Exercise PLAN-206

POVM selector correctness depends on one specified clean projector and normalized coefficient weights, not shared physical ancillas. A union-zero projection works for clean disjoint branches; beta compensation uses the Euclidean norm of the factors.

1. On a normalized alpha preparation the clean map is C_alpha=sum_i(alpha_i/beta_i)|i>K_i; uniform preparation generally reweights, with explicit exceptional/zero-effect boundaries.
2. The successful density map and probabilities follow from T_alpha=sum_i|alpha_i|²P_i/beta_i²; failure completes the physical measurement and conditional normalization is explicit.
3. Correct compensation is beta_i/sqrt(sum beta_j²); common-factor attenuation is a separate explicit construction, not the unnormalized beta_i/beta_max PREP.
4. Disjoint clean ancillas admit a common union-zero projector that kills every active-branch garbage term, refuting the predecessor's forced obstruction.
5. A common carrier/clean initialization must be supplied; max-width sharing is a sufficient alternative, not a necessary or proved optimal physical layout.

Fix register order(index I,global ancilla A,system H), and embed each branch into A before forming W. Let Jψ=|0_A>ψ and J† U_i J=K_i/beta_i, beta_i>0. For a unit ψ write U_iJψ=|0_A>K_iψ/beta_i+g_i(ψ), with (<0_A| tensor I)g_i=0. W is unitary because its index projectors are orthogonal and every embedded U_i is unitary; different U_i need not commute. On a normalized index vector alpha the FULL output is sum_i alpha_i|i>|0_A>K_iψ/beta_i+sum_i alpha_i|i>g_i(ψ). Thus the projected map C_alpha:H->I tensor H is C_alpha=sum_i(alpha_i/beta_i)|i> tensor K_i, with C_alpha†C_alpha=T_alpha=sum_i |alpha_i|^2 P_i/beta_i^2. For input density rho the unnormalized successful joint state is C_alpha rho C_alpha†=sum_ij alpha_i conj(alpha_j)/(beta_i beta_j)|i><j| tensor K_i rho K_j. Its probability is p_alpha=tr(T_alpha rho); conditional joint state is that operator divided by p_alpha only when p_alpha>0. Success-label effects are E_i=|alpha_i|^2P_i/beta_i^2. Compression implies T_alpha<=I, so the full physical measurement includes failure effect I-T_alpha; E_i alone need not sum to I. On successful label i, the system update is K_i rho K_i/tr(P_i rho) if that outcome has nonzero probability. A zero outcome probability supplies no normalized state.

For uniform alpha_i=1/sqrt(L), C_u=L^(-1/2)sum_i beta_i^(-1)|i>K_i, T_u=L^(-1)sum_i P_i/beta_i^2, and conditional label probability is tr(P_i rho)/beta_i^2 divided by sum_j tr(P_j rho)/beta_j^2. This is GENERALLY biased, not an unconditional impossibility theorem. For P_0=P_1=I/2 and(beta_0,beta_1)=(1,2), valid contraction encodings exist and the probabilities become4/5,1/5 rather than1/2,1/2; total uniform success is5/16. But P_0=I,P_1=0 with those factors still gives the desired conditional label0 and unchanged system. Different factors on zero effects, or restricted populated outcomes, do not establish generic failure. In the full finite POVM/uniform case success is positive for every density input, at least1/(L beta_max^2), since sum_i P_i=I.

Let B=sqrt(sum_i beta_i^2)>0 and PREP create sum_i(beta_i/B)|i>. Its coefficients have squared norm1, and C_beta=V/B where V=sum_i|i>K_i satisfies V†V=I. Thus success probability is exactly1/B^2, accepted density is V rho V†/B^2, and the conditional POVM/isometry is the intended one. Existence of the encodings implies P_i<=beta_i^2 I, hence B^2>=1; no probability exceeds1. The proposed denominator beta_max is generally wrong: beta=(1,2) gives norm squared5/4. Known branch phases must be fixed: using e^(i theta_i)beta_i/B inserts the relative label unitary sum_i e^(i theta_i)|i><i| into the coherent target, although label measurement probabilities are unchanged. A common overall phase is different from arbitrary per-label phases. An alternative is beta_*=max beta_i and an extra accepted-block attenuation r_i=beta_i/beta_* in each branch. A clean qubit rotation with00 entry r_i, tensor the old encoding, gives block r_i K_i/beta_i=K_i/beta_*; include its flag in the global success register. Uniform preparation then has success1/(L beta_*^2). Padding/attenuation and preparing the nonuniform vector are construction work, not free or claimed optimal.

Now let the actual ancillas be disjoint A_0,...,A_(L-1). Construct A=their tensor product, prepare every factor in its zero state, and extend branch U_i to act on A_i and H while acting identically on idle A_j. The global projector Q_A=|0_A><0_A| is a SINGLE well-defined projector. Every active dirty vector is killed by <0_(A_i)|, while each idle zero register contributes factor1, not another random success. Therefore the same C_alpha and probabilities above hold exactly. As a witness take L=2,P_0=P_1=I/2,beta=1; branch0 rotates its own one-qubit ancilla by Ry(pi/2), branch1 rotates one of its own TWO ancillas and leaves the other zero. All three ancillas are initialized zero and the system is unchanged. The union-zero output is(|0>+|1>)ψ/2, of norm squared1/2=1/(L beta^2). This has unequal widths and disjoint physical ancillas, so directly refutes the forced part(b) assertion. Relabeling/padding both operators onto a common max-width register is another sufficient interface, not a necessity or a proved cheapest implementation. Without any common carrier the written W is a typing error; with dirty idle registers the union projection need not mean just active-branch success, and their preparation/reset/correlation obligations must be analyzed. The construction supplies CLEAN initialization explicitly; it does not promise this for arbitrary dirty workspace. Positive factors are required even for zero effects, whose zero accepted block can use an extra X flag if a new encoding is constructed. L=1 is allowed and reduces to the single-branch formulas; an empty POVM cannot sum to I on a nonzero physical space. No amplitude-amplification/query-complexity proof for Lemma9.15's final paragraph is supplied here; unknown-state reflection, SELECT control, compiled gate/finite-bit implementation and resource optimality are not declared free or certified.

1. Derives the full alpha-weighted branch state, C_alpha rho C_alpha†, T_alpha, success/failure effects and conditional probabilities, with correct phases, normalization and zero-output boundaries. (1 weight)
2. Computes uniform reweighting and a concrete biased example while refuting the universal 'different beta always fails' claim on an allowed zero-effect case; does not call the successful subeffects alone a complete measurement. (1 weight)
3. Normalizes compensating PREP by B=sqrt(sum beta_i²), checks the beta_max counterexample and relative phases, and constructs common-factor attenuation with its additional clean flag/work explicitly accounted. (1 weight)
4. Constructs the disjoint/unequal-width union-zero projector and exact witness, distinguishes clean initialization from dirty idle workspace/typing, and rejects a necessity or cost-optimality claim for shared physical ancillas. (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: d5bbb6751510de60e219672c2dfd00853e0dd5e7ca9e81d962aedea813e1518b

Item digest: 38f593a23555b0979d0ee1753d3e37256d946e5266044c3469771f7968ddf481

Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a

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

{ "question_version": 2, "predecessor_key": "qcb-core-plan-206-povm-shared-ancilla-beta-assumption-boundary", "predecessor_item_digest": "d38a79b187cf26c01f7885a3a12002f9fe511bfdb2b765bfd886cd809ed7aab2", "predecessor_contract_hash": "649d4e624621233d47f2e87d3362859e93f4b05ff4643ed3d04f7483a81d2d35", "current_contract_hash": "d5bbb6751510de60e219672c2dfd00853e0dd5e7ca9e81d962aedea813e1518b", "changed_contract_fields": [ "question" ], "repair_record": "shared/repairs/plan-206-common-projector-weighted-prep-question-successor.json", "repair_record_digest": "550c9e80fddd7450fa5b98e105811d5a09db11db52ebacc832cf710dcae7fc48", "review_record": "reports/textbook-v1/repairs/plan-206-common-projector-weighted-prep-question-successor-review.json", "review_digest": "1c30634cb35fc5965ae7a395fcdeda20e20e8b800e88e72fc4e861cf3a88a90a", "equivalence_to_predecessor_claimed": false, "additional_core_count": 0 }