104 lines
3.3 KiB
Python
104 lines
3.3 KiB
Python
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"""
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02_werner_qutrit_symbolic.py
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Same check as 01_werner_qubit_symbolic.py, generalized to d=3 (qutrits),
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using the sqrt(3/2)-scaled Gell-Mann convention fixed in the paper's own
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Tiles example (tr(sigma_i sigma_j) = d delta_ij = 3 delta_ij).
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State family: rho(p) = p * P_anti/dim(P_anti) + (1-p) * I/9
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(the natural qutrit "Werner state" built from the antisymmetric subspace
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of C^3 x C^3, dimension 3), invariant under U(x)U for all U in U(3).
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Result (the interesting part): the order-1 shadow-map/correlation-matrix
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criterion gives p_c = 1/2, but the TRUE separability threshold (Werner
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1989, p_sep = 1/(d+1)) is p_c = 1/4. Unlike the qubit case, the criterion
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is here only a valid but NOT tight sufficient condition -- symmetry forces
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"concentration" of the signal (single isotype => correlation matrix
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proportional to identity) but not "sharpening" of the threshold itself.
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Requires: sympy. Runtime: under a minute.
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"""
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import sympy as sp
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from sympy import sqrt, I, simplify, Matrix, eye, zeros, re, symbols, Rational
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d = 3
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lam = [None] * 8
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lam[0] = Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 0]])
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lam[1] = Matrix([[0, -I, 0], [I, 0, 0], [0, 0, 0]])
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lam[2] = Matrix([[1, 0, 0], [0, -1, 0], [0, 0, 0]])
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lam[3] = Matrix([[0, 0, 1], [0, 0, 0], [1, 0, 0]])
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lam[4] = Matrix([[0, 0, -I], [0, 0, 0], [I, 0, 0]])
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lam[5] = Matrix([[0, 0, 0], [0, 0, 1], [0, 1, 0]])
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lam[6] = Matrix([[0, 0, 0], [0, 0, -I], [0, I, 0]])
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lam[7] = (1 / sqrt(3)) * Matrix([[1, 0, 0], [0, 1, 0], [0, 0, -2]])
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c = sqrt(Rational(3, 2))
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sigma = [simplify(c * L) for L in lam]
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for i in range(8):
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for j in range(8):
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val = simplify((sigma[i] * sigma[j]).trace())
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assert val == (3 if i == j else 0), (i, j, val)
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I3 = eye(3)
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def op_A(P):
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return sp.Matrix(sp.kronecker_product(P, I3))
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def op_B(P):
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return sp.Matrix(sp.kronecker_product(I3, P))
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V = zeros(9, 9)
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for a in range(3):
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for b in range(3):
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V[b * 3 + a, a * 3 + b] = 1
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I9 = eye(9)
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P_anti = simplify((I9 - V) / 2)
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dim_anti = simplify(P_anti.trace()) # = 3
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p = symbols('p', real=True)
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rho_p = simplify(p * P_anti / dim_anti + (1 - p) * I9 / 9)
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def entry(rho, ops):
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M = None
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for op in ops:
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M = op if M is None else M * op
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return simplify(re(simplify((rho * M).trace())))
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T = Matrix(8, 8, lambda i, j: entry(rho_p, [op_A(sigma[i]), op_B(sigma[j])]))
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print("Correlation matrix T(p) (should be proportional to I_8):")
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sp.pprint(T)
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norm_const = 1 / sqrt(4) # (d_a-1)(d_bar_a-1) = 2*2 = 4
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Mn = simplify(norm_const * T)
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nuclear_norm = simplify(8 * sp.Abs(Mn[0, 0]))
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print("\nShadow-map nuclear norm:", nuclear_norm)
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print("Shadow-map threshold:", sp.solve(sp.Eq(nuclear_norm, 1), p))
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def partial_transpose_B_d(M, dim):
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Mpt = zeros(dim * dim, dim * dim)
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for a in range(dim):
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for b in range(dim):
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for cc in range(dim):
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for dd in range(dim):
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i, j = a * dim + b, cc * dim + dd
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i2, j2 = a * dim + dd, cc * dim + b
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Mpt[i2, j2] = M[i, j]
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return Mpt
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rho_pt = partial_transpose_B_d(rho_p, 3)
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print("\nPartial-transpose eigenvalues (PPT / true-separability threshold):")
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for ev in rho_pt.eigenvals().keys():
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print(" ", simplify(ev), " = 0 at p =", sp.solve(sp.Eq(ev, 0), p))
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print("\nExpected: shadow-map threshold p=1/2 (NOT tight);"
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" true threshold (Werner 1989, p_sep=1/(d+1)) p=1/4.")
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