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Theoretical analysis demonstrates support localization in finite trace-balanced processes, revealing that reachability coverage is essential for quantum instruments.
The note studies what retained-trace observations determine in a finite trace-balanced process. It proves an exact finite-horizon trace ledger, identifies complementary maps on reachable cones and their complex spans, and develops weighted order-domination bounds. A zero weighted complementary trace localizes a positive map to the complement of the aggregate support; full support gives global vanishing. A rank-one qubit instrument shows why reachability coverage is essential.
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Ritsuki Kawada (2026) studied this question.
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