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 sop_exprs():
"""Prepare a list of super operator algebra expressions"""
hs1 = LocalSpace('q1', dimension=2)
hs2 = LocalSpace('q2', dimension=2)
A = SuperOperatorSymbol("A", hs=hs1)
B = SuperOperatorSymbol("B", hs=hs1)
C = SuperOperatorSymbol("C", hs=hs2)
L = SuperOperatorSymbol("L", hs=1)
M = SuperOperatorSymbol("M", hs=1)
A_op = OperatorSymbol("A", hs=1)
gamma = symbols('gamma')
return [
SuperOperatorSymbol("A", hs=hs1),
SuperOperatorSymbol("A_1", hs=hs1 * hs2),
SuperOperatorSymbol("A", symbols('alpha'), symbols('beta'), hs=hs1),
IdentitySuperOperator,
ZeroSuperOperator,
A + B,
A * B,
A * C,
2 * A,
(1 + 2j) * A,
-(gamma**2) / 2 * A,
SuperAdjoint(A + B),
2 * A - sqrt(gamma) * (B + C),
SPre(A_op),
SPost(A_op),
SuperOperatorTimesOperator(L,
sqrt(gamma) * A_op),
SuperOperatorTimesOperator((L + 2 * M), A_op),
]
def test_tex_bra_elements():
"""Test the tex representation of "atomic" kets"""
hs1 = LocalSpace('q1', basis=('g', 'e'))
hs2 = LocalSpace('q2', basis=('g', 'e'))
alpha, beta = symbols('alpha, beta')
bra = Bra(KetSymbol('Psi', hs=hs1))
assert latex(bra) == r'\left\langle \Psi \right\rvert^{(q_{1})}'
assert latex(Bra(KetSymbol('Psi', alpha, beta, hs=hs1))) == (
r'\left\langle \Psi\left(\alpha, \beta\right) \right\rvert^{(q_{1})}')
assert latex(bra, tex_use_braket=True) == r'\Bra{\Psi}^{(q_{1})}'
assert (latex(bra, tex_use_braket=True,
show_hs_label='subscript') == r'\Bra{\Psi}_{(q_{1})}')
assert (latex(bra, tex_use_braket=True,
show_hs_label=False) == r'\Bra{\Psi}')
assert (latex(Bra(KetSymbol(
'Psi', hs=1))) == r'\left\langle \Psi \right\rvert^{(1)}')
assert (latex(Bra(KetSymbol(
'Psi',
hs=(1, 2)))) == r'\left\langle \Psi \right\rvert^{(1 \otimes 2)}')
assert (latex(Bra(KetSymbol(
'Psi', hs=hs1 *
hs2))) == r'\left\langle \Psi \right\rvert^{(q_{1} \otimes q_{2})}')
assert (latex(KetSymbol(
'Psi', hs=1).dag()) == r'\left\langle \Psi \right\rvert^{(1)}')
assert latex(Bra(ZeroKet)) == '0'
assert latex(Bra(TrivialKet)) == '1'
assert (latex(BasisKet(
'e', hs=hs1).adjoint()) == r'\left\langle e \right\rvert^{(q_{1})}')
assert (latex(BasisKet(
1, hs=1).adjoint()) == r'\left\langle 1 \right\rvert^{(1)}')
assert (latex(CoherentStateKet(
2.0, hs=1).dag()) == r'\left\langle \alpha=2 \right\rvert^{(1)}')
def test_ascii_sop_operations():
"""Test the ascii representation of super operator algebra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
A = SuperOperatorSymbol("A", hs=hs1)
B = SuperOperatorSymbol("B", hs=hs1)
C = SuperOperatorSymbol("C", hs=hs2)
L = SuperOperatorSymbol("L", hs=1)
M = SuperOperatorSymbol("M", hs=1)
A_op = OperatorSymbol("A", hs=1)
gamma = symbols('gamma', positive=True)
assert ascii(A + B) == 'A^(q_1) + B^(q_1)'
assert ascii(A * B) == 'A^(q_1) * B^(q_1)'
assert ascii(A * C) == 'A^(q_1) * C^(q_2)'
assert ascii(2 * A) == '2 * A^(q_1)'
assert ascii(2j * A) == '2j * A^(q_1)'
assert ascii((1 + 2j) * A) == '(1+2j) * A^(q_1)'
assert ascii(gamma**2 * A) == 'gamma**2 * A^(q_1)'
assert ascii(-(gamma**2) / 2 * A) == '-gamma**2/2 * A^(q_1)'
assert ascii(SuperAdjoint(A)) == 'A^(q_1)H'
assert ascii(SuperAdjoint(A + B)) == '(A^(q_1) + B^(q_1))^H'
assert ascii(A - B) == 'A^(q_1) - B^(q_1)'
assert ascii(A - B + C) == 'A^(q_1) - B^(q_1) + C^(q_2)'
assert (
ascii(2 * A - sqrt(gamma) *
(B + C)) == '2 * A^(q_1) - sqrt(gamma) * (B^(q_1) + C^(q_2))')
assert ascii(SPre(A_op)) == 'SPre(A^(1))'
assert ascii(SPost(A_op)) == 'SPost(A^(1))'
assert ascii(SuperOperatorTimesOperator(L, A_op)) == 'L^(1)[A^(1)]'
assert (ascii(SuperOperatorTimesOperator(
L,
sqrt(gamma) * A_op)) == 'L^(1)[sqrt(gamma) * A^(1)]')
assert (ascii(SuperOperatorTimesOperator(
(L + 2 * M), A_op)) == '(L^(1) + 2 * M^(1))[A^(1)]')
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_ascii_ket_elements():
"""Test the ascii representation of "atomic" kets"""
hs1 = LocalSpace('q1', basis=('g', 'e'))
hs2 = LocalSpace('q2', basis=('g', 'e'))
alpha, beta = symbols('alpha, beta')
assert ascii(KetSymbol('Psi', hs=hs1)) == '|Psi>^(q1)'
psi = KetSymbol('Psi', hs=1)
assert ascii(psi) == '|Psi>^(1)'
assert ascii(KetSymbol('Psi', alpha, beta,
hs=1)) == ('|Psi(alpha, beta)>^(1)')
assert ascii(psi, show_hs_label='subscript') == '|Psi>_(1)'
assert ascii(psi, show_hs_label=False) == '|Psi>'
assert ascii(KetSymbol('Psi', hs=(1, 2))) == '|Psi>^(1*2)'
assert ascii(KetSymbol('Psi', hs=hs1 * hs2)) == '|Psi>^(q1*q2)'
with pytest.raises(ValueError):
KetSymbol(r'\Psi', hs=hs1)
assert ascii(KetSymbol('Psi', hs=1)) == '|Psi>^(1)'
assert ascii(KetSymbol('Psi', hs=hs1 * hs2)) == '|Psi>^(q1*q2)'
assert ascii(ZeroKet) == '0'
assert ascii(TrivialKet) == '1'
assert ascii(BasisKet('e', hs=hs1)) == '|e>^(q1)'
assert ascii(BasisKet(1, hs=1)) == '|1>^(1)'
assert ascii(BasisKet(1, hs=hs1)) == '|e>^(q1)'
with pytest.raises(ValueError):
BasisKet('1', hs=hs1)
assert ascii(CoherentStateKet(2.0, hs=1)) == '|alpha=2>^(1)'
assert ascii(CoherentStateKet(2.1, hs=1)) == '|alpha=2.1>^(1)'
def test_ascii_bra_operations():
"""Test the ascii representation of bra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
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)
bra_psi1 = KetSymbol("Psi_1", hs=hs1).dag()
bra_psi2 = KetSymbol("Psi_2", hs=hs1).dag()
bra_psi2 = KetSymbol("Psi_2", hs=hs1).dag()
bra_psi3 = KetSymbol("Psi_3", hs=hs1).dag()
bra_phi = KetSymbol("Phi", hs=hs2).dag()
A = OperatorSymbol("A_0", hs=hs1)
gamma = symbols('gamma', positive=True)
phase = exp(-I * gamma)
assert ascii((psi1 + psi2).dag()) == '<Psi_1|^(q_1) + <Psi_2|^(q_1)'
assert ascii(bra_psi1 + bra_psi2) == '<Psi_1|^(q_1) + <Psi_2|^(q_1)'
assert (ascii(
(psi1 - psi2 +
psi3).dag()) == '<Psi_1|^(q_1) - <Psi_2|^(q_1) + <Psi_3|^(q_1)')
assert (ascii(bra_psi1 - bra_psi2 +
bra_psi3) == '<Psi_1|^(q_1) - <Psi_2|^(q_1) + <Psi_3|^(q_1)')
assert ascii((psi1 * phi).dag()) == '<Psi_1|^(q_1) * <Phi|^(q_2)'
assert ascii(bra_psi1 * bra_phi) == '<Psi_1|^(q_1) * <Phi|^(q_2)'
assert ascii(Bra(phase * psi1)) == 'exp(I*gamma) * <Psi_1|^(q_1)'
assert ascii((A * psi1).dag()) == '<Psi_1|^(q_1) A_0^(q_1)H'
def test_indexed_hs_not_disjoint():
i, j = symbols('i, j', cls=IdxSym)
hs_i = LocalSpace(StrLabel(i))
hs_j = LocalSpace(StrLabel(j))
assert not hs_i.isdisjoint(hs_i)
assert not hs_i.isdisjoint(hs_j)
expr = Create(hs=hs_j) * Destroy(hs=hs_i)
assert expr.args == (Create(hs=hs_j), Destroy(hs=hs_i))
def bell1_expr():
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)
return (ket_e1 * ket_g2 - I * ket_g1 * ket_e2) / sqrt(2)
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_hilbert_tree():
"""Test tree representation of a Hilbert space algebra expression"""
H1 = LocalSpace(1)
H2 = LocalSpace(2)
tree = tree_str(H1 * H2)
assert (tree == dedent(r'''
. ProductSpace(ℌ₁, ℌ₂)
├─ LocalSpace(1)
└─ LocalSpace(2)
''').strip())
def test_quantum_symbols_with_indexedhs():
"""Test the free_symbols method for objects that have a Hilbert space with a
sybmolic label, for the example of an OperatorSymbol"""
i, j = symbols('i, j', cls=IdxSym)
hs_i = LocalSpace(StrLabel(i))
hs_j = LocalSpace(StrLabel(j))
A = OperatorSymbol("A", hs=hs_i * hs_j)
assert A.free_symbols == {i, j}
expr = Create(hs=hs_i) * Destroy(hs=hs_i)
assert expr.free_symbols == {
i,
}
def test_two_hs_symbol_sum():
"""Test sum_{ij} a_{ij} Psi_{ij}"""
i = IdxSym('i')
j = IdxSym('j')
a = IndexedBase('a')
hs1 = LocalSpace('1', dimension=3)
hs2 = LocalSpace('2', dimension=3)
hs = hs1 * hs2
Psi = IndexedBase('Psi')
a_ij = a[i, j]
Psi_ij = Psi[i, j]
KetPsi_ij = KetSymbol(StrLabel(Psi_ij), hs=hs)
term = a_ij * KetPsi_ij
expr1 = KetIndexedSum(
term,
ranges=(IndexOverFockSpace(i, hs=hs1), IndexOverFockSpace(j, hs=hs2)),
)
expr2 = KetIndexedSum(term,
ranges=(IndexOverRange(i, 0,
2), IndexOverRange(j, 0, 2)))
assert expr1.term.free_symbols == set(
[i, j, symbols('a'), symbols('Psi'), a_ij, Psi_ij])
assert expr1.free_symbols == set(
[symbols('a'), symbols('Psi'), a_ij, Psi_ij])
assert expr1.variables == [i, j]
assert (
ascii(expr1) == 'Sum_{i in H_1} Sum_{j in H_2} a_ij * |Psi_ij>^(1*2)')
assert unicode(expr1) == '∑_{i ∈ ℌ₁} ∑_{j ∈ ℌ₂} a_ij |Ψ_ij⟩^(1⊗2)'
assert (latex(expr1) ==
r'\sum_{i \in \mathcal{H}_{1}} \sum_{j \in \mathcal{H}_{2}} '
r'a_{i j} \left\lvert \Psi_{i j} \right\rangle^{(1 \otimes 2)}')
assert ascii(expr2) == 'Sum_{i,j=0}^{2} a_ij * |Psi_ij>^(1*2)'
assert unicode(expr2) == '∑_{i,j=0}^{2} a_ij |Ψ_ij⟩^(1⊗2)'
assert (latex(expr2) == r'\sum_{i,j=0}^{2} a_{i j} '
r'\left\lvert \Psi_{i j} \right\rangle^{(1 \otimes 2)}')
assert expr1.doit() == expr2.doit()
assert expr1.doit() == KetPlus(
a[0, 0] * KetSymbol('Psi_00', hs=hs),
a[0, 1] * KetSymbol('Psi_01', hs=hs),
a[0, 2] * KetSymbol('Psi_02', hs=hs),
a[1, 0] * KetSymbol('Psi_10', hs=hs),
a[1, 1] * KetSymbol('Psi_11', hs=hs),
a[1, 2] * KetSymbol('Psi_12', hs=hs),
a[2, 0] * KetSymbol('Psi_20', hs=hs),
a[2, 1] * KetSymbol('Psi_21', hs=hs),
a[2, 2] * KetSymbol('Psi_22', hs=hs),
)
def test_unicode_bra_operations():
"""Test the unicode representation of bra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
psi1 = KetSymbol("Psi_1", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
phi = KetSymbol("Phi", hs=hs2)
gamma = symbols('gamma', positive=True)
phase = exp(-I * gamma)
assert unicode((psi1 + psi2).dag()) == '⟨Ψ₁|^(q₁) + ⟨Ψ₂|^(q₁)'
assert unicode((psi1 * phi).dag()) == '⟨Ψ₁|^(q₁) ⊗ ⟨Φ|^(q₂)'
assert unicode(Bra(phase * psi1)) == 'exp(ⅈ γ) ⟨Ψ₁|^(q₁)'
def test_unicode_operator_operations():
"""Test the unicode representation of operator algebra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
A = OperatorSymbol("A", hs=hs1)
B = OperatorSymbol("B", hs=hs1)
C = OperatorSymbol("C", hs=hs2)
psi = KetSymbol('Psi', hs=hs1)
gamma = symbols('gamma', positive=True)
assert unicode(A + B) == 'A\u0302^(q\u2081) + B\u0302^(q\u2081)'
# Â^(q₁) + B̂^(q₁)
assert unicode(A * B) == 'A\u0302^(q\u2081) B\u0302^(q\u2081)'
# Â^(q₁) B̂^(q₁)
assert unicode(A * C) == 'A\u0302^(q\u2081) C\u0302^(q\u2082)'
# Â^(q₁) Ĉ^(q₂)
assert unicode(2 * A) == '2 A\u0302^(q\u2081)' # 2 Â^(q₁)
assert unicode(2j * A) == '2j A\u0302^(q\u2081)'
# 2j Â^(q₁)
assert unicode((1 + 2j) * A) == '(1+2j) A\u0302^(q\u2081)'
# (1+2j) Â^(q₁)
assert unicode(gamma**2 * A) == '\u03b3\xb2 A\u0302^(q\u2081)'
# γ² Â^(q₁)
assert unicode(-(gamma**2) / 2 * A) == '-\u03b3\xb2/2 A\u0302^(q\u2081)'
# -γ²/2 Â^(q₁)
assert (unicode(tr(
A * C,
over_space=hs2)) == 'tr_(q\u2082)[C\u0302^(q\u2082)] A\u0302^(q\u2081)'
)
# tr_(q₂)[Ĉ^(q₂)] Â^(q₁)
assert unicode(Adjoint(A)) == 'A\u0302^(q\u2081)\u2020'
# Â^(q₁)†
assert unicode(Adjoint(Create(hs=1))) == 'a\u0302\u207d\xb9\u207e'
# â⁽¹⁾
assert unicode(PseudoInverse(A)) == '(A\u0302^(q\u2081))^+'
# (Â^(q₁))^+
assert unicode(NullSpaceProjector(A)) == 'P\u0302_Ker(A\u0302^(q\u2081))'
# P̂_Ker(Â^(q₁))
assert unicode(A - B) == 'A\u0302^(q\u2081) - B\u0302^(q\u2081)'
# Â^(q₁) - B̂^(q₁)
assert unicode(2 * A - sqrt(gamma) * (B + C)) in [
'2 A\u0302^(q\u2081) - \u221a\u03b3 (B\u0302^(q\u2081) '
'+ C\u0302^(q\u2082))',
'2 A\u0302^(q\u2081) - sqrt(\u03b3) (B\u0302^(q\u2081) '
'+ C\u0302^(q\u2082))',
]
# 2 Â^(q₁) - √γ (B̂^(q₁) + Ĉ^(q₂))
assert (unicode(Commutator(A,
B)) == '[A\u0302^(q\u2081), B\u0302^(q\u2081)]')
# [Â^(q₁), B̂^(q₁)]
expr = (Commutator(A, B) * psi).dag()
assert unicode(expr, show_hs_label=False) == r'⟨Ψ| [Â, B̂]^†'
def test_operator_times_order():
A1 = OperatorSymbol("A", hs=1)
B1 = OperatorSymbol("B", hs=1)
A2 = OperatorSymbol("A", hs=2)
A3 = OperatorSymbol("A", hs=3)
B4 = OperatorSymbol("B", hs=4)
B1_m = OperatorSymbol("B", hs=LocalSpace(1, order_index=2))
B2_m = OperatorSymbol("B", hs=LocalSpace(2, order_index=1))
assert A1 * A2 == A2 * A1
assert A1 * B1 != B1 * A1
assert (A2 * A1).operands == (A1, A2)
assert (B2_m * B1_m).operands == (B2_m, B1_m)
assert ((B4 + A3) * (A2 + A1)).operands == (A1 + A2, A3 + B4)
def test_unicode_operator_elements():
"""Test the unicode representation of "atomic" operator algebra elements"""
hs1 = LocalSpace('q1', dimension=2)
hs2 = LocalSpace('q2', dimension=2)
alpha, beta = symbols('alpha, beta')
assert unicode(OperatorSymbol("A", hs=hs1)) == 'A\u0302^(q\u2081)'
# Â^(q₁)
assert (unicode(OperatorSymbol('A', hs=1),
show_hs_label='subscript') == 'A\u0302\u208d\u2081\u208e'
) # Â₍₁₎
assert (unicode(
OperatorSymbol("A", hs=hs1),
unicode_op_hats=False,
unicode_sub_super=False,
) == 'A^(q_1)')
assert (unicode(OperatorSymbol(
"A_1",
hs=hs1 * hs2)) == 'A\u0302_1^(q\u2081\u2297q\u2082)') # Â_1^(q₁⊗q₂)
assert (unicode(OperatorSymbol(
"Xi_2", hs=('q1', 'q2'))) == '\u039e\u0302_2^(q\u2081\u2297q\u2082)'
) # Ξ̂_2^(q₁⊗q₂)
assert unicode(OperatorSymbol("Xi", alpha, beta, hs=1)) == ('Ξ̂⁽¹⁾(α, β)')
assert unicode(IdentityOperator) == "𝟙"
assert unicode(ZeroOperator) == "0"
assert unicode(Create(hs=1)) == 'a\u0302^(1)\u2020' # â^(1)†
assert unicode(Destroy(hs=1)) == 'a\u0302\u207d\xb9\u207e' # â⁽¹⁾
assert unicode(Destroy(hs=1), unicode_sub_super=False) == 'a\u0302^(1)'
assert unicode(Destroy(hs=1), unicode_op_hats=False) == 'a\u207d\xb9\u207e'
assert (unicode(Destroy(hs=1),
unicode_op_hats=False,
unicode_sub_super=False) == 'a^(1)')
assert (unicode(Squeeze(Rational(1, 2),
hs=1)) == 'Squeeze\u207d\xb9\u207e(1/2)')
# Squeeze⁽¹⁾(1/2)
hs_tls = LocalSpace('1', basis=('g', 'e'))
sig_e_g = LocalSigma('e', 'g', hs=hs_tls)
assert unicode(sig_e_g) == '|e⟩⟨g|⁽¹⁾'
assert unicode(sig_e_g, unicode_sub_super=False) == '|e⟩⟨g|^(1)'
assert unicode(sig_e_g, show_hs_label=False) == '|e⟩⟨g|'
assert (unicode(sig_e_g, sig_as_ketbra=False) == '\u03c3\u0302_e,g^(1)'
) # σ̂_e,g^(1)
sig_e_e = LocalProjector('e', hs=hs_tls)
assert unicode(sig_e_e) == '|e⟩⟨e|⁽¹⁾'
assert (unicode(
sig_e_e,
sig_as_ketbra=False) == '\u03a0\u0302\u2091\u207d\xb9\u207e') # Π̂ₑ⁽¹⁾
assert (unicode(BasisKet(0, hs=1) * BasisKet(0, hs=2) *
BasisKet(0, hs=3)) == '|0,0,0⟩^(1⊗2⊗3)')
assert unicode(BasisKet(0, hs=hs1) * BasisKet(0, hs=hs2)) == '|00⟩^(q₁⊗q₂)'
def test_tex_operator_operations():
"""Test the tex representation of operator algebra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
A = OperatorSymbol("A", hs=hs1)
B = OperatorSymbol("B", hs=hs1)
C = OperatorSymbol("C", hs=hs2)
psi = KetSymbol('Psi', hs=hs1)
gamma = symbols('gamma', positive=True)
assert latex(A.dag()) == r'\hat{A}^{(q_{1})\dagger}'
assert latex(A + B) == r'\hat{A}^{(q_{1})} + \hat{B}^{(q_{1})}'
assert latex(A * B) == r'\hat{A}^{(q_{1})} \hat{B}^{(q_{1})}'
assert latex(A * C) == r'\hat{A}^{(q_{1})} \hat{C}^{(q_{2})}'
assert latex(2 * A) == r'2 \hat{A}^{(q_{1})}'
assert latex(2j * A) == r'2i \hat{A}^{(q_{1})}'
assert latex((1 + 2j) * A) == r'(1+2i) \hat{A}^{(q_{1})}'
assert latex(gamma**2 * A) == r'\gamma^{2} \hat{A}^{(q_{1})}'
assert (latex(-(gamma**2) / 2 *
A) == r'- \frac{\gamma^{2}}{2} \hat{A}^{(q_{1})}')
assert (latex(tr(
A * C,
over_space=hs2)) == r'{\rm tr}_{q_{2}}\left[\hat{C}^{(q_{2})}\right] '
r'\hat{A}^{(q_{1})}')
assert latex(Adjoint(A)) == r'\hat{A}^{(q_{1})\dagger}'
assert (latex(Adjoint(
A**2)) == r'\left(\hat{A}^{(q_{1})} \hat{A}^{(q_{1})}\right)^\dagger')
assert (latex(
Adjoint(A)**2) == r'\hat{A}^{(q_{1})\dagger} \hat{A}^{(q_{1})\dagger}')
assert latex(Adjoint(Create(hs=1))) == r'\hat{a}^{(1)}'
assert (latex(Adjoint(A + B)) ==
r'\left(\hat{A}^{(q_{1})} + \hat{B}^{(q_{1})}\right)^\dagger')
assert latex(PseudoInverse(A)) == r'\left(\hat{A}^{(q_{1})}\right)^+'
assert (latex(
PseudoInverse(A)**2
) == r'\left(\hat{A}^{(q_{1})}\right)^+ \left(\hat{A}^{(q_{1})}\right)^+')
assert (latex(NullSpaceProjector(A)) ==
r'\hat{P}_{Ker}\left(\hat{A}^{(q_{1})}\right)')
assert latex(A - B) == r'\hat{A}^{(q_{1})} - \hat{B}^{(q_{1})}'
assert (latex(A - B + C) ==
r'\hat{A}^{(q_{1})} - \hat{B}^{(q_{1})} + \hat{C}^{(q_{2})}')
assert (latex(2 * A - sqrt(gamma) * (B + C)) ==
r'2 \hat{A}^{(q_{1})} - \sqrt{\gamma} \left(\hat{B}^{(q_{1})} + '
r'\hat{C}^{(q_{2})}\right)')
assert (latex(Commutator(
A, B)) == r'\left[\hat{A}^{(q_{1})}, \hat{B}^{(q_{1})}\right]')
expr = (Commutator(A, B) * psi).dag()
assert (latex(expr, show_hs_label=False) ==
r'\left\langle \Psi \right\rvert \left[\hat{A}, '
r'\hat{B}\right]^{\dagger}')
def test_ascii_sop_elements():
"""Test the ascii representation of "atomic" Superoperators"""
hs1 = LocalSpace('q1', dimension=2)
hs2 = LocalSpace('q2', dimension=2)
alpha, beta = symbols('alpha, beta')
assert ascii(SuperOperatorSymbol("A", hs=hs1)) == 'A^(q1)'
assert ascii(SuperOperatorSymbol("A_1", hs=hs1 * hs2)) == 'A_1^(q1*q2)'
assert (ascii(SuperOperatorSymbol("Xi_2",
hs=('q1', 'q2'))) == 'Xi_2^(q1*q2)')
assert (ascii(SuperOperatorSymbol("Xi", alpha, beta,
hs=hs1)) == 'Xi^(q1)(alpha, beta)')
assert ascii(SuperOperatorSymbol("Xi_full", hs=1)) == 'Xi_full^(1)'
with pytest.raises(ValueError):
SuperOperatorSymbol(r'\Xi^2', hs='a')
assert ascii(IdentitySuperOperator) == "1"
assert ascii(ZeroSuperOperator) == "0"
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_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_invalid_spin_basis_ket():
"""Test that trying to instantiate invalid :func:`SpinBasisKet` raises the
appropriate exceptions"""
hs1 = SpinSpace('s', spin='3/2')
hs2 = SpinSpace('s', spin=1)
with pytest.raises(TypeError) as exc_info:
SpinBasisKet(1, 2, hs=LocalSpace(1))
assert "must be a SpinSpace" in str(exc_info.value)
with pytest.raises(TypeError) as exc_info:
SpinBasisKet(1, hs=hs1)
assert "exactly two positional arguments" in str(exc_info.value)
with pytest.raises(TypeError) as exc_info:
SpinBasisKet(1, 2, hs=hs2)
assert "exactly one positional argument" in str(exc_info.value)
with pytest.raises(ValueError) as exc_info:
SpinBasisKet(1, 3, hs=hs1)
assert "must be 2" in str(exc_info.value)
with pytest.raises(ValueError) as exc_info:
SpinBasisKet(-5, 2, hs=hs1)
assert "must be in range" in str(exc_info.value)
with pytest.raises(ValueError) as exc_info:
SpinBasisKet(5, 2, hs=hs1)
assert "must be in range" in str(exc_info.value)
with pytest.raises(ValueError) as exc_info:
SpinBasisKet(-5, hs=hs2)
assert "must be in range" in str(exc_info.value)
with pytest.raises(ValueError) as exc_info:
SpinBasisKet(5, hs=hs2)
assert "must be in range" in str(exc_info.value)
def test_coherent_state():
"""Test fock representation of coherent state"""
alpha = symbols('alpha')
hs0 = LocalSpace(0)
hs1 = LocalSpace(1, dimension=3)
i = IdxSym('i')
n = IdxSym('n')
psi = CoherentStateKet(alpha, hs=hs0)
psi_focksum_3 = psi.to_fock_representation(max_terms=3)
assert len(psi_focksum_3.term) == 3
for n_val in (0, 1, 2):
assert n_val in psi_focksum_3.term.ranges[0]
psi_focksum_inf = psi.to_fock_representation()
with pytest.raises(InfiniteSumError):
len(psi_focksum_inf.term)
for n_val in (0, 1, 2, 3):
assert n_val in psi_focksum_inf.term.ranges[0]
assert psi_focksum_inf.term.term.free_symbols == set([n, symbols('alpha')])
assert psi_focksum_inf.free_symbols == set([symbols('alpha')])
assert psi_focksum_inf.term.variables == [n]
assert psi_focksum_inf.substitute({n: i}).term.variables == [i]
assert (psi_focksum_inf.substitute({hs0: hs1}) == CoherentStateKet(
alpha, hs=hs1).to_fock_representation())
assert (psi.to_fock_representation(index_symbol='i').substitute(
{i: n}) == psi_focksum_inf)
assert (psi.to_fock_representation(index_symbol=i).substitute(
{i: n}) == psi_focksum_inf)
assert psi_focksum_3.doit([IndexedSum
]) == psi_focksum_inf.doit([IndexedSum],
max_terms=3)
psi_expanded_3 = psi_focksum_3.doit([IndexedSum])
assert psi_expanded_3 == (
sympy.exp(-alpha * alpha.conjugate() / 2) * KetPlus(
BasisKet(0, hs=LocalSpace(0)),
ScalarTimesKet(alpha, BasisKet(1, hs=LocalSpace(0))),
ScalarTimesKet(alpha**2 / sympy.sqrt(2),
BasisKet(2, hs=LocalSpace(0))),
))
psi = CoherentStateKet(alpha, hs=hs1)
assert psi.to_fock_representation().doit(
[IndexedSum]) == psi_expanded_3.substitute({hs0: hs1})
def test_cached_srepr(bell1_expr):
"""Test that we can get simplified expressions by passing a cache, and that
the cache is updated appropriately while printing"""
hs1 = LocalSpace('q_1', basis=('g', 'e'))
hs2 = LocalSpace('q_2', basis=('g', 'e'))
ket_g1 = BasisKet('g', hs=hs1)
cache = {hs1: 'hs1', hs2: 'hs2', 1 / sqrt(2): '1/sqrt(2)', -I: '-I'}
res = srepr(bell1_expr, cache=cache)
expected = (
"ScalarTimesKet(1/sqrt(2), KetPlus(TensorKet(BasisKet('e', hs=hs1), "
"BasisKet('g', hs=hs2)), ScalarTimesKet(-I, "
"TensorKet(BasisKet('g', hs=hs1), BasisKet('e', hs=hs2)))))")
assert res == expected
assert ket_g1 in cache
assert cache[ket_g1] == "BasisKet('g', hs=hs1)"
cache = {hs1: 'hs1', hs2: 'hs2', 1 / sqrt(2): '1/sqrt(2)', -I: '-I'}
# note that we *must* use a different cache
res = srepr(bell1_expr, cache=cache, indented=True)
expected = dedent(r'''
ScalarTimesKet(
1/sqrt(2),
KetPlus(
TensorKet(
BasisKet(
'e',
hs=hs1),
BasisKet(
'g',
hs=hs2)),
ScalarTimesKet(
-I,
TensorKet(
BasisKet(
'g',
hs=hs1),
BasisKet(
'e',
hs=hs2)))))''').strip()
assert res == expected
assert ket_g1 in cache
assert cache[ket_g1] == "BasisKet(\n 'g',\n hs=hs1)"
def test_tex_sop_elements():
"""Test the tex representation of "atomic" Superoperators"""
hs1 = LocalSpace('q1', dimension=2)
hs2 = LocalSpace('q2', dimension=2)
alpha, beta = symbols('alpha, beta')
assert latex(SuperOperatorSymbol("A", hs=hs1)) == r'\mathrm{A}^{(q_{1})}'
assert (latex(SuperOperatorSymbol(
"A_1", hs=hs1 * hs2)) == r'\mathrm{A}_{1}^{(q_{1} \otimes q_{2})}')
assert (latex(SuperOperatorSymbol(
"Xi", alpha, beta,
hs=hs1)) == r'\mathrm{\Xi}^{(q_{1})}\left(\alpha, \beta\right)')
assert (latex(SuperOperatorSymbol(
"Xi_2",
hs=('q1', 'q2'))) == r'\mathrm{\Xi}_{2}^{(q_{1} \otimes q_{2})}')
assert (latex(SuperOperatorSymbol(
"Xi_full", hs=1)) == r'\mathrm{\Xi}_{\text{full}}^{(1)}')
assert latex(IdentitySuperOperator) == r'\mathbb{1}'
assert latex(ZeroSuperOperator) == r'\mathbb{0}'
def test_ascii_operator_operations():
"""Test the ascii representation of operator algebra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
A = OperatorSymbol("A", hs=hs1)
B = OperatorSymbol("B", hs=hs1)
C = OperatorSymbol("C", hs=hs2)
D = OperatorSymbol("D", hs=hs1)
psi = KetSymbol('Psi', hs=hs1)
gamma = symbols('gamma', positive=True)
assert ascii(A + B) == 'A^(q_1) + B^(q_1)'
assert ascii(A * B) == 'A^(q_1) * B^(q_1)'
assert ascii(A * C) == 'A^(q_1) * C^(q_2)'
assert ascii(A * (B + D)) == 'A^(q_1) * (B^(q_1) + D^(q_1))'
assert ascii(A * (B - D)) == 'A^(q_1) * (B^(q_1) - D^(q_1))'
assert (ascii(
(A + B) *
(-2 * B - D)) == '(A^(q_1) + B^(q_1)) * (-D^(q_1) - 2 * B^(q_1))')
assert ascii(OperatorTimes(A, -B)) == 'A^(q_1) * (-B^(q_1))'
assert ascii(OperatorTimes(A, -B), show_hs_label=False) == 'A * (-B)'
assert ascii(2 * A) == '2 * A^(q_1)'
assert ascii(2j * A) == '2j * A^(q_1)'
assert ascii((1 + 2j) * A) == '(1+2j) * A^(q_1)'
assert ascii(gamma**2 * A) == 'gamma**2 * A^(q_1)'
assert ascii(-(gamma**2) / 2 * A) == '-gamma**2/2 * A^(q_1)'
assert ascii(tr(A * C, over_space=hs2)) == 'tr_(q_2)[C^(q_2)] * A^(q_1)'
expr = A + OperatorPlusMinusCC(B * D)
assert ascii(expr, show_hs_label=False) == 'A + (B * D + c.c.)'
expr = A + OperatorPlusMinusCC(B + D)
assert ascii(expr, show_hs_label=False) == 'A + (B + D + c.c.)'
expr = A * OperatorPlusMinusCC(B * D)
assert ascii(expr, show_hs_label=False) == 'A * (B * D + c.c.)'
assert ascii(Adjoint(A)) == 'A^(q_1)H'
assert ascii(Adjoint(Create(hs=1))) == 'a^(1)'
assert ascii(Adjoint(A + B)) == '(A^(q_1) + B^(q_1))^H'
assert ascii(PseudoInverse(A)) == '(A^(q_1))^+'
assert ascii(NullSpaceProjector(A)) == 'P_Ker(A^(q_1))'
assert ascii(A - B) == 'A^(q_1) - B^(q_1)'
assert ascii(A - B + C) == 'A^(q_1) - B^(q_1) + C^(q_2)'
expr = 2 * A - sqrt(gamma) * (B + C)
assert ascii(expr) == '2 * A^(q_1) - sqrt(gamma) * (B^(q_1) + C^(q_2))'
assert ascii(Commutator(A, B)) == r'[A^(q_1), B^(q_1)]'
expr = (Commutator(A, B) * psi).dag()
assert ascii(expr, show_hs_label=False) == r'<Psi| [A, B]^H'
def test_tex_ket_elements():
"""Test the tex representation of "atomic" kets"""
hs1 = LocalSpace('q1', basis=('g', 'e'))
hs2 = LocalSpace('q2', basis=('g', 'e'))
alpha, beta = symbols('alpha, beta')
psi = KetSymbol('Psi', hs=hs1)
assert latex(psi) == r'\left\lvert \Psi \right\rangle^{(q_{1})}'
assert (latex(KetSymbol('Psi', alpha, beta, hs=1)) ==
r'\left\lvert \Psi\left(\alpha, \beta\right) \right\rangle^{(1)}')
assert latex(psi, tex_use_braket=True) == r'\Ket{\Psi}^{(q_{1})}'
assert (latex(psi, tex_use_braket=True,
show_hs_label='subscript') == r'\Ket{\Psi}_{(q_{1})}')
assert (latex(psi, tex_use_braket=True,
show_hs_label=False) == r'\Ket{\Psi}')
assert (latex(KetSymbol('Psi',
hs=1)) == r'\left\lvert \Psi \right\rangle^{(1)}')
assert (latex(KetSymbol(
'Psi',
hs=(1, 2))) == r'\left\lvert \Psi \right\rangle^{(1 \otimes 2)}')
assert (latex(KetSymbol(
'Psi', hs=hs1 *
hs2)) == r'\left\lvert \Psi \right\rangle^{(q_{1} \otimes q_{2})}')
assert (latex(KetSymbol('Psi',
hs=1)) == r'\left\lvert \Psi \right\rangle^{(1)}')
assert latex(ZeroKet) == '0'
assert latex(TrivialKet) == '1'
assert (latex(BasisKet(
'e', hs=hs1)) == r'\left\lvert e \right\rangle^{(q_{1})}')
hs_tls = LocalSpace('1', basis=('excited', 'ground'))
assert (latex(BasisKet(
'excited',
hs=hs_tls)) == r'\left\lvert \text{excited} \right\rangle^{(1)}')
assert latex(BasisKet(1, hs=1)) == r'\left\lvert 1 \right\rangle^{(1)}'
spin = SpinSpace('s', spin=(3, 2))
assert (latex(SpinBasisKet(
-3, 2, hs=spin)) == r'\left\lvert -3/2 \right\rangle^{(s)}')
assert (latex(SpinBasisKet(
1, 2, hs=spin)) == r'\left\lvert +1/2 \right\rangle^{(s)}')
assert (latex(SpinBasisKet(-3, 2, hs=spin), tex_frac_for_spin_labels=True)
== r'\left\lvert -\frac{3}{2} \right\rangle^{(s)}')
assert (latex(SpinBasisKet(1, 2, hs=spin), tex_frac_for_spin_labels=True)
== r'\left\lvert +\frac{1}{2} \right\rangle^{(s)}')
assert (latex(CoherentStateKet(
2.0, hs=1)) == r'\left\lvert \alpha=2 \right\rangle^{(1)}')
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_tex_bra_operations():
"""Test the tex representation of bra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
psi1 = KetSymbol("Psi_1", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
psi2 = KetSymbol("Psi_2", hs=hs1)
bra_psi1 = KetSymbol("Psi_1", hs=hs1).dag()
bra_psi2 = KetSymbol("Psi_2", hs=hs1).dag()
bra_psi2 = KetSymbol("Psi_2", hs=hs1).dag()
bra_psi3 = KetSymbol("Psi_3", hs=hs1).dag()
bra_phi = KetSymbol("Phi", hs=hs2).dag()
A = OperatorSymbol("A_0", hs=hs1)
gamma = symbols('gamma', positive=True)
phase = exp(-I * gamma)
assert (latex(
(psi1 +
psi2).dag()) == r'\left\langle \Psi_{1} \right\rvert^{(q_{1})} + '
r'\left\langle \Psi_{2} \right\rvert^{(q_{1})}')
assert (latex((psi1 + psi2).dag(), tex_use_braket=True) ==
r'\Bra{\Psi_{1}}^{(q_{1})} + \Bra{\Psi_{2}}^{(q_{1})}')
assert (
latex(bra_psi1 +
bra_psi2) == r'\left\langle \Psi_{1} \right\rvert^{(q_{1})} + '
r'\left\langle \Psi_{2} \right\rvert^{(q_{1})}')
assert (
latex(bra_psi1 - bra_psi2 +
bra_psi3) == r'\left\langle \Psi_{1} \right\rvert^{(q_{1})} - '
r'\left\langle \Psi_{2} \right\rvert^{(q_{1})} + '
r'\left\langle \Psi_{3} \right\rvert^{(q_{1})}')
assert (latex(
bra_psi1 *
bra_phi) == r'\left\langle \Psi_{1} \right\rvert^{(q_{1})} \otimes '
r'\left\langle \Phi \right\rvert^{(q_{2})}')
assert (latex(bra_psi1 * bra_phi, tex_use_braket=True) ==
r'\Bra{\Psi_{1}}^{(q_{1})} \otimes \Bra{\Phi}^{(q_{2})}')
assert (latex(Bra(
phase *
psi1)) == r'e^{i \gamma} \left\langle \Psi_{1} \right\rvert^{(q_{1})}')
assert (latex(
(A * psi1).dag()) == r'\left\langle \Psi_{1} \right\rvert^{(q_{1})} '
r'\hat{A}_{0}^{(q_{1})\dagger}')
def test_tex_sop_operations():
"""Test the tex representation of super operator algebra operations"""
hs1 = LocalSpace('q_1', dimension=2)
hs2 = LocalSpace('q_2', dimension=2)
A = SuperOperatorSymbol("A", hs=hs1)
B = SuperOperatorSymbol("B", hs=hs1)
C = SuperOperatorSymbol("C", hs=hs2)
L = SuperOperatorSymbol("L", hs=1)
M = SuperOperatorSymbol("M", hs=1)
A_op = OperatorSymbol("A", hs=1)
gamma = symbols('gamma', positive=True)
assert latex(A + B) == r'\mathrm{A}^{(q_{1})} + \mathrm{B}^{(q_{1})}'
assert latex(A * B) == r'\mathrm{A}^{(q_{1})} \mathrm{B}^{(q_{1})}'
assert latex(A * C) == r'\mathrm{A}^{(q_{1})} \mathrm{C}^{(q_{2})}'
assert latex(2 * A) == r'2 \mathrm{A}^{(q_{1})}'
assert latex(2j * A) == r'2i \mathrm{A}^{(q_{1})}'
assert latex((1 + 2j) * A) == r'(1+2i) \mathrm{A}^{(q_{1})}'
assert latex(gamma**2 * A) == r'\gamma^{2} \mathrm{A}^{(q_{1})}'
assert (latex(-(gamma**2) / 2 *
A) == r'- \frac{\gamma^{2}}{2} \mathrm{A}^{(q_{1})}')
assert latex(SuperAdjoint(A)) == r'\mathrm{A}^{(q_{1})\dagger}'
assert (latex(SuperAdjoint(A + B)) == r'\left(\mathrm{A}^{(q_{1})} + '
r'\mathrm{B}^{(q_{1})}\right)^\dagger')
assert latex(A - B) == r'\mathrm{A}^{(q_{1})} - \mathrm{B}^{(q_{1})}'
assert (latex(A - B +
C) == r'\mathrm{A}^{(q_{1})} - \mathrm{B}^{(q_{1})} + '
r'\mathrm{C}^{(q_{2})}')
assert (latex(2 * A - sqrt(gamma) *
(B + C)) == r'2 \mathrm{A}^{(q_{1})} - \sqrt{\gamma} '
r'\left(\mathrm{B}^{(q_{1})} + \mathrm{C}^{(q_{2})}\right)')
assert latex(SPre(A_op)) == r'\mathrm{SPre}\left(\hat{A}^{(1)}\right)'
assert latex(SPost(A_op)) == r'\mathrm{SPost}\left(\hat{A}^{(1)}\right)'
assert (latex(SuperOperatorTimesOperator(
L, A_op)) == r'\mathrm{L}^{(1)}\left[\hat{A}^{(1)}\right]')
assert (latex(SuperOperatorTimesOperator(
L,
sqrt(gamma) *
A_op)) == r'\mathrm{L}^{(1)}\left[\sqrt{\gamma} \hat{A}^{(1)}\right]')
assert (latex(SuperOperatorTimesOperator(
(L + 2 * M),
A_op)) == r'\left(\mathrm{L}^{(1)} + 2 \mathrm{M}^{(1)}\right)'
r'\left[\hat{A}^{(1)}\right]')