Existence conditions for alternative Bayesian inverses

Determine conditions, such as conditions formulated in terms of the modular group, under which Bayesian inverses defined using alternatives to the GNS inner product exist, including the Bayesian inverse associated with the inner product proposed in Ref. [PaFu24TSC].

Background

The paper notes that existence conditions for GNS Bayesian inverses are well known, whereas the existence theory for other quantum generalizations of Bayesian inversion is less developed. One example is the Bayesian inverse defined as an adjoint with respect to the symmetrized inner product $$\langle\!\langle a_1,a_2\rangle\!\rangle_\omega^{\mathrm{J}}=\frac{1}{2}\omega(a_1^*a_2+a_2a_1^*)$$.

The unresolved issue is to identify structural conditions—particularly conditions involving modular groups—that guarantee existence for such alternative Bayesian inverses. The authors connect this problem to distinguishing correlation from causation in measurements of Pauli observables for multiqubit systems.