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arXiv:2609.10072

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

Masahito Hayashi, Ayanava Dasgupta, Naqueeb Ahmad Warsi

quant-phcs.ITSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
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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 $((n+1)^2+d-1)/(n+d)^2$.

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We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on nnn 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 EEE, the corresponding posterior predictive state is μE=(I+nρE)/(n+d)μ_E=(I+nρ_E)/(n+d)μE​=(I+nρE​)/(n+d), where ρEρ_EρE​ is the normalized one-particle marginal of EEE. 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))h_d((n+1)/(n+d))hd​((n+1)/(n+d)), where hd(r)=−rlog⁡r−(1−r)log⁡((1−r)/(d−1))h_d(r)=-r\log r-(1-r)\log((1-r)/(d-1))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((n+1)^2+d-1)/(n+d)^2((n+1)2+d−1)/(n+d)2.
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An analytical solution for the posterior estimate in Bayesian tomography of the unknown quantum state of an arbitrary quantum system (with a finite-dimensional Hilbert space) is found. First, we derive the Bayesian estimate for a pure quantum state measured by a set of arbitrary rank-1 POVMs under the uninformative (i.e. the unitary invariant or Haar) prior. The expression for the estimate involves the matrix permanents of the Gram matrices with repeated rows and columns, with the matrix elements being the scalar products of vectors giving the measurement outcomes. Second, an unknown mixed state is treated by the Hilbert-Schmidt purification. In this case, under the uninformative prior for the combined pure state, the posterior estimate of the mixed state of the system is expressed through the matrix ααα-permanents of the Gram matrices of scalar products of vectors giving the measurement outcomes. In the mixed case, there is also a free integer parameter -- the Schmidt number -- which can be used to optimise the Bayesian reconstruction (for instance, in case of Schimdt number being equal to 1, the mixed state estimates reduces to the pure state estimate). We also discuss the perspectives of approximate numerical computation and asymptotic analytical evaluation of the Bayesian estimate using the derived formula.
We study estimation of the von Neumann entropy of a ddd-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let κν∈[1,d+1]κ_ν\in[1,d+1]κν​∈[1,d+1] denote the normalized trace-zero frame spread; κν=1κ_ν=1κν​=1 for exact complex projective 222-designs. For any fixed rank-one measurement and n≤c2d2/κνn\le c_2d^2/κ_νn≤c2​d2/κν​, the minimax mean-squared error is at least c1log⁡2{d2/(κνn)}c_1\log^2\{d^2/(κ_νn)\}c1​log2{d2/(κν​n)}. For bounded frame spread and every fixed 0<a<20<a<20<a<2, the minimax risk is of order log⁡2d\log^2 dlog2d throughout n≲d2−an\lesssim d^{2-a}n≲d2−a; hence fixed uniform accuracy requires Ω(d2)Ω(d^2)Ω(d2) observations. Existing tomography bounds give bounded risk from O(d3log⁡3d)O(d^3\log^3 d)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.
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Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.