We study prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after independently measuring the observed copies with the one-copy Haar POVM. For the resulting fixed separable observation, the Bayes predictive state under quantum relative-entropy loss is the full-rank posterior mean. We evaluate it exactly through permanents of minors of the outcome Gram matrix, prove positivity and normalization of the permanent representation, and establish minimaxity among decision rules based on these fixed outcomes. Using Hilbert--Schmidt projection of the posterior mean and an exactly analyzed linear-inversion competitor, we prove a finite-sample posterior-purity bound and an explicit relative-entropy guarantee with sample count $O((d^2/ε)\log(d/ε))$. Using the exact collective benchmarks proved in a companion paper, we also obtain entropy and posterior-purity comparisons. To connect the fine product data to a regular ray-valued experiment without assuming deterministic finite-sample uniqueness, we draw one random orientation for a finite projective mesh, retain it for the entire sample, and apply a selected finite-alphabet MLE in the resulting reference experiment. A nested recovering mesh uniformly recovers the fine Fisher information. Consequently every sufficiently fine fixed mesh gives a regular finite-alphabet model. Its selected MLE has Haar-averaged inverse-Fisher coefficient $\overline a_k$, and refining the mesh after the fixed-mesh large-sample limit makes $\overline a_k$ approach $d-1$. This yields epsilon-optimal leading relative-entropy and posterior-overlap bounds for the fine posterior and the sharper fixed-dimensional asymptotic scale $(d/ε)\log(d/ε)$. The latter is not asserted as a uniform finite-sample guarantee. No deterministic finite-sample MLE uniqueness or triangular mesh schedule is used.
We study prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after independently measuring the observed copies with the one-copy Haar POVM. For the resulting fixed separable observation, the Bayes predictive state under quantum relative-entropy loss is the full-rank posterior mean. We evaluate it exactly through permanents of minors of the outcome Gram matrix, prove positivity and normalization of the permanent representation, and establish minimaxity among decision rules based on these fixed outcomes. Using Hilbert--Schmidt projection of the posterior mean and an exactly analyzed linear-inversion competitor, we prove a finite-sample posterior-purity bound and an explicit relative-entropy guarantee with sample count O((d2/ε)log(d/ε)). Using the exact collective benchmarks proved in a companion paper, we also obtain entropy and posterior-purity comparisons. To connect the fine product data to a regular ray-valued experiment without assuming deterministic finite-sample uniqueness, we draw one random orientation for a finite projective mesh, retain it for the entire sample, and apply a selected finite-alphabet MLE in the resulting reference experiment. A nested recovering mesh uniformly recovers the fine Fisher information. Consequently every sufficiently fine fixed mesh gives a regular finite-alphabet model. Its selected MLE has Haar-averaged inverse-Fisher coefficient ak, and refining the mesh after the fixed-mesh large-sample limit makes ak approach d−1. This yields epsilon-optimal leading relative-entropy and posterior-overlap bounds for the fine posterior and the sharper fixed-dimensional asymptotic scale (d/ε)log(d/ε). The latter is not asserted as a uniform finite-sample guarantee. No deterministic finite-sample MLE uniqueness or triangular mesh schedule is used.
We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on n observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect E, the corresponding posterior predictive state is μE=(I+nρE)/(n+d), where ρE is the normalized one-particle marginal of E. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is hd((n+1)/(n+d)), where hd(r)=−rlogr−(1−r)log((1−r)/(d−1)). The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum ((n+1)2+d−1)/(n+d)2.
We study estimation of the von Neumann entropy of a d-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let κν∈[1,d+1] denote the normalized trace-zero frame spread; κν=1 for exact complex projective 2-designs. For any fixed rank-one measurement and n≤c2d2/κν, the minimax mean-squared error is at least c1log2{d2/(κνn)}. For bounded frame spread and every fixed 0<a<2, the minimax risk is of order log2d throughout n≲d2−a; hence fixed uniform accuracy requires Ω(d2) observations. Existing tomography bounds give bounded risk from O(d3log3d) observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length η. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For N copies, inverse-Fisher A loss has the exact value 2/[Nf(η)], where $
f(η)=
\frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely A- and D-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have A-loss coefficient 4/η2, whereas collective measurements attain 2(1+η)/η2. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.
We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least (d/n)log2{n/(4d)} when d≥C and n≥Cd, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius r around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by ε<1 from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most d3r2/n+d4/n2+d6r6. Lower bounds under the same frame control yield the local minimax rate d3r2/n+d4/n2 when n≥Cd2 and r lies in an explicit matching range. For every fixed upper bound on ε, approximation changes only the constants, not the powers of d,n,r. At the critical radius r=n−1/2, the local risk is asymptotically negligible relative to the global risk when nlog2(n/d)≫d3. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.
Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues d∣χφ(u)∣2, turning stability into an explicit worst-direction design problem; write λ for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while E[λ−1]=∞, and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly d2 outcomes in every integer dimension has floors Θ(d−3) for odd d and Θ(d−5) for even d; a finite-field family for q=2m obeys the uniform bound λ≥4/9. Our main result treats every prime-power dimension of characteristic p≥5. A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to [Lq,Uq] with Uq/Lq→1. Its SIC-normalized minimum tends to one, and λ(φq)/Λq⋆→1 for the global finite-field WH max--min optimum Λq⋆, without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at I/d, while its lower edge controls local Fisher efficiency and canonical-shadow bounds.
We study when conjugation by a product Haar unitary on (ℂ^d)^{⊗D} reproduces the scalar mixed-moment behavior of a Haar unitary on the full space as d→∞, with D fixed. On the fully traceless sector—matrices whose one-leg partial traces vanish—we prove sharp global-Haar-scale cancellation: for N=d^D, fixed alternating words have bias O(N^-2) and fixed-L^q fluctuations O(N^-1). If this partial-trace condition is asymptotically stable under centered polynomial evaluation, it yields quantitative scalar asymptotic freeness. For liberation from every bounded deterministic test family, the corresponding partial-trace defect is also necessary at the first nontrivial order and can be reconstructed from D fixed probes, giving a local-unitary second-moment diagnostic related to randomized-measurement methods. For arbitrary bounded deterministic coefficients, we derive a self-contained fixed-order expansion in which finitely many normalized permutation-tuple contractions determine every product-Haar mixed moment up to O(d^-1). Using the exact multi-leg contraction exponent of Collins–Yuan, currently available as a preprint, we obtain a finite necessary-and-sufficient criterion for matching global-Haar mixed moments at every fixed (D,k), and hence an all-orders characterization. In the bipartite k=4 case exactly four tensor-transport contractions survive: the traceless SU(d) exchange interaction activates the two inverse-cycle modes. For Bell-diagonal generators the same modes become explicit ordinary and symplectically twisted autocorrelation energies: centered random sign masks satisfy all four asymptotically almost surely, whereas a classical circulant two-level mask has an exact surviving pairing coordinate 1/3+8/(3d^2). Finally, exact vanishing of the polynomial partial-trace defect is equivalent to local complementarity, yielding a d^2 algebra-dimension bound and extremal connections with distance-two quantum MDS codes and absolutely maximally entangled states.