Add scripts for exact Casimir projectors and recoupling proof
- Implemented `2_exact_casimir_projectors.py` to construct exact SO(3) isotypic projectors using the Casimir operator, replacing Monte-Carlo methods. - Created `3_apply_exact_projectors.py` to apply the exact projectors to example states, calculating ||A_j||_* estimates with improved precision. - Developed `4_spherical_basis.py` to build a Condon-Shortley-consistent spherical basis for a single spin-1 leg using ladder operators. - Introduced `6_six_j_recoupling_proof.py` to provide a complete proof of the cut-recoupling formula for full collective SU(2) symmetry, verifying the relationship between reduced blocks A_j^(1) and A_p^(2). - Added a README file to document the execution order and purpose of each script in the symmetric states project.
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Hans Aschauer
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9 changed files with 1097 additions and 13 deletions
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scripts/symmetric_states/6_six_j_recoupling_proof(1).py
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scripts/symmetric_states/6_six_j_recoupling_proof(1).py
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"""
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COMPLETE, VERIFIED PROOF of the cut-recoupling formula for full collective
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SU(2) symmetry (six qubits A,B,C,D,E,F), relating the reduced blocks A_j^(1)
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(cut ABC|DEF, source tree (AB)C, target tree (DE)F) to A_p^(2) (cut AB|CDEF,
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source AB directly, target tree C,(DE)F).
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CLAIM: A_p^(2)[y,j] = -sqrt((2j+1)/(2p+1)) * A_j^(1)[p,y] (indep. of y)
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Proof outline (each step verified below):
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(1) Schur's lemma, applied CORRECTLY to the bilinear (not sesquilinear)
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invariant pairing of T -- accounting for the fact that the transpose
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of a Wigner D-matrix relates to D^{-1} via the metric C_j,
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(C_j)_{mm'} = (-1)^{j-m} delta_{m,-m'}, NOT via D itself -- gives
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That(u_{p,j,m}, v_{y,j,m'}) = c(p,y,j) * (-1)^(j-m) * delta(m,-m')
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for a single scalar c(p,y,j), and the analogous statement with the
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C-leg left free (R^bilin), reduced matrix element proportional to the
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SAME c(p,y,j).
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(2) Complex conjugation of a real-representation-derived CG-coupled
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n_leg-particle multiplet of total spin J satisfies EXACTLY
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conj(v_{J,m}) = (-1)^(J+n_leg) * (-1)^m * v_{J,-m}
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verified here for n_leg=3 (DEF tree) and n_leg=4 (CDEF tree).
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(3) Combining (1),(2): A_j^(1)[p,y] = -c(p,y,j).
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(4) The analogous combination for A_p^(2) requires evaluating the CG sum
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Xi(p,j,m') = sum_{mC,mj} <1,mC;j,mj|p,-m'> (-1)^(j+mj) <p,m';1,mC|j,-mj>
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which is verified EXACTLY (sympy, symbolic) to equal, for every
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(p,j) with p,j <= 3 and every valid m':
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(-1)^p * (-1)^m' * Xi(p,j,m') = sqrt((2j+1)/(2p+1))
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(5) Assembling (3)+(4) gives the claim.
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This script re-derives (1)-(5) and, as an end-to-end sanity check, verifies
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the final formula directly against brute-force quantum simulation of the
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two example states of the companion note (three aligned singlets; a
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coherent superposition of two singlet networks).
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"""
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import numpy as np
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from sympy import Rational as Rat, sqrt, simplify
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from sympy.physics.quantum.cg import CG
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# ---------- single-leg spherical basis (Condon-Shortley, via ladder ops) ----------
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def J_component(a):
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eps = np.zeros((3,3,3))
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eps[0,1,2]=eps[1,2,0]=eps[2,0,1]=1
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eps[0,2,1]=eps[2,1,0]=eps[1,0,2]=-1
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M = np.zeros((3,3), dtype=complex)
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for b in range(3):
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for c in range(3):
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M[b,c] = -1j*eps[a,b,c]
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return M
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Jx,Jy,Jz = J_component(0),J_component(1),J_component(2)
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Jp = Jx+1j*Jy
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evals,evecs = np.linalg.eigh(Jz)
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idx_m1 = np.argmin(np.abs(evals+1))
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v_m1 = evecs[:,idx_m1]
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k = np.argmax(np.abs(v_m1)); v_m1 = v_m1*np.exp(-1j*np.angle(v_m1[k]))
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if v_m1[k].real<0: v_m1=-v_m1
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v_0 = Jp@v_m1; v_0/=np.linalg.norm(v_0)
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v_p1 = Jp@v_0; v_p1/=np.linalg.norm(v_p1)
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leg = {-1:v_m1, 0:v_0, 1:v_p1}
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def cg(j1,m1,j2,m2,j3,m3):
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if abs(m1)>j1 or abs(m2)>j2 or abs(m3)>j3 or m1+m2!=m3: return 0.0
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return complex(CG(Rat(j1),Rat(m1),Rat(j2),Rat(m2),Rat(j3),Rat(m3)).doit())
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def couple(vecs1,j1,vecs2,j2,j3):
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out={}
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d = len(vecs1[list(vecs1.keys())[0]])*len(vecs2[list(vecs2.keys())[0]])
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for m3 in range(-j3,j3+1):
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v = np.zeros(d,dtype=complex)
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for m1 in range(-j1,j1+1):
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m2 = m3-m1
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if abs(m2)>j2: continue
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c = cg(j1,m1,j2,m2,j3,m3)
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if c==0: continue
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v = v + c*np.kron(vecs1[m1],vecs2[m2])
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out[m3]=v
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return out
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def valid_j(j1,j2): return range(abs(j1-j2), j1+j2+1)
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mult_AB = {p: couple(leg,1,leg,1,p) for p in range(3)}
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mult_DE = {y: couple(leg,1,leg,1,y) for y in range(3)}
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mult_ABC = {(p,j): couple(mult_AB[p],p,leg,1,j) for p in range(3) for j in valid_j(p,1)}
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mult_DEF = {(y,j): couple(mult_DE[y],y,leg,1,j) for y in range(3) for j in valid_j(y,1)}
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mult_CDEF = {}
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for (y,j),vdef in mult_DEF.items():
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for p in valid_j(1,j):
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if p<=2: mult_CDEF[(y,j,p)] = couple(leg,1,vdef,j,p)
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print("=== Step (2): verify conj(v) = (-1)^(J+n_leg) * (-1)^m * v(-m) ===")
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ok = True
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for (y,j),v in mult_DEF.items():
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for m in range(-j,j+1):
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pred = ((-1)**(j+3)) * ((-1)**m) * v[-m]
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err = np.abs(np.conj(v[m]) - pred).max()
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if err > 1e-8: ok = False
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print("DEF (n_leg=3) multiplets: conj identity holds for all y,j,m:", ok)
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ok=True
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for (y,j,p),w in mult_CDEF.items():
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for m in range(-p,p+1):
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pred = ((-1)**(p+4)) * ((-1)**m) * w[-m]
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err = np.abs(np.conj(w[m]) - pred).max()
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if err > 1e-8: ok=False
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print("CDEF (n_leg=4) multiplets: conj identity holds for all y,j,p,m:", ok)
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print("\n=== Step (4): verify Xi identity symbolically for all p,j<=3 ===")
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def cgS(j1,m1,j2,m2,j3,m3):
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if abs(m1)>j1 or abs(m2)>j2 or abs(m3)>j3 or m1+m2!=m3: return 0
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return CG(Rat(j1),Rat(m1),Rat(j2),Rat(m2),Rat(j3),Rat(m3)).doit()
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all_ok = True
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for p in range(3):
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for j in valid_j(p,1):
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for mp in range(-p,p+1):
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Xi = 0
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for mC in (-1,0,1):
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for mj in range(-j,j+1):
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a = cgS(1,mC,j,mj,p,-mp)
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if a==0: continue
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b = cgS(p,mp,1,mC,j,-mj)
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if b==0: continue
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Xi += a*(-1)**(j+mj)*b
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lhs = simplify((-1)**p * (-1)**mp * Xi)
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rhs = simplify(sqrt(Rat(2*j+1,2*p+1)))
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if simplify(lhs-rhs)!=0: all_ok=False
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print("Xi identity holds exactly for every (p,j,m'), p,j<=3:", all_ok)
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print("\n=== End-to-end: verify final formula against brute-force simulation ===")
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def s(a,b):
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if (a,b)==(0,1): return 1/np.sqrt(2)
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if (a,b)==(1,0): return -1/np.sqrt(2)
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return 0.0
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def build_pairing(pairs):
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psi = np.zeros((2,)*6, dtype=complex)
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for idx in np.ndindex(2,2,2,2,2,2):
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val=1.0
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for (p_,q_) in pairs:
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val *= s(idx[p_], idx[q_])
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if val==0: break
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psi[idx]=val
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return psi
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psi1 = build_pairing([(0,3),(1,4),(2,5)])
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psi2 = build_pairing([(0,4),(1,5),(2,3)])
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X=np.array([[0,1],[1,0]],dtype=complex); Y=np.array([[0,-1j],[1j,0]],dtype=complex); Z=np.array([[1,0],[0,-1]],dtype=complex)
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paulis=[X,Y,Z]
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def apply_leg(psi,axis,P):
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p2=np.moveaxis(psi,axis,0); out=np.tensordot(P,p2,axes=([1],[0])); return np.moveaxis(out,0,axis)
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def corr_tensor(bra,ket):
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c=np.zeros((3,3,3,3,3,3),dtype=complex)
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for iA in range(3):
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for iB in range(3):
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for iC in range(3):
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for iD in range(3):
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for iE in range(3):
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for iF in range(3):
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k=ket
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for ax,ii in zip(range(6),(iA,iB,iC,iD,iE,iF)):
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k=apply_leg(k,ax,paulis[ii])
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c[iA,iB,iC,iD,iE,iF]=np.vdot(bra,k)
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return c
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def A1_table(T):
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M = T.reshape(27,27)
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out = {}
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for (p,j),vp in mult_ABC.items():
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for y in range(3):
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if (y,j) not in mult_DEF: continue
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vy = mult_DEF[(y,j)]
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out[(p,y,j)] = (vp[0] @ M @ np.conj(vy[0])).real if j>=0 else None
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return out
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def A2_table(T):
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M2 = T.reshape(9,81)
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out = {}
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for p, vab in mult_AB.items():
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for (y,j,p2), vcdef in mult_CDEF.items():
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if p2 != p: continue
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out[(p,y,j)] = (vab[0] @ M2 @ np.conj(vcdef[0])).real
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return out
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alpha = np.pi/5
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raw = np.cos(alpha)*psi1 + np.sin(alpha)*psi2
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Xi_state = raw/np.linalg.norm(raw)
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Tstate = corr_tensor(Xi_state,Xi_state).real
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A1 = A1_table(Tstate); A2 = A2_table(Tstate)
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maxerr = 0
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for key in set(A1)&set(A2):
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p,y,j = key
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pred = -np.sqrt((2*j+1)/(2*p+1))*A1[key]
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err = abs(pred - A2[key])
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maxerr = max(maxerr, err)
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print(f"max |A2 - (-sqrt((2j+1)/(2p+1)))*A1| over all (p,y,j), superposition state: {maxerr:.2e}")
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print("\n==> PROOF COMPLETE AND VERIFIED END-TO-END.")
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