Exercise PLAN-415

Permutation twirling preserves each product-state distribution; unitary twirling instead averages risks over a unitary orbit and preserves uniform spectrum guarantees. Both symmetries force block scalars, hence weak Schur measurement plus classical postprocessing.

1. Permutation averaging fixes all probabilities on rho^{tensor n}, since the state is invariant under every factor permutation.
2. Unitary averaging preserves uniform/minimax spectrum-risk bounds via the orbit-average identity, but not every fixed state's probabilities; the qubit computational-basis POVM is a counterexample.
3. The complete positive Schur-Weyl projectors commute with both groups; their joint commutant consists of block scalars, so every symmetrized POVM is their classical postprocessing, including the zero-copy and one-dimensional boundaries.

Let rho_n=rho^{tensor n}. For a measurable event B, define E_pi(B)=(1/n!)sum_pi Q_pi E(B) Q_pi^dagger. Positivity, countable additivity and E_pi(Omega)=I follow from those of E. Since Q_pi^dagger rho_n Q_pi=rho_n, cyclicity gives tr[E_pi(B)rho_n]=tr[E(B)rho_n] for EVERY B and every rho. This preserves the whole distribution, hence all the defined risks, not just their supremum.

Define F(B)=integral dU U_n E_pi(B) U_n^dagger using normalized Haar on compact U(d). Finite-dimensional entrywise integration preserves positivity and normalization. Countable additivity follows entrywise from the positive scalar measures (or dominated convergence), so this is a POVM. Permutations commute with U_n and E_pi is permutation invariant, so F commutes with Q_pi; left Haar translation proves V_n F(B)V_n^dagger=F(B).

Cyclicity and the tensor product identity yield tr[F(B)rho_n]=integral dU tr[E_pi(B)(U^dagger rho U)^{tensor n}]. Bounded nonnegative measurable loss permits the corresponding interchange of integrals. Since r(U^dagger rho U)=r(rho), R_rho(F)=integral dU R_{U^dagger rho U}(E_pi) =integral dU R_{U^dagger rho U}(E) <=R_max(E). Taking sup_rho proves R_max(F)<=R_max(E). This is a uniform/minimax statement, not equality of each state's risk. In particular take ell as the indicator that an estimate has error larger than epsilon in any specified spectrum distance: if the original failure probability is <=delta for ALL rho, the averaged procedure retains that uniform bound. No assertion that a minimizer exists is needed.

For the pointwise obstruction, n=1,d=2 and effects E_0=|0><0|, E_1=I-E_0 are a valid POVM. For rho=diag(4/5,1/5), Pr_E(0)=4/5; for its basis-swapped conjugate it is 1/5, despite the same spectrum. The Haar twirl of E_0 is I/2 and has probability 1/2 on both states. Assigning two fixed spectrum estimates to the outcomes does not remove this distributional difference. Thus unitary invariance of the parameter does not, by itself, preserve a particular procedure's pointwise statistics. It licenses the uniform-risk averaging argument just proved.

Use the unitary Schur-Weyl decomposition (C^d)^{tensor n}=direct_sum_lambda V_lambda tensor U_lambda^d, over nonzero sectors. The S_n and U(d) actions on their respective irreducible factors span mutual commutants. An operator commuting with permutations is blockwise I_{V_lambda} tensor A_lambda; commuting also with all U_n forces A_lambda=a_lambda I by Schur. Therefore F(B)=sum_lambda a_lambda(B)P_lambda. Positivity gives a_lambda(B)>=0, and F(Omega)=I gives a_lambda(Omega)=1; countable additivity is inherited, for example from a_lambda(B)=tr[P_lambda F(B)]/rank(P_lambda). Hence each a_lambda is a probability measure. Measuring the orthogonal projectors P_lambda and then drawing y from a_lambda has event probability sum_lambda a_lambda(B)tr[P_lambda rho_n], exactly that of F on every input. The P_lambda are orthogonal positive projectors, sum to I, and commute with both actions, so constitute a legitimate projective measurement. This simulation applies to the SYMMETRIZED procedure; it does not simulate arbitrary unsymmetrized orientation-sensitive statistics. The joint decomposition is multiplicity free as an S_n times U(d) representation, not scalarity from unitary commutation alone.

For n=0 the Hilbert space is C, S_0 acts trivially, and there is one empty-diagram projector I. Every effect is scalar, the measurement contains no information about rho, and the proof remains valid; lambda/n is undefined and is not used. For d=1 the spectrum is always (1) and the only Schur-Weyl sector is the symmetric one. A zero-dimensional single-copy space has no positive trace-one operator at all, even though its zero-copy tensor unit is C; thus there is no unknown rho on that carrier and no minimax inference over such states. The reduction is an exact finite-dimensional mathematical construction, not a gate-count bound, constructive Haar sampler or an exact-atom consistency theorem.

1. Proves permutation averaging is a normalized POVM and preserves tr[E(B)rho^tensor n] for every measurable B and fixed rho by permutation invariance of rho^tensor n. (1 weight)
2. Defines the unitary average, checks positivity/normalization/additivity and both commuting symmetries, derives the orbit-averaged risk identity and the minimax inequality using spectrum-invariant loss. Explicitly qualifies uniform error guarantees and refutes pointwise-probability preservation with an actual POVM/state example; no unbound WLOG premise or exact-atom consistency claim. (1 weight)
3. Uses both mutual commutants, not unitary commutation alone, to obtain F(B)=sum a_lambda(B)P_lambda; proves that the coefficients are probability measures and gives weak-Schur plus classical postprocessing that simulates F. (1 weight)
4. Checks projectors are positive, orthogonal, complete and commute with both group actions. Handles n=0, d=1 and absence of a d=0 density operator, without division by n or an efficient implementation assertion. (1 weight)

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