Exercise PLAN-204

Correct complex conjugation fixes the forced-weight contradiction for conclusive-only unambiguous discrimination; the positive inconclusive effect provides the slack for a valid three-outcome POVM.

1. PSD zero expectation implies E_psi annihilates phi; in the two-dimensional span its support is the normalized phi_perp line, and similarly for the other effect.
2. With phi_perp=(psi-conj(c)phi)/sqrt(1-|c|^2), the correct overlap is sqrt(1-|c|^2); completeness forces both weights to1/(1-|c|^2).
3. The forced two-effect sum has trace2/(1-|c|^2)>2 for0<|c|<1, contradicting completeness on the two-dimensional space.
4. Thus overlapping distinct states cannot have a valid unambiguous measurement with only the two conclusive labels.
5. The given smaller weights make the conclusive sum a positive contraction with spectrum1 and(1-|c|)/(1+|c|); its positive complement restores completeness while preserving unambiguity.

Use the bra-ket convention of the question: c=<psi|phi>, conjugate-linear in the bra and linear in the ket. Put r=|c| and d=1-r^2>0. Let u=phi_perp=(psi-conj(c)*phi)/sqrt(d) and v=psi_perp=(phi-c*psi)/sqrt(d). Expanding the inner products gives ||u||=||v||=1, <phi|u>=0 and <psi|v>=0. In particular <psi|u>=(1-conj(c)*c)/sqrt(d)=sqrt(d), and <phi|v>=sqrt(d). The coefficient here is conj(c), not c; replacing it by c would incorrectly equate 1-c^2 with 1-|c|^2 for a complex overlap. No rephasing assumption is imposed.

Suppose a two-element POVM with just the two conclusive labels existed. Positivity and <phi|E_psi|phi>=0 imply E_psi*phi=0, since that expectation is ||sqrt(E_psi)*phi||^2. Self-adjointness then places its range in phi's orthogonal complement. In the specified two-dimensional span this forces E_psi=a*|u><u| for some a>=0. Likewise E_phi=b*|v><v|. Completeness evaluated on psi, with the other expectation zero by unambiguity, gives a*d=1; evaluating on phi gives b*d=1. Hence a=b=1/d. But the resulting sum has trace2/d>2, whereas the identity on the two-dimensional space has trace2. This is a contradiction for every 0<r<1. A sharp alternate check is E_psi*psi=psi from completeness and E_phi*psi=0; then self-adjointness and E_psi*phi=0 would force c=<psi|phi>=<psi|E_psi*phi>=0, contradicting the same assumption.

For the proposed three effects, let F be the linear map whose two columns are u and v. Their overlap is <u|v>=-c, so F^dagger*F has diagonal entries1 and off-diagonal entries -c,-conj(c). Its characteristic polynomial is (1-lambda)^2-r^2; the eigenvalues are 1+r and 1-r. Since r<1, F is invertible on the two-dimensional span and F*F^dagger has the same eigenvalues. Therefore S=P_psi+P_phi=(F*F^dagger)/(1+r) has eigenvalues1 and (1-r)/(1+r). It is positive and at most the identity, so P_?=I-S is positive, with eigenvalues0 and 2r/(1+r). Each conclusive effect is positive and completeness holds by definition. The orthogonality relations above give zero probability for either wrong conclusive label.

On input psi, the correct conclusive probability is d/(1+r)=1-r, the wrong probability is zero and the inconclusive probability is r; the same holds with psi and phi exchanged. Introducing P_? supplies exactly the positive slack missing from a conclusive-only complete measurement. This verifies the stated construction, not uniqueness or optimality among every possible prior-weighted discrimination scheme. At r=0 the obstruction disappears and two orthogonal projectors suffice; r=1 is excluded because the complement formulas divide by zero. The question's actual regime is 0<r<1. No circuit, finite-bit cost or new Lean closure is inferred from these operator calculations.

1. Uses positivity/zero expectation to annihilate the wrong state and two-dimensional self-adjointness to obtain a|phi_perp><phi_perp| and b|psi_perp><psi_perp|. (1 weight)
2. Computes <psi|phi_perp>=(1-conj(c)c)/sqrt(1-|c|^2)=sqrt(1-|c|^2), with the actual complex conjugate, and derives a=b=1/(1-|c|^2) from completeness without restricting c to real. (1 weight)
3. Obtains trace2/(1-|c|^2)>2, or an equivalent self-adjointness contradiction forcing c=0, to rule out every conclusive-only two-element POVM for overlapping states. (1 weight)
4. Verifies the given three effects are PSD and complete, with zero wrong-label probabilities. Derives the rank-one-sum spectrum1±|c| and explains the positive inconclusive slack; does not assert a unique measurement or arbitrary-prior optimum. (1 weight)

No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.

Read the Lean track and existing diagnostic warmups

• website/research/sources.json

  lin-wiebe-2026-qasc

  Source SHA-256: 9f85f45f20ed5a46931ad0dec3b94ee09748bc56e39fae29b0aefe910440b0ae · MIT metadata only
• Primary source

Contract: b1569085f93b8ecb592f938cc36e92f9283b3fe8ac30012a7073869eb995671f

Item digest: 978453b6eb90931fa2ea361e9bf82185e9a84e1604d41632767f0242a529507e

Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a

Authored support steps: 5. This is not measured graph distance or model difficulty.

{ "predecessor_item_digest": "d925946e833792c3cf95b2b993cd3b820be35abfd5c259fdb68384b3f29d3558", "predecessor_key": "qcb-core-plan-204-unambiguous-discrimination-inconclusive-necessity", "repair_record": "shared/repairs/plan-204-complex-conjugation-repair.json", "repair_record_digest": "bf3c01e5a75f04338bc73b1803fe7dbd7447f815747aee41380175ead74b16fb", "review_record": "reports/textbook-v1/repairs/plan-204-complex-conjugation-repair-migration-revalidation-review.json", "review_digest": "79dfb46418a93cef899a657194add6a16e21e6257ffd1ce97b40ae92fa68cde9" }