Geometric Rényi divergence and RLD Fisher information under quantum processes without definite causal order

Abstract

Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in $(0,1)\cup(1,2]$ and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.

The paper asks how much channel distinguishability and parameter information an experiment can extract without a definite causal order. It considers finite-dimensional processes whose action on tensor products of slot Choi operators is completely positive and produces a normalized channel or state for every independent choice of channels. This class includes fixed-order circuits and the quantum switch, but is not assumed entirely physically realizable. For geometric Rényi divergence, it proves $\widehat D_\alpha(\mathcal W(\mathcal N_1,\ldots,\mathcal N_n)\|\mathcal W(\mathcal M_1,\ldots,\mathcal M_n))\leq\sum_i\widehat D_\alpha(\mathcal N_i\|\mathcal M_i)$ for $\alpha\in(0,1)\cup(1,2]$. Optimizing over channel-to-state processes gives equality, approached by independent reference-assisted parallel probes. The proof combines operator-perspective Jensen inequalities with a normalization lemma bounding process weights of positive product operators by products of their input marginals’ extremal eigenvalues. The lower bound uses reciprocal input effects, affine extension of channel normalization, and Cauchy–Schwarz. For differentiable channel families and a parameter-independent process, it also proves $I_{\rm RLD}^{\rm out}\leq\sum_i I_{\rm RLD}^{(i)}$. Affine Choi perturbations establish this for singular channels. Consequently, locally unbiased estimators satisfy $\operatorname{MSE}\geq1/\sum_i I_{\rm RLD}^{(i)}$; estimating an erasure probability attains $p(1-p)/n$. Applications give finite-use discrimination and exclusion converses, plus communication and resource-generation bounds conditional on additional comparison-output preservation. Divergence optimization does not characterize the full finite-use discrimination region or optimal Stein threshold. RLD bounds need not be attainable and become vacuous for nontrivial unitary families. Infinite-dimensional and multiparameter extensions are left open.