Exercise PLAN-011

SVD gives an exact one-ancilla unitary dilation of every contraction. Its chosen branch has probability||A b||^2, but exact-operator nonunitarity does not force probabilistic normalized state preparation: cI has the identity as its normalized target procedure.

1. M=[[Sigma,S],[S,-Sigma]] is Hermitian-unitary because S^2=I-Sigma^2 and Sigma,S commute.
2. The full product is unitary with clean block A and failure block S V^dagger; its selected probability is||A b||^2, with conditioning undefined on the kernel and probability1 precisely on normalized vectors in ker(I-A^dagger A).
3. Nonunitary A equals no system-only unitary as an exact operator, and its unnormalized branch is not trace preserving on all densities. But cI refutes universal ancilla/postselection necessity for normalized output rays; no generic deterministic normalized procedure or efficient gate synthesis is claimed.

Since the singular values lie in[0,1], S is a well-defined nonnegative real diagonal matrix, S^2=I-Sigma^2, and S Sigma=Sigma S. The middle matrix M=[[Sigma,S],[S,-Sigma]] is Hermitian, and block multiplication gives M^dagger M=M^2=[[Sigma^2+S^2,Sigma S-S Sigma],[S Sigma-Sigma S,S^2+Sigma^2]]=diag(I,I). It is therefore unitary. Both outer block-diagonal factors are unitary, so U_A is unitary. Multiplying the blocks explicitly gives U_A=[[W Sigma V^dagger,W S],[S V^dagger,-Sigma]]=[[A,W S],[S V^dagger,-Sigma]]. Thus the clean block is exactly A, an exact unit-factor one-ancilla block encoding.

For normalized b this same full matrix gives U_A(|0> tensor b)=|0> tensor A b+|1> tensor S V^dagger b. The two ancillary sectors are orthogonal. Their squared norms add to1, with p_b=||A b||^2=<b|A^dagger A|b> and failure probability1-p_b=||S V^dagger b||^2. If p_b>0, conditioning on outcome0 gives the normalized system vector A b/sqrt(p_b). The amplitude A b before conditioning is generally subnormalized; it is not itself a normalized physical output state. If A b=0, p_b=0 and the selected outcome never occurs, so its conditioned state is UNDEFINED; one must not divide by zero or call the zero vector a state. In particular A=0 has no defined normalized target on any input.

Because A is a contraction, I-A^dagger A is positive semidefinite. For normalized b, p_b=1 exactly when <b|(I-A^dagger A)|b>=0. A positive square root shows that this happens exactly when (I-A^dagger A)b=0, i.e. b belongs to K=ker(I-A^dagger A). This is the norm-one subspace, spanned by right singular vectors of singular value1. On it A preserves norms and inner products; the displayed branch is certain and the failure component vanishes. A may be nonunitary while K is nonzero: A=diag(1,1/2) on a qubit has p_|0>=1. This does not make A unitary on the entire system, nor claim that an arbitrary input lies in K. A strictly contractive example cI has K={0} even though its normalized-state target is simple.

The asserted universal necessity for NORMALIZED state preparation is false. Take A=cI with real0<c<1. For every normalized b, A b=c b is nonzero and A b/||A b||=b. The system-only identity prepares that normalized target deterministically, with no ancillary qubit and no postselection. Yet the chosen unit-factor dilation branch above has success probability c^2. That probability characterizes this compression/selected unnormalized branch, not all procedures producing the same output ray. The identity does NOT implement the unnormalized operator cI as an exact linear map: it discards its scalar norm loss, which normalized state preparation does not retain. This one counterexample refutes universal necessity; it does not establish a deterministic normalized-state implementation for every nonunitary A. In general normalization is input dependent, and this construction asserts no such generic procedure.

The correct exact-operator obstruction remains. If a system-only unitary U equaled A on every input vector, then U=A as a matrix and A^dagger A=U^dagger U=I, which for a square finite-dimensional A implies unitarity. Thus a nonunitary A cannot equal any such U. Equivalently, some right singular value of a nonunitary square contraction is less than1, and on a corresponding unit vector ||A b||<1 whereas U preserves norm. This is different from equality of normalized rays. More generally a deterministic trace-preserving operation has trace1 on every input density rho. The branch rho->A rho A^dagger has trace tr(rho A^dagger A), equal to1 for every density rho only if A^dagger A=I: apply the condition to every pure projector, so the Hermitian A^dagger A-I has zero quadratic form and is zero. Conversely A^dagger A=I makes this square-matrix branch unitary and trace preserving. For nonunitary A the branch is a trace-nonincreasing operation, not a deterministic normalized output channel. Its conditioned density A rho A^dagger/tr(A rho A^dagger), when the denominator is positive, is a different normalized expression and is undefined at zero denominator.

All these statements allow singular and zero A; no nonzero-target or norm-loss premise was smuggled into the original contraction domain. When A itself is unitary, S=0 and no system-only unitarity obstruction exists; the construction still supplies a valid but unnecessary dilation. Example9.3/Eq9.14 proves finite-dimensional existence, and Section9.1/Eq9.4 describes the selected branch. Neither gives an efficient decomposition of a general supplied dense A or this U_A into simple one-/two-qubit gates, an efficient SVD-access oracle, finite-bit costs or a new local Lean certificate. This expressly versioned question corrects a false universal normalized-preparation claim while preserving the exact dilation and exact-operator statements.

1. Proves M is Hermitian-unitary using S^2=I-Sigma^2 and commutativity, and concludes the full product is unitary. (1 weight)
2. Multiplies the full blocks correctly, obtains clean block A and failure component S V^dagger b, and derives selected probability ||A b||^2 and the normalized conditioned state only when positive; handles kernel/zero A and norm-one subspace K, including a nonunitary p=1 example. (1 weight)
3. Explicitly refutes universal ancilla/postselection necessity for normalized state preparation using A=cI,0<c<1 and the deterministic system identity; distinguishes its output ray from the exact unnormalized map and the chosen dilation's c^2 success probability. Does not claim generic deterministic normalized preparation. (1 weight)
4. Proves no nonunitary A equals a system-only unitary on every input, and identifies A^dagger A=I as the exact trace-preservation condition for the branch on all density inputs. Separates subnormalized/conditioned operations and states the absence of an efficient one-/two-qubit implementation guarantee. (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.

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