def test_unicode_equation():
"""Test printing of the Eq class"""
eq_1 = Eq(lhs=OperatorSymbol('H', hs=0), rhs=Create(hs=0) * Destroy(hs=0))
# fmt: off
eq = (eq_1.apply_to_lhs(lambda expr: expr + 1).apply_to_rhs(
lambda expr: expr + 1).apply_to_rhs(lambda expr: expr**2).tag(
3).transform(lambda eq: eq + 1).tag(4).apply_to_rhs('expand').
apply_to_lhs(lambda expr: expr**2).tag(5).apply(
'expand').apply_to_lhs(lambda expr: expr**2).tag(6).apply_to_lhs(
'expand').apply_to_rhs(lambda expr: expr + 1))
# fmt: on
assert unicode(eq_1) == 'Ĥ⁽⁰⁾ = â^(0)† â⁽⁰⁾'
assert unicode(eq_1.tag(1)) == 'Ĥ⁽⁰⁾ = â^(0)† â⁽⁰⁾ (1)'
unicode_lines = unicode(eq, show_hs_label=False).split("\n")
expected = [
' Ĥ = â^† â',
' 𝟙 + Ĥ = â^† â',
' = 𝟙 + â^† â',
' = (𝟙 + â^† â) (𝟙 + â^† â) (3)',
' 2 + Ĥ = 𝟙 + (𝟙 + â^† â) (𝟙 + â^† â) (4)',
' = 2 + â^† â^† â â + 3 â^† â',
' (2 + Ĥ) (2 + Ĥ) = 2 + â^† â^† â â + 3 â^† â (5)',
' 4 + 4 Ĥ + Ĥ Ĥ = 2 + â^† â^† â â + 3 â^† â',
' (4 + 4 Ĥ + Ĥ Ĥ) (4 + 4 Ĥ + Ĥ Ĥ) = 2 + â^† â^† â â + 3 â^† â (6)',
'16 + 32 Ĥ + Ĥ Ĥ Ĥ Ĥ + 8 Ĥ Ĥ Ĥ + 24 Ĥ Ĥ = 2 + â^† â^† â â + 3 â^† â',
' = 3 + â^† â^† â â + 3 â^† â',
]
for i, line in enumerate(unicode_lines):
assert line == expected[i]
eq = Eq(OperatorSymbol('H', hs=0), 0, eq_sym_str='→')
assert unicode(eq, show_hs_label=False) == 'Ĥ → 0'
def test_render_head_repr_tuple_list():
"""Test that render_head_repr works for lists and tuples"""
a, b = symbols('a, b')
A = OperatorSymbol('A', hs=0)
B = OperatorSymbol('B', hs=0)
expr = ((a, 2), (b, 1))
assert render_head_repr(expr) == "((Symbol('a'), 2), (Symbol('b'), 1))"
expr = [[a, 2], [b, 1]]
assert render_head_repr(expr) == "[[Symbol('a'), 2], [Symbol('b'), 1]]"
expr = [(a, 2), (b, 1)]
assert render_head_repr(expr) == "[(Symbol('a'), 2), (Symbol('b'), 1)]"
expr = ([a, 2], [b, 1])
assert render_head_repr(expr) == "([Symbol('a'), 2], [Symbol('b'), 1])"
expr = (A, (b, 1))
assert (render_head_repr(expr) ==
"(OperatorSymbol('A', hs=LocalSpace('0')), (Symbol('b'), 1))")
expr = [(a, 2), (A, B)]
assert (render_head_repr(expr) ==
"[(Symbol('a'), 2), (OperatorSymbol('A', hs=LocalSpace('0')), "
"OperatorSymbol('B', hs=LocalSpace('0')))]")
def matrix_expr():
A = OperatorSymbol("A", hs=1)
B = OperatorSymbol("B", hs=1)
C = OperatorSymbol("C", hs=1)
D = OperatorSymbol("D", hs=1)
gamma = symbols('gamma')
phase = exp(-I * gamma / 2)
return Matrix([[phase * A, B], [C, phase.conjugate() * D]])
def test_apply_mtd():
H_0 = OperatorSymbol('H_0', hs=0)
H = OperatorSymbol('H', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq = eq0.apply('substitute', {H_0: H, E0: 0}).tag('new')
assert eq.lhs == H
assert eq.rhs == ω * Create(hs=0) * Destroy(hs=0)
assert eq._tag == 'new'
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_eq_sub_eq():
ω, E0 = sympy.symbols('omega, E_0')
H_0 = OperatorSymbol('H_0', hs=0)
H_1 = OperatorSymbol('H_1', hs=0)
mu = OperatorSymbol('mu', hs=0)
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq1 = Eq(H_1, mu + E0, tag='1')
eq = eq0 - eq1
assert eq.lhs == H_0 - H_1
assert eq.rhs == ω * Create(hs=0) * Destroy(hs=0) - mu
assert eq._tag is None
def test_apply():
"""Test the apply method"""
A = OperatorSymbol('A', hs=0)
def raise_to_power(x, y):
return x**y
def plus_n(expr, *, n):
return expr + n
assert A.apply(raise_to_power, 2).apply(plus_n, n=1) == A**2 + 1
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_ascii_matrix():
"""Test ascii representation of the Matrix class"""
A = OperatorSymbol("A", hs=1)
B = OperatorSymbol("B", hs=1)
C = OperatorSymbol("C", hs=1)
D = OperatorSymbol("D", hs=1)
assert (ascii(Matrix([[A, B], [C,
D]])) == '[[A^(1), B^(1)], [C^(1), D^(1)]]')
assert (ascii(Matrix([A, B, C,
D])) == '[[A^(1)], [B^(1)], [C^(1)], [D^(1)]]')
assert ascii(Matrix([[A, B, C, D]])) == '[[A^(1), B^(1), C^(1), D^(1)]]'
assert ascii(Matrix([[0, 1], [-1, 0]])) == '[[0, 1], [-1, 0]]'
assert ascii(Matrix([[], []])) == '[[], []]'
assert ascii(Matrix([])) == '[[], []]'
def matrix_exprs():
"""Prepare a list of Matrix expressions"""
A = OperatorSymbol("A", hs=1)
B = OperatorSymbol("B", hs=1)
C = OperatorSymbol("C", hs=1)
D = OperatorSymbol("D", hs=1)
return [
Matrix([[A, B], [C, D]]),
Matrix([A, B, C, D]),
Matrix([[A, B, C, D]]),
Matrix([[0, 1], [-1, 0]]),
# Matrix([[], []]), # see issue #8316 in numpy
# Matrix([]),
]
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_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_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_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_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_eq_substitute():
H_0 = OperatorSymbol('H_0', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq1 = eq0.apply('substitute', {E0: 0}).reset()
eq2 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0))
assert eq1 == eq2
def test_ascii_equation():
"""Test printing of the Eq class"""
eq_1 = Eq(lhs=OperatorSymbol('H', hs=0), rhs=Create(hs=0) * Destroy(hs=0))
# fmt: off
eq = (eq_1.apply_to_lhs(lambda expr: expr + 1).apply_to_rhs(
lambda expr: expr + 1).apply_to_rhs(lambda expr: expr**2).tag(
3).transform(lambda eq: eq + 1).tag(4).apply_to_rhs('expand').
apply_to_lhs(lambda expr: expr**2).tag(5).apply(
'expand').apply_to_lhs(lambda expr: expr**2).tag(6).apply_to_lhs(
'expand').apply_to_rhs(lambda expr: expr + 1))
# fmt: on
assert ascii(eq_1) == 'H^(0) = a^(0)H * a^(0)'
assert ascii(eq_1.tag(1).reset()) == 'H^(0) = a^(0)H * a^(0) (1)'
assert ascii(eq, show_hs_label=False).strip() == (r'''
H = a^H * a
1 + H = a^H * a
= 1 + a^H * a
= (1 + a^H * a) * (1 + a^H * a) (3)
2 + H = 1 + (1 + a^H * a) * (1 + a^H * a) (4)
= 2 + a^H * a^H * a * a + 3 * a^H * a
(2 + H) * (2 + H) = 2 + a^H * a^H * a * a + 3 * a^H * a (5)
4 + 4 * H + H * H = 2 + a^H * a^H * a * a + 3 * a^H * a
(4 + 4 * H + H * H) * (4 + 4 * H + H * H) = 2 + a^H * a^H * a * a + 3 * a^H * a (6)
16 + 32 * H + H * H * H * H + 8 * H * H * H + 24 * H * H = 2 + a^H * a^H * a * a + 3 * a^H * a
= 3 + a^H * a^H * a * a + 3 * a^H * a
'''.strip())
def test_eq_copy():
H_0 = OperatorSymbol('H_0', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq = eq0.copy()
assert eq == eq0
assert eq is not eq0
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_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_matrixis_zero():
"""Test that zero-matrices can be identified"""
m = Matrix([[0, 0], [0, 0]])
assert m.is_zero
m = Matrix([[Zero, Zero], [Zero, Zero]])
assert m.is_zero
m = Matrix([[ZeroOperator, ZeroOperator], [ZeroOperator, ZeroOperator]])
assert m.is_zero
m = Matrix([[ZeroOperator, 0], [Zero, ZeroOperator]])
assert m.is_zero
m = Matrix([[0, 0], [1, 0]])
assert not m.is_zero
m = Matrix([[Zero, Zero], [One, Zero]])
assert not m.is_zero
A = OperatorSymbol("A", hs=0)
m = Matrix([[ZeroOperator, ZeroOperator], [A, ZeroOperator]])
assert not m.is_zero
m = Matrix([[ZeroOperator, 0], [symbols('alpha'), ZeroOperator]])
assert not m.is_zero
def test_ascii_operator_elements():
"""Test the ascii representation of "atomic" operator algebra elements"""
hs1 = LocalSpace('q1', dimension=2)
hs2 = LocalSpace('q2', dimension=2)
alpha, beta = symbols('alpha, beta')
assert ascii(OperatorSymbol("A", hs=hs1)) == 'A^(q1)'
A_1 = OperatorSymbol("A_1", hs=1)
assert ascii(A_1, show_hs_label='subscript') == 'A_1,(1)'
assert ascii(OperatorSymbol("A", hs=hs1), show_hs_label=False) == 'A'
assert ascii(OperatorSymbol("A_1", hs=hs1 * hs2)) == 'A_1^(q1*q2)'
assert ascii(OperatorSymbol("Xi_2", hs=('q1', 'q2'))) == 'Xi_2^(q1*q2)'
assert ascii(OperatorSymbol("Xi_full", hs=1)) == 'Xi_full^(1)'
assert ascii(OperatorSymbol("Xi", alpha, beta,
hs=1)) == ('Xi^(1)(alpha, beta)')
with pytest.raises(ValueError):
OperatorSymbol(r'\Xi^2', hs='a')
assert ascii(IdentityOperator) == "1"
assert ascii(ZeroOperator) == "0"
assert ascii(Create(hs=1)) == "a^(1)H"
assert ascii(Create(hs=1), show_hs_label=False) == "a^H"
assert ascii(Create(hs=1), show_hs_label='subscript') == "a_(1)^H"
assert ascii(Destroy(hs=1)) == "a^(1)"
fock1 = LocalSpace(1,
local_identifiers={
'Create': 'b',
'Destroy': 'b',
'Phase': 'Ph'
})
spin1 = SpinSpace(1,
spin=1,
local_identifiers={
'Jz': 'Z',
'Jplus': 'Jp',
'Jminus': 'Jm'
})
assert ascii(Create(hs=fock1)) == "b^(1)H"
assert ascii(Destroy(hs=fock1)) == "b^(1)"
assert ascii(Jz(hs=SpinSpace(1, spin=1))) == "J_z^(1)"
assert ascii(Jz(hs=spin1)) == "Z^(1)"
assert ascii(Jplus(hs=spin1)) == "Jp^(1)"
assert ascii(Jminus(hs=spin1)) == "Jm^(1)"
assert ascii(Phase(0.5, hs=1)) == 'Phase^(1)(0.5)'
assert ascii(Phase(0.5, hs=fock1)) == 'Ph^(1)(0.5)'
assert ascii(Displace(0.5, hs=1)) == 'D^(1)(0.5)'
assert ascii(Squeeze(0.5, hs=1)) == 'Squeeze^(1)(0.5)'
hs_tls = LocalSpace('1', basis=('g', 'e'))
sig_e_g = LocalSigma('e', 'g', hs=hs_tls)
assert ascii(sig_e_g) == '|e><g|^(1)'
assert ascii(sig_e_g, sig_as_ketbra=False) == 'sigma_e,g^(1)'
sig_e_e = LocalProjector('e', hs=hs_tls)
assert ascii(sig_e_e, sig_as_ketbra=False) == 'Pi_e^(1)'
assert (ascii(BasisKet(0, hs=1) * BasisKet(0, hs=2) *
BasisKet(0, hs=3)) == '|0,0,0>^(1*2*3)')
assert ascii(BasisKet(0, hs=hs1) * BasisKet(0, hs=hs2)) == '|00>^(q1*q2)'
assert (ascii(
BasisKet(0, hs=LocalSpace(0, dimension=20)) *
BasisKet(0, hs=LocalSpace(1, dimension=20))) == '|0,0>^(0*1)')
def test_repr_latex():
H_0 = OperatorSymbol('H_0', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
Eq.latex_renderer = staticmethod(latex)
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
repr1 = eq0._repr_latex_()
repr2 = latex(eq0)
assert repr1 == repr2
def test_eq_div_const():
H_0 = OperatorSymbol('H_0', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq = eq0 / 2
assert eq.lhs == eq0.lhs / 2
assert eq.rhs == eq0.rhs / 2
assert eq._tag is None
def test_apply_to_lhs():
H_0 = OperatorSymbol('H_0', hs=0)
ω, E0 = sympy.symbols('omega, E_0')
eq0 = Eq(H_0, ω * Create(hs=0) * Destroy(hs=0) + E0, tag='0')
eq = eq0.apply_to_lhs(lambda expr: expr + E0).tag('new')
assert eq.lhs == H_0 + E0
assert eq.rhs == eq0.rhs
assert eq._tag == 'new'
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_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_indented_srepr_tuple_list():
"""Test that indendented srepr of a list or tuple is same as unindented
Because these occur as kwargs values, they are always printed unindented.
"""
a, b = symbols('a, b')
A = OperatorSymbol('A', hs=0)
B = OperatorSymbol('B', hs=0)
exprs = [
((a, 2), (b, 1)),
[[a, 2], [b, 1]],
[(a, 2), (b, 1)],
([a, 2], [b, 1]),
(A, (b, 1)),
[(a, 2), (A, B)],
]
for expr in exprs:
assert (srepr(expr, indented=True) == srepr(expr) ==
render_head_repr(expr))
def test_unicode_matrix():
"""Test unicode representation of the Matrix class"""
A = OperatorSymbol("A", hs=1)
B = OperatorSymbol("B", hs=1)
C = OperatorSymbol("C", hs=1)
D = OperatorSymbol("D", hs=1)
assert (unicode(Matrix(
[[A, B],
[C, D]])) == '[[A\u0302\u207d\xb9\u207e, B\u0302\u207d\xb9\u207e], '
'[C\u0302\u207d\xb9\u207e, D\u0302\u207d\xb9\u207e]]')
# '[[Â⁽¹⁾, B̂⁽¹⁾], [Ĉ⁽¹⁾, D̂⁽¹⁾]]')
assert (unicode(Matrix(
[A, B, C,
D])) == '[[A\u0302\u207d\xb9\u207e], [B\u0302\u207d\xb9\u207e], '
'[C\u0302\u207d\xb9\u207e], [D\u0302\u207d\xb9\u207e]]')
# '[Â⁽¹⁾], [B̂⁽¹⁾], [Ĉ⁽¹⁾], [D̂⁽¹⁾]]')
assert unicode(Matrix([[0, 1], [-1, 0]])) == '[[0, 1], [-1, 0]]'
assert unicode(Matrix([[], []])) == '[[], []]'
assert unicode(Matrix([])) == '[[], []]'
def test_tex_matrix():
"""Test tex representation of the Matrix class"""
A = OperatorSymbol("A", hs=1)
B = OperatorSymbol("B", hs=1)
C = OperatorSymbol("C", hs=1)
D = OperatorSymbol("D", hs=1)
assert latex(OperatorSymbol("A", hs=1)) == r'\hat{A}^{(1)}'
assert (latex(Matrix(
[[A, B], [C,
D]])) == r'\begin{pmatrix}\hat{A}^{(1)} & \hat{B}^{(1)} \\'
r'\hat{C}^{(1)} & \hat{D}^{(1)}\end{pmatrix}')
assert (latex(Matrix(
[A, B, C, D])) == r'\begin{pmatrix}\hat{A}^{(1)} \\\hat{B}^{(1)} \\'
r'\hat{C}^{(1)} \\\hat{D}^{(1)}\end{pmatrix}')
assert (latex(Matrix(
[[A, B, C, D]])) == r'\begin{pmatrix}\hat{A}^{(1)} & \hat{B}^{(1)} & '
r'\hat{C}^{(1)} & \hat{D}^{(1)}\end{pmatrix}')
assert (latex(Matrix([[0, 1], [-1, 0]
])) == r'\begin{pmatrix}0 & 1 \\-1 & 0\end{pmatrix}')
assert latex(Matrix([[], []])) == r'\begin{pmatrix} \\\end{pmatrix}'
assert latex(Matrix([])) == r'\begin{pmatrix} \\\end{pmatrix}'
