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000001 | COMPLETE COVARIANT INSTRUMENTS FOR THE NATIVE SIX-SIMPLEX POVM
000002 | ================================================================
000003 | 
000004 | Scope
000005 | -----
000006 | 
000007 | This theorem stays on the original native H5 and uses the six rank-one ovoid
000008 | effects and the sixty marked G31 actions already reconstructed by
000009 | doily_quartic_bridge.py.  An instrument means six completely positive maps
000010 | 
000011 |     I_a : End(H5) -> End(H5)
000012 | 
000013 | whose effects are the fixed operators E_a and which obey the actual outcome
000014 | covariance
000015 | 
000016 |     I_{pi_g(a)}(rho_g X rho_g^dagger)
000017 |       = rho_g I_a(X) rho_g^dagger.
000018 | 
000019 | The result classifies every such finite instrument.  It does not add a new
000020 | carrier, select a physical value of its parameter, or turn the instrument into
000021 | physical time evolution.
000022 | 
000023 | Why rank one forces measure and prepare
000024 | ---------------------------------------
000025 | 
000026 | For one outcome let J_a be the positive Choi matrix of I_a.  Its input
000027 | partial trace is E_a^T.  Since E_a has rank one, positivity forces J_a to have
000028 | support only on
000029 | 
000030 |     support(E_a^T) tensor H5.
000031 | 
000032 | Indeed, for x in the kernel of E_a^T,
000033 | 
000034 |     Tr[(|x><x| tensor I) J_a] = <x|E_a^T|x> = 0.
000035 | 
000036 | A positive matrix with zero expectation on that whole subspace has no row or
000037 | column there.  Therefore there is a unique density matrix sigma_a such that
000038 | 
000039 |     J_a = E_a^T tensor sigma_a,
000040 | 
000041 | or equivalently
000042 | 
000043 |     I_a(rho) = Tr(E_a rho) sigma_a.
000044 | 
000045 | This is a theorem for every completely positive operation with the fixed
000046 | rank-one effect.  It is not a chosen Kraus ansatz.
000047 | 
000048 | Exact stabilizer commutant
000049 | --------------------------
000050 | 
000051 | Write
000052 | 
000053 |     P_a = (6/5) E_a.
000054 | 
000055 | Each P_a is a rank-one orthogonal projector.  The sixty original marked
000056 | actions are transitive on the six labels.  For a fixed label a, forty marked
000057 | reflections fix a; their outcome permutations contain the ten transpositions
000058 | of the other five labels.  Solving the full rational commutator system on
000059 | End(H5) gives rank 23 and hence
000060 | 
000061 |     dim_C Comm(Stab(a)) = 2.
000062 | 
000063 | Both I5 and P_a lie in this commutant and are independent, so they form its
000064 | complete basis.  Covariance makes sigma_a stabilizer invariant.  Positivity
000065 | and trace one therefore give exactly
000066 | 
000067 |     sigma_a(p) = p P_a + (1-p)/4 (I5-P_a),
000068 |     0 <= p <= 1.
000069 | 
000070 | Transitivity forces the same p for all six outcomes.  Conversely the checker
000071 | verifies this family against every one of the sixty native actions.  Thus the
000072 | displayed interval is the complete covariant instrument space.
000073 | 
000074 | Kraus ranks and the extra minimality rule
000075 | -----------------------------------------
000076 | 
000077 | Because J_a=E_a^T tensor sigma_a, the minimal Kraus rank of one outcome is
000078 | rank(sigma_a):
000079 | 
000080 |     p=1:       rank 1,
000081 |     p=0:       rank 4,
000082 |     0<p<1:     rank 5.
000083 | 
000084 | With the classical outcome retained, the six blocks have total minimal Kraus
000085 | ranks 6, 24, and 30 respectively.  Requiring one Kraus operator per outcome,
000086 | pure prepared output, or minimal tagged environment dimension selects p=1
000087 | uniquely.  Each is an additional instrument principle.  None is implied by
000088 | the word primitive in the P1 source.
000089 | 
000090 | Channel after forgetting the outcome
000091 | ------------------------------------
000092 | 
000093 | Let Phi_p=sum_a I_a and let Phi_L be the square-root Lueders channel of the
000094 | same POVM.  Since E_a=(5/6)P_a and P_a rho P_a=Tr(P_a rho)P_a,
000095 | 
000096 |     Phi_L(rho) = sum_a Tr(E_a rho) P_a.
000097 | 
000098 | Expanding sigma_a(p) now gives the exact identity
000099 | 
000100 |     Phi_p = (5p-1)/4 Phi_L + (1-p)/4 Tr(rho) I5.
000101 | 
000102 | The existing Lueders decomposition of End(H5) has dimensions 1+5+19 and
000103 | eigenvalues 1,4/5,0.  The replacement term acts only on the identity line.
000104 | Consequently
000105 | 
000106 |     spec(Phi_p) = 1^1, (p-1/5)^5, 0^19.
000107 | 
000108 | For 0<=p<=1 the second eigenvalue lies in [-1/5,4/5].  The unique stationary
000109 | density is therefore I5/5 for every member of the classified family.
000110 | 
000111 | The Choi matrix of the forgotten-outcome channel is
000112 | 
000113 |     J_p = (5p-1)/4 J_L + (1-p)/4 I25.
000114 | 
000115 | For this real symmetric rank-one POVM the numeric J_L array equals the
000116 | Lueders Liouville array, although their tensor-index meanings differ.  Its
000117 | exact Choi spectrum is
000118 | 
000119 |     p^1, ((15p+1)/20)^5, ((1-p)/4)^19.
000120 | 
000121 | Thus the forgotten-outcome channel has Choi rank 6 at p=1, rank 24 at p=0,
000122 | and rank 25 for 0<p<1.  These ranks differ from the tagged instrument ranks
000123 | 6,24,30 because forgetting the classical label can merge Kraus spans.
000124 | 
000125 | Repeated outcome statistics
000126 | ---------------------------
000127 | 
000128 | After observing a, the prepared state is sigma_a(p).  A second use of the
000129 | same POVM is therefore the six-state Markov chain
000130 | 
000131 |     Pr(b=a | a) = 5p/6,
000132 |     Pr(b!=a | a) = (6-5p)/30.
000133 | 
000134 | The transition matrix is doubly stochastic, so its stationary distribution
000135 | is uniform.  The stationary two-step law has
000136 | 
000137 |     Pr(a,a) = 5p/36,
000138 |     Pr(a,b) = (6-5p)/180,  a!=b.
000139 | 
000140 | At p=1 these numbers are 5/36 and 1/180, the same numbers obtained from the
000141 | Omega state in RELATIONAL-CHANNEL.txt.  Here they arise from a separable
000142 | measure-and-prepare time sequence.  Equality of these two tables is therefore
000143 | not evidence for entanglement or for a common physical preparation.
000144 | 
000145 | Exact failure of state tomography
000146 | ---------------------------------
000147 | 
000148 | The six effects span a six dimensional subspace of the full 25 dimensional
000149 | operator space.  Repeating the instrument does not recover the missing
000150 | coherences.  This is witnessed by two explicit positive states in the real
000151 | isometric H5 chart.
000152 | 
000153 | Let
000154 | 
000155 |     X = i(|0><1| - |1><0|),
000156 |     rho_+ = I5/5 + X/10,
000157 |     rho_- = I5/5 - X/10.
000158 | 
000159 | X is Hermitian and traceless.  Both density matrices have the exact spectrum
000160 | 
000161 |     {1/10, 1/5, 1/5, 1/5, 3/10},
000162 | 
000163 | so they are distinct strictly positive states.  Every E_a is real symmetric,
000164 | while X is imaginary antisymmetric.  Therefore
000165 | 
000166 |     Tr(E_a X) = 0
000167 | 
000168 | for all six outcomes and
000169 | 
000170 |     Tr(E_a rho_+) = Tr(E_a rho_-) = 1/6.
000171 | 
000172 | The measure-and-prepare classification gives the probability of every finite
000173 | outcome word directly:
000174 | 
000175 |     Pr(a_1...a_n | rho)
000176 |       = Tr(E_{a_1} rho)
000177 |         product_{k=2}^n T_p[a_{k-1},a_k].
000178 | 
000179 | The two states have the same first factor, and the transition matrix is fixed
000180 | by p and the previous outcome.  Induction in n proves that rho_+ and rho_-
000181 | have identical probabilities for every finite outcome word.  Thus this
000182 | instrument family is not informationally complete and cannot reconstruct the
000183 | unmeasured GNS coherences or a complete GNS crossword from its outcome history.
000184 | This is an explicit counterexample, rather than a dimension-only inference.
000185 | 
000186 | Rational endpoint Choi certificates
000187 | ------------------------------------
000188 | 
000189 | The checker also represents the same native H5 as the zero-sum subspace of
000190 | Q^6.  This is a redundant rational frame for the existing six rays, not a new
000191 | Hilbert space.  With
000192 | 
000193 |     Q = I6 - (1/6) 11^T,
000194 |     u_a = e_a - (1/6)1,
000195 | 
000196 | one has exactly
000197 | 
000198 |     Gram(u_a) = Gram(q_a)/12,
000199 |     E_a = u_a u_a^T,
000200 |     P_a = (6/5)u_a u_a^T.