Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality

• URL: http://arxiv.org/abs/2609.10072v1
• Date: Wed, 09 Sep 2026 11:53:28 GMT
• Title: Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality
• Abstract summary: We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state.<n>Performance is evaluated by quantum relative entropy.
• Score: 42.694152897125726
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $μ_E=(I+nρ_E)/(n+d)$, where $ρ_E$ 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 $h_d((n+1)/(n+d))$, where $h_d(r)=-r\log r-(1-r)\log((1-r)/(d-1))$. 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 $((n+1)^2+d-1)/(n+d)^2$.

Related papers

• Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails [47.36630149538994]
  For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)2$, where $w(A)$ is the Gaussian width of the normalized descent cone.<n>We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form.<n>Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.
  arXiv  Detail & Related papers  (2026-09-03T08:05:05Z)
• Global Minimax Readout of a Qubit Direction [5.246986428523558]
  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 $$.<n>Every fixed-local architecture, including recorded classical randomization, reduces exactly to a zero-barycenter probability measure on the Bloch ball.<n>For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid.
  arXiv  Detail & Related papers  (2026-08-17T17:26:32Z)
• Estimation of the sub-Gaussian parameter [7.4076878426925035]
  The sub-Gaussian random variable $X$ is defined as $2_* = s_upin mathbbR L()$ where $L() = frac22 log mathbbE eX$ is a weighted cumulant generating function.<n>Despite the ubiquity of sub-Gaussian random variables, the estimation of $2_*$ has received little attention and is not yet well understood.
  arXiv  Detail & Related papers  (2026-06-04T16:48:31Z)
• Sufficient support size of measurements for quantum estimation [0.0]
  We prove that it suffices to consider POVMs with at most $left(rm dim, mathcalHright)2+d(d+1)/2-1$ outcomes, and that an optimal measurement can be chosen to be rank-one.
  arXiv  Detail & Related papers  (2026-04-23T06:26:07Z)
• Optimal Unconstrained Self-Distillation in Ridge Regression: Strict Improvements, Precise Asymptotics, and One-Shot Tuning [61.07540493350384]
  Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth and the teacher's own predictions.<n>We show that for any prediction risk, the optimally mixed student improves upon the ridge teacher for every regularization level.<n>We propose a consistent one-shot tuning method to estimate $star$ without grid search, sample splitting, or refitting.
  arXiv  Detail & Related papers  (2026-02-19T17:21:15Z)
• Further Understanding of a Local Gaussian Process Approximation: Characterising Convergence in the Finite Regime [1.3518297878940662]
  We show that common choices of kernel functions for a highly accurate and massively scalable GPnn regression model exhibit gradual convergence to behaviour as dataset-size $n$ increases. Similar bounds can be found under model misspecification and combined to give overall rates of convergence of both MSE and an important calibration metric.
  arXiv  Detail & Related papers  (2024-04-09T10:47:01Z)
• On the $O(\frac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm [54.28350823319057]
  This paper considers the RMSProp and its momentum extension and establishes the convergence rate of $frac1Tsum_k=1T.<n>Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$.<n>Our convergence rate can be considered to be analogous to the $frac1Tsum_k=1T.
  arXiv  Detail & Related papers  (2024-02-01T07:21:32Z)
• Near Sample-Optimal Reduction-based Policy Learning for Average Reward MDP [58.13930707612128]
  This work considers the sample complexity of obtaining an $varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP) We prove an upper bound of $widetilde O(H varepsilon-3 ln frac1delta)$ samples per state-action pair, where $H := sp(h*)$ is the span of bias of any optimal policy, $varepsilon$ is the accuracy and $delta$ is the failure probability.
  arXiv  Detail & Related papers  (2022-12-01T15:57:58Z)
• Phase estimation with limited coherence [0.0]
  For pure states, we give the minimum estimation variance attainable, $V(C)$, and the optimal state, in the limit when the probe system size, $n$, is large. We show that the variance exhibits a Heisenberg-like scaling, $V(C) sim a_n/C2$, where $a_n$ decreases to $pi2/3$ as $n$ increases, leading to a dimension-independent relation.
  arXiv  Detail & Related papers  (2022-07-12T16:36:59Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.