In the quantum state tomography, which is the task of estimating the unknown quantum state on a Hilbert space from measurement data, the standard Bayesian technique is often used.As is well known in Bayesian statistics, it is a nontrivial problem to define a noninformative prior in gen- eral statistical models.Although some definitions for statistical models of mixed states are given by straightforward extension of classical results, there is no counterpart for those of pure states.Recently the author proposed a conceptual extension of the famous reference prior by Bemardo and defined noninformative priors for statistical models of pure states.While the formal extension of the reference prior corresponds to the prior maximizing the von Neumann entropy, our proposed prior is shown to maximize the limit of the quantum Renyi entropy.We show some relations between two principles.
In the present paper, we consider a suitable definition of a noninformative prior on the quantum statistical model of pure states. While the full pure-states model is invariant under unitary rotation and admits the Haar measure, restricted models, which we often see in quantum channel estimation and quantum process tomography, have less symmetry and no compelling rationale for any choice. We adopt a game-theoretic approach that is applicable to classical Bayesian statistics and yields a noninformative prior for a general class of probability distributions. We define the quantum detection game and show that there exist noninformative priors for a general class of a pure-states model. Theoretically, it gives one of the ways that we represent ignorance on the given quantum system with partial information. Practically, our method proposes a default distribution on the model in order to use the Bayesian technique in the quantum-state tomography with a small sample.
The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (σ_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_α). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},σ_{1}^2,α) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < α< \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featured. Our model can be contrasted with the three-parameter (b_{q}, σ_{q}^2,q) one of Abe and Rajagopal, which employs a q(Tsallis)-parameter rather than
α, and has simply q-invariant HS separability probabilities of 1/2. Our results emerge in a study initially focused on embedding certain information metrics over the two-level quantum systems into a q-framework. We find evidence that Srednicki's recently-stated biasedness criterion for noninformative priors yields rankings of priors fully consistent with an information-theoretic test of Clarke, previously applied to quantum systems by Slater.