TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/COVARIANT-INSTRUMENTS.txt
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Inclusive original lines: 127-185
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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 |