C)) * Tr(D * Dagger(D)))
t = Tr(d, [0])
assert t.doit() == (0.5 * Tr(A * Dagger(A)) * B * Dagger(B) +
0.5 * Tr(C * Dagger(C)) * D * Dagger(D))
#Density with mixed states
d = Density([t2 + t3, 1.0])
t = Tr(d)
assert t.doit() == (1.0 * Tr(A * Dagger(A)) * Tr(B * Dagger(B)) +
1.0 * Tr(A * Dagger(C)) * Tr(B * Dagger(D)) +
1.0 * Tr(C * Dagger(A)) * Tr(D * Dagger(B)) +
1.0 * Tr(C * Dagger(C)) * Tr(D * Dagger(D)))
t = Tr(d, [1])
assert t.doit() == (1.0 * A * Dagger(A) * Tr(B * Dagger(B)) +
1.0 * A * Dagger(C) * Tr(B * Dagger(D)) +
1.0 * C * Dagger(A) * Tr(D * Dagger(B)) +
1.0 * C * Dagger(C) * Tr(D * Dagger(D)))