def test_sum_instantiator():
"""Test use of Sum instantiator."""
i = IdxSym('i')
j = IdxSym('j')
ket_i = BasisKet(FockIndex(i), hs=0)
ket_j = BasisKet(FockIndex(j), hs=0)
A_i = OperatorSymbol(StrLabel(IndexedBase('A')[i]), hs=0)
hs0 = LocalSpace('0')
sum = Sum(i)(ket_i)
ful = KetIndexedSum(ket_i, ranges=IndexOverFockSpace(i, hs=hs0))
assert sum == ful
assert sum == Sum(i, hs0)(ket_i)
assert sum == Sum(i, hs=hs0)(ket_i)
sum = Sum(i, 1, 10)(ket_i)
ful = KetIndexedSum(ket_i, ranges=IndexOverRange(i, 1, 10))
assert sum == ful
assert sum == Sum(i, 1, 10, 1)(ket_i)
assert sum == Sum(i, 1, to=10, step=1)(ket_i)
assert sum == Sum(i, 1, 10, step=1)(ket_i)
sum = Sum(i, (1, 2, 3))(ket_i)
ful = KetIndexedSum(ket_i, ranges=IndexOverList(i, (1, 2, 3)))
assert sum == KetIndexedSum(ket_i, ranges=IndexOverList(i, (1, 2, 3)))
assert sum == Sum(i, [1, 2, 3])(ket_i)
sum = Sum(i)(Sum(j)(ket_i * ket_j.dag()))
ful = OperatorIndexedSum(
ket_i * ket_j.dag(),
ranges=(IndexOverFockSpace(i, hs0), IndexOverFockSpace(j, hs0)),
)
assert sum == ful
def test_unicode_ket_operations():
"""Test the unicode representation of ket operations"""
hs1 = LocalSpace('q_1', basis=('g', 'e'))
hs2 = LocalSpace('q_2', basis=('g', 'e'))
ket_g1 = BasisKet('g', hs=hs1)
ket_e1 = BasisKet('e', hs=hs1)
ket_g2 = BasisKet('g', hs=hs2)
ket_e2 = BasisKet('e', hs=hs2)
psi1 = KetSymbol("Psi_1", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
phi = KetSymbol("Phi", hs=hs2)
A = OperatorSymbol("A_0", hs=hs1)
gamma = symbols('gamma', positive=True)
alpha = symbols('alpha')
beta = symbols('beta')
phase = exp(-I * gamma)
i = IdxSym('i')
assert unicode(psi1 + psi2) == '|Ψ₁⟩^(q₁) + |Ψ₂⟩^(q₁)'
assert unicode(psi1 * phi) == '|Ψ₁⟩^(q₁) ⊗ |Φ⟩^(q₂)'
assert unicode(phase * psi1) == 'exp(-ⅈ γ) |Ψ₁⟩^(q₁)'
assert unicode((alpha + 1) * KetSymbol('Psi', hs=0)) == '(α + 1) |Ψ⟩⁽⁰⁾'
assert (unicode(
A * psi1) == 'A\u0302_0^(q\u2081) |\u03a8\u2081\u27e9^(q\u2081)')
# Â_0^(q₁) |Ψ₁⟩^(q₁)
assert unicode(BraKet(psi1, psi2)) == '⟨Ψ₁|Ψ₂⟩^(q₁)'
expr = BraKet(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert unicode(expr) == '⟨Ψ₁(α)|Ψ₂(β)⟩^(q₁)'
assert unicode(ket_e1.dag() * ket_e1) == '1'
assert unicode(ket_g1.dag() * ket_e1) == '0'
assert unicode(KetBra(psi1, psi2)) == '|Ψ₁⟩⟨Ψ₂|^(q₁)'
expr = KetBra(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert unicode(expr) == '|Ψ₁(α)⟩⟨Ψ₂(β)|^(q₁)'
bell1 = (ket_e1 * ket_g2 - I * ket_g1 * ket_e2) / sqrt(2)
bell2 = (ket_e1 * ket_e2 - ket_g1 * ket_g2) / sqrt(2)
assert unicode(bell1) == '1/√2 (|eg⟩^(q₁⊗q₂) - ⅈ |ge⟩^(q₁⊗q₂))'
assert (unicode(BraKet.create(
bell1,
bell2)) == r'1/2 (⟨eg|^(q₁⊗q₂) + ⅈ ⟨ge|^(q₁⊗q₂)) (|ee⟩^(q₁⊗q₂) - '
r'|gg⟩^(q₁⊗q₂))')
assert (unicode(KetBra.create(
bell1,
bell2)) == r'1/2 (|eg⟩^(q₁⊗q₂) - ⅈ |ge⟩^(q₁⊗q₂))(⟨ee|^(q₁⊗q₂) - '
r'⟨gg|^(q₁⊗q₂))')
assert (unicode(
KetBra.create(bell1, bell2),
show_hs_label=False) == r'1/2 (|eg⟩ - ⅈ |ge⟩)(⟨ee| - ⟨gg|)')
expr = KetBra(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert unicode(expr) == "|Ψ⟩⟨i|⁽⁰⁾"
expr = KetBra(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert unicode(expr) == "|i⟩⟨Ψ|⁽⁰⁾"
expr = BraKet(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert unicode(expr) == "⟨Ψ|i⟩⁽⁰⁾"
expr = BraKet(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert unicode(expr) == "⟨i|Ψ⟩⁽⁰⁾"
def test_operator_kronecker_sum():
"""Test that Kronecker delta are eliminiated from indexed sums over
operators"""
i = IdxSym('i')
j = IdxSym('j')
alpha = symbols('alpha')
delta_ij = KroneckerDelta(i, j)
delta_0i = KroneckerDelta(0, i)
delta_1j = KroneckerDelta(1, j)
delta_0j = KroneckerDelta(0, j)
delta_1i = KroneckerDelta(1, i)
def A(i, j):
return OperatorSymbol(StrLabel(IndexedBase('A')[i, j]), hs=0)
term = delta_ij * A(i, j)
sum = OperatorIndexedSum.create(term,
ranges=(IndexOverList(i, (1, 2)),
IndexOverList(j, (1, 2))))
assert sum == OperatorIndexedSum.create(A(i, i),
ranges=(IndexOverList(i,
(1, 2)), ))
assert sum.doit() == (OperatorSymbol("A_11", hs=0) +
OperatorSymbol("A_22", hs=0))
term = alpha * delta_ij * A(i, j)
range_i = IndexOverList(i, (1, 2))
range_j = IndexOverList(j, (1, 2))
sum = OperatorIndexedSum.create(term, ranges=(range_i, range_j))
assert isinstance(sum, ScalarTimesOperator)
expected = alpha * OperatorIndexedSum.create(
A(i, i), ranges=(IndexOverList(i, (1, 2)), ))
assert sum == expected
hs = LocalSpace('0', basis=('g', 'e'))
i_range = IndexOverFockSpace(i, hs)
j_range = IndexOverFockSpace(j, hs)
sig_ij = LocalSigma(FockIndex(i), FockIndex(j), hs=hs)
sig_0j = LocalSigma('g', FockIndex(j), hs=hs)
sig_i1 = LocalSigma(FockIndex(i), 'e', hs=hs)
term = delta_0i * delta_1j * sig_ij
sum = OperatorIndexedSum.create(term, ranges=(i_range, ))
expected = delta_1j * sig_0j
assert sum == expected
sum = OperatorIndexedSum.create(term, ranges=(j_range, ))
expected = delta_0i * sig_i1
assert sum == expected
term = (delta_0i * delta_1j + delta_0j * delta_1i) * sig_ij
sum = OperatorIndexedSum.create(term, ranges=(i_range, j_range))
expected = LocalSigma('g', 'e', hs=hs) + LocalSigma('e', 'g', hs=hs)
assert sum == expected
def test_qubit_state_bra():
"""Test sum_i alpha_i <i| for TLS"""
i = IdxSym('i')
alpha = IndexedBase('alpha')
alpha_i = alpha[i]
hs_tls = LocalSpace('tls', basis=('g', 'e'))
term = alpha_i * BasisKet(FockIndex(i), hs=hs_tls).dag()
expr = KetIndexedSum.create(term, ranges=IndexOverFockSpace(i, hs=hs_tls))
assert IndexOverFockSpace(i, hs=hs_tls) in expr.ket.kwargs['ranges']
assert ascii(expr) == "Sum_{i in H_tls} alpha_i * <i|^(tls)"
assert expr.ket.term.free_symbols == set([i, symbols('alpha'), alpha_i])
assert expr.free_symbols == set([symbols('alpha'), alpha_i])
assert expr.ket.variables == [i]
assert expr.space == hs_tls
assert len(expr.ket.args) == 1
assert len(expr.ket.operands) == 1
assert len(expr.ket.kwargs) == 1
assert expr.ket.args[0] == term.ket
assert expr.ket.term == term.ket
assert len(expr.kwargs) == 0
expr_expand = Bra.create(expr.ket.doit().substitute({
alpha[0]: alpha['g'],
alpha[1]: alpha['e']
}))
assert expr_expand == (alpha['g'] * BasisKet('g', hs=hs_tls).dag() +
alpha['e'] * BasisKet('e', hs=hs_tls).dag())
assert ascii(expr_expand) == 'alpha_e * <e|^(tls) + alpha_g * <g|^(tls)'
def test_create_on_fock_expansion():
"""Test ``Create * sum_i alpha_i |i> = sqrt(i+1) * alpha_i * |i+1>``"""
i = IdxSym('i')
alpha = IndexedBase('alpha')
hs = LocalSpace('0', dimension=3)
expr = Create(hs=hs) * KetIndexedSum(
alpha[i] * BasisKet(FockIndex(i), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
assert expr == KetIndexedSum(
sympy.sqrt(i + 1) * alpha[i] * BasisKet(FockIndex(i + 1), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
assert expr.doit() == (alpha[0] * BasisKet(1, hs=hs) +
sympy.sqrt(2) * alpha[1] * BasisKet(2, hs=hs))
def test_ascii_symbolic_labels():
"""Test ascii representation of symbols with symbolic labels"""
i = IdxSym('i')
j = IdxSym('j')
hs0 = LocalSpace(0)
hs1 = LocalSpace(1)
Psi = IndexedBase('Psi')
assert ascii(BasisKet(FockIndex(2 * i), hs=hs0)) == '|2*i>^(0)'
assert ascii(KetSymbol(StrLabel(2 * i), hs=hs0)) == '|2*i>^(0)'
assert (ascii(KetSymbol(StrLabel(Psi[i, j]),
hs=hs0 * hs1)) == '|Psi_ij>^(0*1)')
expr = BasisKet(FockIndex(i), hs=hs0) * BasisKet(FockIndex(j), hs=hs1)
assert ascii(expr) == '|i,j>^(0*1)'
assert ascii(Bra(BasisKet(FockIndex(2 * i), hs=hs0))) == '<2*i|^(0)'
assert (ascii(LocalSigma(FockIndex(i), FockIndex(j),
hs=hs0)) == '|i><j|^(0)')
expr = CoherentStateKet(symbols('alpha'), hs=1).to_fock_representation()
assert (ascii(expr) == 'exp(-alpha*conjugate(alpha)/2) * '
'(Sum_{n in H_1} alpha**n/sqrt(n!) * |n>^(1))')
tls = SpinSpace(label='s', spin='1/2', basis=('down', 'up'))
Sig = IndexedBase('sigma')
n = IdxSym('n')
Sig_n = OperatorSymbol(StrLabel(Sig[n]), hs=tls)
assert ascii(Sig_n, show_hs_label=False) == 'sigma_n'
def test_tex_symbolic_labels():
"""Test tex representation of symbols with symbolic labels"""
i = IdxSym('i')
j = IdxSym('j')
hs0 = LocalSpace(0)
hs1 = LocalSpace(1)
Psi = IndexedBase('Psi')
with configure_printing(tex_use_braket=True):
assert latex(BasisKet(FockIndex(2 * i), hs=hs0)) == r'\Ket{2 i}^{(0)}'
assert latex(KetSymbol(StrLabel(2 * i), hs=hs0)) == r'\Ket{2 i}^{(0)}'
assert (latex(KetSymbol(StrLabel(Psi[i, j]), hs=hs0 *
hs1)) == r'\Ket{\Psi_{i j}}^{(0 \otimes 1)}')
expr = BasisKet(FockIndex(i), hs=hs0) * BasisKet(FockIndex(j), hs=hs1)
assert latex(expr) == r'\Ket{i,j}^{(0 \otimes 1)}'
assert (latex(Bra(BasisKet(FockIndex(2 * i),
hs=hs0))) == r'\Bra{2 i}^{(0)}')
assert (latex(LocalSigma(FockIndex(i), FockIndex(j),
hs=hs0)) == r'\Ket{i}\!\Bra{j}^{(0)}')
alpha = symbols('alpha')
expr = CoherentStateKet(alpha, hs=1).to_fock_representation()
assert (latex(expr) == r'e^{- \frac{\alpha \overline{\alpha}}{2}} '
r'\left(\sum_{n \in \mathcal{H}_{1}} '
r'\frac{\alpha^{n}}{\sqrt{n!}} \Ket{n}^{(1)}\right)')
assert (latex(
expr, conjg_style='star') == r'e^{- \frac{\alpha {\alpha}^*}{2}} '
r'\left(\sum_{n \in \mathcal{H}_{1}} '
r'\frac{\alpha^{n}}{\sqrt{n!}} \Ket{n}^{(1)}\right)')
tls = SpinSpace(label='s', spin='1/2', basis=('down', 'up'))
Sig = IndexedBase('sigma')
n = IdxSym('n')
Sig_n = OperatorSymbol(StrLabel(Sig[n]), hs=tls)
assert latex(Sig_n, show_hs_label=False) == r'\hat{\sigma}_{n}'
def test_braket_indexed_sum():
"""Test braket product of sums"""
i = IdxSym('i')
hs = LocalSpace(1, dimension=5)
alpha = IndexedBase('alpha')
psi = KetSymbol('Psi', hs=hs)
psi1 = KetIndexedSum(
alpha[1, i] * BasisKet(FockIndex(i), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
psi2 = KetIndexedSum(
alpha[2, i] * BasisKet(FockIndex(i), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
expr = Bra.create(psi1) * psi2
assert expr.space == TrivialSpace
assert expr == ScalarIndexedSum.create(
alpha[1, i].conjugate() * alpha[2, i],
ranges=(IndexOverFockSpace(i, hs), ),
)
assert BraKet.create(psi1, psi2) == expr
expr = psi.dag() * psi2
assert expr == ScalarIndexedSum(
alpha[2, i] * BraKet(psi, BasisKet(FockIndex(i), hs=hs)),
ranges=IndexOverFockSpace(i, hs),
)
assert BraKet.create(psi, psi2) == expr
expr = psi1.dag() * psi
assert expr == ScalarIndexedSum(
alpha[1, i].conjugate() * BraKet(BasisKet(FockIndex(i), hs=hs), psi),
ranges=IndexOverFockSpace(i, hs),
)
assert BraKet.create(psi1, psi) == expr
def test_tls_norm():
"""Test that calculating the norm of a TLS state results in 1"""
hs = LocalSpace('tls', dimension=2)
i = IdxSym('i')
ket_i = BasisKet(FockIndex(i), hs=hs)
nrm = BraKet.create(ket_i, ket_i)
assert nrm == 1
psi = KetIndexedSum((1 / sympy.sqrt(2)) * ket_i,
ranges=IndexOverFockSpace(i, hs))
nrm = BraKet.create(psi, psi)
assert nrm == 1
def test_ket_indexed_sum_simplify_scalar():
"""Test calling the simplify_scalar method of an KetIndexedSum."""
# This tests originates from some broken behavior when IndexedSum received
# `ranges` as a positional argument instead of a keyword argument.
a, b, ϕ = symbols('a, b, phi')
factor = (a + b) * sympy.exp(I * ϕ)
factor_expand = factor.expand()
hs = LocalSpace(0)
n = symbols('n', cls=IdxSym)
psi_n = hs.basis_state(FockIndex(n))
expr = KetIndexedSum(factor * psi_n,
ranges=(IndexOverFockSpace(n, hs=hs), ))
expr_expand = expr.simplify_scalar(sympy.expand)
expected = factor_expand * KetIndexedSum(
psi_n, ranges=(IndexOverFockSpace(n, hs=hs), ))
assert expr_expand != expected.term # happened when ranges was an argument
assert expr_expand == expected
def test_ketbra_indexed_sum():
"""Test ketbra product of sums"""
i = IdxSym('i')
hs = LocalSpace(1, dimension=5)
alpha = IndexedBase('alpha')
psi = KetSymbol('Psi', hs=hs)
psi1 = KetIndexedSum(
alpha[1, i] * BasisKet(FockIndex(i), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
psi2 = KetIndexedSum(
alpha[2, i] * BasisKet(FockIndex(i), hs=hs),
ranges=IndexOverFockSpace(i, hs),
)
expr = psi1 * psi2.dag()
assert expr.space == hs
expected = OperatorIndexedSum(
alpha[2, i.prime].conjugate() * alpha[1, i] *
KetBra.create(BasisKet(FockIndex(i), hs=hs),
BasisKet(FockIndex(i.prime), hs=hs)),
ranges=(IndexOverFockSpace(i, hs), IndexOverFockSpace(i.prime, hs)),
)
assert expr == expected
assert KetBra.create(psi1, psi2) == expr
expr = psi * psi2.dag()
assert expr.space == hs
expected = OperatorIndexedSum(
alpha[2, i].conjugate() *
KetBra.create(psi, BasisKet(FockIndex(i), hs=hs)),
ranges=IndexOverFockSpace(i, hs),
)
assert expr == expected
assert KetBra.create(psi, psi2) == expr
expr = psi1 * psi.dag()
assert expr.space == hs
expected = OperatorIndexedSum(
alpha[1, i] * KetBra.create(BasisKet(FockIndex(i), hs=hs), psi),
ranges=IndexOverFockSpace(i, hs),
)
assert expr == expected
assert KetBra.create(psi1, psi) == expr
def test_tensor_indexed_sum():
"""Test tensor product of sums"""
i = IdxSym('i')
hs1 = LocalSpace(1)
hs2 = LocalSpace(2)
alpha = IndexedBase('alpha')
psi1 = KetIndexedSum(
alpha[1, i] * BasisKet(FockIndex(i), hs=hs1),
ranges=IndexOverFockSpace(i, hs1),
)
psi2 = KetIndexedSum(
alpha[2, i] * BasisKet(FockIndex(i), hs=hs2),
ranges=IndexOverFockSpace(i, hs2),
)
expr = psi1 * psi2
assert expr.space == hs1 * hs2
rhs = KetIndexedSum(
alpha[1, i] * alpha[2, i.prime] *
(BasisKet(FockIndex(i), hs=hs1) *
BasisKet(FockIndex(i.prime), hs=hs2)),
ranges=(IndexOverFockSpace(i, hs1), IndexOverFockSpace(i.prime, hs2)),
)
assert expr == rhs
psi0 = KetSymbol('Psi', hs=0)
psi3 = KetSymbol('Psi', hs=3)
expr2 = psi0 * psi1 * psi2 * psi3
rhs = KetIndexedSum(
alpha[1, i] * alpha[2, i.prime] *
(psi0 * BasisKet(FockIndex(i), hs=hs1) *
BasisKet(FockIndex(i.prime), hs=hs2) * psi3),
ranges=(IndexOverFockSpace(i, hs1), IndexOverFockSpace(i.prime, hs2)),
)
assert expr2 == rhs
assert TensorKet.create(psi0, psi1, psi2, psi3) == expr2
def test_unicode_symbolic_labels():
"""Test unicode representation of symbols with symbolic labels"""
i = IdxSym('i')
j = IdxSym('j')
hs0 = LocalSpace(0)
hs1 = LocalSpace(1)
Psi = IndexedBase('Psi')
assert unicode(BasisKet(FockIndex(2 * i), hs=hs0)) == '|2 i⟩⁽⁰⁾'
assert unicode(KetSymbol(StrLabel(2 * i), hs=hs0)) == '|2 i⟩⁽⁰⁾'
assert (unicode(KetSymbol(StrLabel(Psi[i, j]),
hs=hs0 * hs1)) == '|Ψ_ij⟩^(0⊗1)')
expr = BasisKet(FockIndex(i), hs=hs0) * BasisKet(FockIndex(j), hs=hs1)
assert unicode(expr) == '|i,j⟩^(0⊗1)'
assert unicode(Bra(BasisKet(FockIndex(2 * i), hs=hs0))) == '⟨2 i|⁽⁰⁾'
assert (unicode(LocalSigma(FockIndex(i), FockIndex(j),
hs=hs0)) == '|i⟩⟨j|⁽⁰⁾')
expr = CoherentStateKet(symbols('alpha'), hs=1).to_fock_representation()
assert unicode(expr) == 'exp(-α α ⃰/2) (∑_{n ∈ ℌ₁} αⁿ/√n! |n⟩⁽¹⁾)'
tls = SpinSpace(label='s', spin='1/2', basis=('down', 'up'))
Sig = IndexedBase('sigma')
n = IdxSym('n')
Sig_n = OperatorSymbol(StrLabel(Sig[n]), hs=tls)
assert unicode(Sig_n, show_hs_label=False) == 'σ̂ₙ'
{},
ScalarTimesOperator(gamma, OpA),
),
# State Algebra
# ...
(
KetIndexedSum,
'R001',
(KetSymbol(StrLabel(i), hs=0) - KetSymbol(StrLabel(i), hs=0), ),
dict(ranges=(IndexOverFockSpace(i, hs=LocalSpace(0)), )),
ZeroKet,
),
(
KetIndexedSum,
'R002',
(symbols('a') * BasisKet(FockIndex(i), hs=0), ),
dict(ranges=(IndexOverRange(i, 0, 1), )),
symbols('a') * KetIndexedSum(BasisKet(FockIndex(i), hs=0),
ranges=(IndexOverRange(i, 0, 1), )),
),
]
@pytest.mark.parametrize("cls, rule, args, kwargs, expected", TESTS)
def test_rule(cls, rule, args, kwargs, expected, caplog):
"""Check that for the given `cls` and `rule` name (which must be a key in
``cls._rules`` or ``cls._binary_rules``), if we instantiate
``cls(*args, **kwargs)``, `rule` is applied and we obtain the `expected`
result.
In order to review the log of how all test expressions are created, call
def test_ascii_ket_operations():
"""Test the ascii representation of ket operations"""
hs1 = LocalSpace('q_1', basis=('g', 'e'))
hs2 = LocalSpace('q_2', basis=('g', 'e'))
ket_g1 = BasisKet('g', hs=hs1)
ket_e1 = BasisKet('e', hs=hs1)
ket_g2 = BasisKet('g', hs=hs2)
ket_e2 = BasisKet('e', hs=hs2)
psi1 = KetSymbol("Psi_1", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi3 = KetSymbol("Psi_3", hs=hs1)
phi = KetSymbol("Phi", hs=hs2)
A = OperatorSymbol("A_0", hs=hs1)
gamma = symbols('gamma', positive=True)
alpha = symbols('alpha')
beta = symbols('beta')
phase = exp(-I * gamma)
i = IdxSym('i')
assert ascii(psi1 + psi2) == '|Psi_1>^(q_1) + |Psi_2>^(q_1)'
assert (ascii(psi1 - psi2 +
psi3) == '|Psi_1>^(q_1) - |Psi_2>^(q_1) + |Psi_3>^(q_1)')
with pytest.raises(UnequalSpaces):
psi1 + phi
with pytest.raises(AttributeError):
(psi1 * phi).label
assert ascii(psi1 * phi) == '|Psi_1>^(q_1) * |Phi>^(q_2)'
with pytest.raises(OverlappingSpaces):
psi1 * psi2
assert ascii(phase * psi1) == 'exp(-I*gamma) * |Psi_1>^(q_1)'
assert (ascii(
(alpha + 1) * KetSymbol('Psi', hs=0)) == '(alpha + 1) * |Psi>^(0)')
assert ascii(A * psi1) == 'A_0^(q_1) |Psi_1>^(q_1)'
with pytest.raises(SpaceTooLargeError):
A * phi
assert ascii(BraKet(psi1, psi2)) == '<Psi_1|Psi_2>^(q_1)'
expr = BraKet(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert ascii(expr) == '<Psi_1(alpha)|Psi_2(beta)>^(q_1)'
assert ascii(psi1.dag() * psi2) == '<Psi_1|Psi_2>^(q_1)'
assert ascii(ket_e1.dag() * ket_e1) == '1'
assert ascii(ket_g1.dag() * ket_e1) == '0'
assert ascii(KetBra(psi1, psi2)) == '|Psi_1><Psi_2|^(q_1)'
expr = KetBra(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert ascii(expr) == '|Psi_1(alpha)><Psi_2(beta)|^(q_1)'
bell1 = (ket_e1 * ket_g2 - I * ket_g1 * ket_e2) / sqrt(2)
bell2 = (ket_e1 * ket_e2 - ket_g1 * ket_g2) / sqrt(2)
assert ascii(bell1) == '1/sqrt(2) * (|eg>^(q_1*q_2) - I * |ge>^(q_1*q_2))'
assert ascii(bell2) == '1/sqrt(2) * (|ee>^(q_1*q_2) - |gg>^(q_1*q_2))'
expr = BraKet.create(bell1, bell2)
expected = (
r'1/2 * (<eg|^(q_1*q_2) + I * <ge|^(q_1*q_2)) * (|ee>^(q_1*q_2) '
r'- |gg>^(q_1*q_2))')
assert ascii(expr) == expected
assert (ascii(KetBra.create(bell1, bell2)) ==
'1/2 * (|eg>^(q_1*q_2) - I * |ge>^(q_1*q_2))(<ee|^(q_1*q_2) '
'- <gg|^(q_1*q_2))')
expr = KetBra(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert ascii(expr) == "|Psi><i|^(0)"
expr = KetBra(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert ascii(expr) == "|i><Psi|^(0)"
expr = BraKet(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert ascii(expr) == "<Psi|i>^(0)"
expr = BraKet(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert ascii(expr) == "<i|Psi>^(0)"
def test_qubit_state():
"""Test sum_i alpha_i |i> for TLS"""
i = IdxSym('i')
alpha = IndexedBase('alpha')
alpha_i = alpha[i]
hs_tls = LocalSpace('tls', basis=('g', 'e'))
term = alpha_i * BasisKet(FockIndex(i), hs=hs_tls)
expr1 = KetIndexedSum.create(term, ranges=IndexOverFockSpace(i, hs=hs_tls))
expr2 = KetIndexedSum.create(term, ranges=IndexOverList(i, [0, 1]))
expr3 = KetIndexedSum.create(term,
ranges=IndexOverRange(i, start_from=0, to=1))
assert IndexOverFockSpace(i, hs=hs_tls) in expr1.kwargs['ranges']
assert ascii(expr1) == "Sum_{i in H_tls} alpha_i * |i>^(tls)"
assert unicode(expr1) == "∑_{i ∈ ℌ_tls} α_i |i⟩⁽ᵗˡˢ⁾"
assert (
srepr(expr1) ==
"KetIndexedSum(ScalarTimesKet(ScalarValue(Indexed(IndexedBase(Symbol('alpha')), IdxSym('i', integer=True))), BasisKet(FockIndex(IdxSym('i', integer=True)), hs=LocalSpace('tls', basis=('g', 'e')))), ranges=(IndexOverFockSpace(IdxSym('i', integer=True), LocalSpace('tls', basis=('g', 'e'))),))"
)
with configure_printing(tex_use_braket=True):
assert (latex(expr1) ==
r'\sum_{i \in \mathcal{H}_{tls}} \alpha_{i} \Ket{i}^{(tls)}')
assert ascii(expr2) == 'Sum_{i in {0,1}} alpha_i * |i>^(tls)'
assert unicode(expr2) == '∑_{i ∈ {0,1}} α_i |i⟩⁽ᵗˡˢ⁾'
assert (
srepr(expr2) ==
"KetIndexedSum(ScalarTimesKet(ScalarValue(Indexed(IndexedBase(Symbol('alpha')), IdxSym('i', integer=True))), BasisKet(FockIndex(IdxSym('i', integer=True)), hs=LocalSpace('tls', basis=('g', 'e')))), ranges=(IndexOverList(IdxSym('i', integer=True), (0, 1)),))"
)
with configure_printing(tex_use_braket=True):
assert (
latex(expr2) == r'\sum_{i \in \{0,1\}} \alpha_{i} \Ket{i}^{(tls)}')
assert ascii(expr3) == 'Sum_{i=0}^{1} alpha_i * |i>^(tls)'
assert unicode(expr3) == '∑_{i=0}^{1} α_i |i⟩⁽ᵗˡˢ⁾'
assert (
srepr(expr3) ==
"KetIndexedSum(ScalarTimesKet(ScalarValue(Indexed(IndexedBase(Symbol('alpha')), IdxSym('i', integer=True))), BasisKet(FockIndex(IdxSym('i', integer=True)), hs=LocalSpace('tls', basis=('g', 'e')))), ranges=(IndexOverRange(IdxSym('i', integer=True), 0, 1),))"
)
with configure_printing(tex_use_braket=True):
assert latex(expr3) == r'\sum_{i=0}^{1} \alpha_{i} \Ket{i}^{(tls)}'
for expr in (expr1, expr2, expr3):
assert expr.term.free_symbols == set([i, symbols('alpha'), alpha_i])
assert expr.term.bound_symbols == set()
assert expr.free_symbols == set([symbols('alpha'), alpha_i])
assert expr.variables == [i]
assert expr.bound_symbols == set([i])
assert len(expr) == len(expr.ranges[0]) == 2
assert 0 in expr.ranges[0]
assert 1 in expr.ranges[0]
assert expr.space == hs_tls
assert len(expr.args) == 1
assert len(expr.kwargs) == 1
assert len(expr.operands) == 1
assert expr.args[0] == term
assert expr.term == term
expr_expand = expr.doit().substitute({
alpha[0]: alpha['g'],
alpha[1]: alpha['e']
})
assert expr_expand == (alpha['g'] * BasisKet('g', hs=hs_tls) +
alpha['e'] * BasisKet('e', hs=hs_tls))
assert (
ascii(expr_expand) == 'alpha_e * |e>^(tls) + alpha_g * |g>^(tls)')
with pytest.raises(TypeError) as exc_info:
KetIndexedSum.create(alpha_i * BasisKet(i, hs=hs_tls),
IndexOverFockSpace(i, hs=hs_tls))
assert "label_or_index must be an instance of" in str(exc_info.value)
def test_tex_ket_operations():
"""Test the tex representation of ket operations"""
hs1 = LocalSpace('q_1', basis=('g', 'e'))
hs2 = LocalSpace('q_2', basis=('g', 'e'))
ket_g1 = BasisKet('g', hs=hs1)
ket_e1 = BasisKet('e', hs=hs1)
ket_g2 = BasisKet('g', hs=hs2)
ket_e2 = BasisKet('e', hs=hs2)
psi1 = KetSymbol("Psi_1", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi3 = KetSymbol("Psi_3", hs=hs1)
phi = KetSymbol("Phi", hs=hs2)
A = OperatorSymbol("A_0", hs=hs1)
gamma = symbols('gamma', positive=True)
alpha = symbols('alpha')
beta = symbols('beta')
phase = exp(-I * gamma)
i = IdxSym('i')
assert (latex(psi1 +
psi2) == r'\left\lvert \Psi_{1} \right\rangle^{(q_{1})} + '
r'\left\lvert \Psi_{2} \right\rangle^{(q_{1})}')
assert (latex(psi1 - psi2 +
psi3) == r'\left\lvert \Psi_{1} \right\rangle^{(q_{1})} - '
r'\left\lvert \Psi_{2} \right\rangle^{(q_{1})} + '
r'\left\lvert \Psi_{3} \right\rangle^{(q_{1})}')
assert (latex(
psi1 * phi) == r'\left\lvert \Psi_{1} \right\rangle^{(q_{1})} \otimes '
r'\left\lvert \Phi \right\rangle^{(q_{2})}')
assert (latex(phase * psi1) ==
r'e^{- i \gamma} \left\lvert \Psi_{1} \right\rangle^{(q_{1})}')
assert (latex((alpha + 1) * KetSymbol('Psi', hs=0)) ==
r'\left(\alpha + 1\right) \left\lvert \Psi \right\rangle^{(0)}')
assert (latex(
A * psi1
) == r'\hat{A}_{0}^{(q_{1})} \left\lvert \Psi_{1} \right\rangle^{(q_{1})}')
braket = BraKet(psi1, psi2)
assert (
latex(braket, show_hs_label='subscript') ==
r'\left\langle \Psi_{1} \middle\vert \Psi_{2} \right\rangle_{(q_{1})}')
assert (latex(braket, show_hs_label=False) ==
r'\left\langle \Psi_{1} \middle\vert \Psi_{2} \right\rangle')
expr = BraKet(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert (latex(expr) ==
r'\left\langle \Psi_{1}\left(\alpha\right) \middle\vert '
r'\Psi_{2}\left(\beta\right) \right\rangle^{(q_{1})}')
assert (latex(
ket_e1 *
ket_e2) == r'\left\lvert ee \right\rangle^{(q_{1} \otimes q_{2})}')
assert latex(ket_e1.dag() * ket_e1) == r'1'
assert latex(ket_g1.dag() * ket_e1) == r'0'
ketbra = KetBra(psi1, psi2)
assert (latex(ketbra) == r'\left\lvert \Psi_{1} \middle\rangle\!'
r'\middle\langle \Psi_{2} \right\rvert^{(q_{1})}')
assert (latex(
ketbra,
show_hs_label='subscript') == r'\left\lvert \Psi_{1} \middle\rangle\!'
r'\middle\langle \Psi_{2} \right\rvert_{(q_{1})}')
assert (latex(
ketbra,
show_hs_label=False) == r'\left\lvert \Psi_{1} \middle\rangle\!'
r'\middle\langle \Psi_{2} \right\rvert')
expr = KetBra(KetSymbol('Psi_1', alpha, hs=hs1),
KetSymbol('Psi_2', beta, hs=hs1))
assert (
latex(expr) ==
r'\left\lvert \Psi_{1}\left(\alpha\right) \middle\rangle\!'
r'\middle\langle \Psi_{2}\left(\beta\right) \right\rvert^{(q_{1})}')
bell1 = (ket_e1 * ket_g2 - I * ket_g1 * ket_e2) / sqrt(2)
bell2 = (ket_e1 * ket_e2 - ket_g1 * ket_g2) / sqrt(2)
assert (latex(bell1) ==
r'\frac{1}{\sqrt{2}} \left(\left\lvert eg \right\rangle^{(q_{1} '
r'\otimes q_{2})} - i \left\lvert ge \right\rangle'
r'^{(q_{1} \otimes q_{2})}\right)')
assert (latex(bell2) ==
r'\frac{1}{\sqrt{2}} \left(\left\lvert ee \right\rangle^{(q_{1} '
r'\otimes q_{2})} - \left\lvert gg \right\rangle'
r'^{(q_{1} \otimes q_{2})}\right)')
assert (latex(bell2, show_hs_label=False) ==
r'\frac{1}{\sqrt{2}} \left(\left\lvert ee \right\rangle - '
r'\left\lvert gg \right\rangle\right)')
assert BraKet.create(bell1, bell2).expand() == 0
assert (latex(BraKet.create(
bell1, bell2)) == r'\frac{1}{2} \left(\left\langle eg \right\rvert'
r'^{(q_{1} \otimes q_{2})} + i \left\langle ge \right\rvert'
r'^{(q_{1} \otimes q_{2})}\right) '
r'\left(\left\lvert ee \right\rangle^{(q_{1} \otimes q_{2})} '
r'- \left\lvert gg \right\rangle^{(q_{1} \otimes q_{2})}\right)')
assert (
latex(KetBra.create(
bell1, bell2)) == r'\frac{1}{2} \left(\left\lvert eg \right\rangle'
r'^{(q_{1} \otimes q_{2})} - i \left\lvert ge \right\rangle'
r'^{(q_{1} \otimes q_{2})}\right)\left(\left\langle ee \right\rvert'
r'^{(q_{1} \otimes q_{2})} - \left\langle gg \right\rvert'
r'^{(q_{1} \otimes q_{2})}\right)')
with configure_printing(tex_use_braket=True):
expr = KetBra(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert latex(expr) == r'\Ket{\Psi}\!\Bra{i}^{(0)}'
expr = KetBra(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert latex(expr) == r'\Ket{i}\!\Bra{\Psi}^{(0)}'
expr = BraKet(KetSymbol('Psi', hs=0), BasisKet(FockIndex(i), hs=0))
assert latex(expr) == r'\Braket{\Psi | i}^(0)'
expr = BraKet(BasisKet(FockIndex(i), hs=0), KetSymbol('Psi', hs=0))
assert latex(expr) == r'\Braket{i | \Psi}^(0)'
def test_scalar_indexed_sum(braket):
"""Test instantiation and behavior of a ScalarIndexedSum"""
i = IdxSym('i')
ip = i.prime
ipp = ip.prime
alpha = IndexedBase('alpha')
a = symbols('a')
hs = LocalSpace(0)
ket_sum = KetIndexedSum(
alpha[1, i] * BasisKet(FockIndex(i), hs=hs),
ranges=(IndexOverRange(i, 1, 2), ),
)
bra = KetSymbol('Psi', hs=hs).dag()
expr = bra * ket_sum
half = sympify(1) / 2
assert isinstance(expr, ScalarIndexedSum)
assert isinstance(expr.term, ScalarTimes)
assert expr.term == bra * ket_sum.term
assert expr.ranges == ket_sum.ranges
assert expr.doit() == (alpha[1, 1] * bra * BasisKet(1, hs=hs) +
alpha[1, 2] * bra * BasisKet(2, hs=hs))
expr = ScalarIndexedSum.create(i, ranges=(IndexOverRange(i, 1, 2), ))
assert expr == ScalarIndexedSum(i, ranges=(IndexOverRange(i, 1, 2), ))
assert isinstance(expr.doit(), ScalarValue)
assert expr.doit() == 3
assert expr.real == expr
assert expr.imag == Zero
assert expr.conjugate() == expr
assert 3 * expr == expr * 3 == Sum(i, 1, 2)(3 * i)
assert a * expr == expr * a == Sum(i, 1, 2)(a * i)
assert braket * expr == ScalarTimes(braket, Sum(i, 1, 2)(i))
assert expr * braket == ScalarTimes(braket, Sum(i, 1, 2)(i))
assert (2 * i) * expr == 2 * expr * i
assert (2 * i) * expr == Sum(i, 1, 2)(2 * i * i.prime)
assert expr * expr == ScalarIndexedSum(
ScalarValue(i * ip),
ranges=(IndexOverRange(i, 1, 2), IndexOverRange(ip, 1, 2)),
)
sum3 = expr**3
assert sum3 == ScalarIndexedSum(
ScalarValue(i * ip * ipp),
ranges=(
IndexOverRange(i, 1, 2),
IndexOverRange(ip, 1, 2),
IndexOverRange(ipp, 1, 2),
),
)
assert expr**0 is One
assert expr**1 is expr
assert (expr**alpha).exp == alpha
assert expr**-1 == 1 / expr
assert (1 / expr).exp == -1
assert (expr**-alpha).exp == -alpha
sqrt_sum = sqrt(expr)
assert sqrt_sum == ScalarPower(expr, ScalarValue(half))
expr = ScalarIndexedSum.create(I * i, ranges=(IndexOverRange(i, 1, 2), ))
assert expr.real == Zero
assert expr.imag == ScalarIndexedSum.create(i,
ranges=(IndexOverRange(
i, 1, 2), ))
assert expr.conjugate() == -expr
def test_evaluate_symbolic_labels():
"""Test the behavior of the `substitute` method for evaluation of symbolic
labels"""
i, j = symbols('i j', cls=IdxSym)
A = IndexedBase('A')
lbl = FockIndex(i + j)
assert lbl.substitute({i: 1, j: 2}) == 3
assert lbl.substitute({i: 1}) == FockIndex(1 + j)
assert lbl.substitute({j: 2}) == FockIndex(i + 2)
assert lbl.substitute({i: 1}).substitute({j: 2}) == 3
assert lbl.substitute({}) == lbl
lbl = StrLabel(A[i, j])
assert lbl.substitute({i: 1, j: 2}) == 'A_12'
assert lbl.substitute({i: 1}) == StrLabel(A[1, j])
assert lbl.substitute({j: 2}) == StrLabel(A[i, 2])
assert lbl.substitute({i: 1}).substitute({j: 2}) == 'A_12'
assert lbl.substitute({}) == lbl
hs = SpinSpace('s', spin=3)
lbl = SpinIndex(i + j, hs)
assert lbl.substitute({i: 1, j: 2}) == '+3'
assert lbl.substitute({i: 1}) == SpinIndex(1 + j, hs)
assert lbl.substitute({j: 2}) == SpinIndex(i + 2, hs)
assert lbl.substitute({i: 1}).substitute({j: 2}) == '+3'
assert lbl.substitute({}) == lbl
hs = SpinSpace('s', spin='3/2')
lbl = SpinIndex((i + j) / 2, hs=hs)
assert lbl.substitute({i: 1, j: 2}) == '+3/2'
assert lbl.substitute({i: 1}) == SpinIndex((1 + j) / 2, hs)
assert lbl.substitute({j: 2}) == SpinIndex((i + 2) / 2, hs)
assert lbl.substitute({i: 1}).substitute({j: 2}) == '+3/2'
assert lbl.substitute({}) == lbl