Fundamental Limits of Joint Target Detection and Parameter Estimation - Characterizing Mixed-State Sensing Limits via Posterior Entropy Volume
The development of integrated sensing and communication calls for a unified theoretical foundation for sensing. This paper models target-presence patterns and continuous physical parameters as a mixed discrete-continuous state $Ξ$ on a branched reference measure, and treats posterior entropy volume and joint mutual information as two complementary representations of the same limit. Entropy volume carries physical units, can be compared with engineering scales such as resolution cells, and remains meaningful when the number of active targets varies; mutual information is dimensionless, invariant to coordinates and units, and additive through the chain rule. We define entropy number, entropy volume, and mixed entropy volume for discrete, continuous, and mixed states, respectively. The maximum-entropy principle for the uniform distribution on a support of fixed measure explains the measure-theoretic meaning of the exponential entropy scale, while the main limit follows from a mixed asymptotic equipartition property and a posterior probability-volume inequality through conditional typical sets. For asymptotically reliable high-probability sensing regions, the minimum achievable first-order posterior mixed entropy volume equals the prior mixed entropy volume multiplied by $2^{-I(Ξ;Y)}$, where $I(Ξ;Y)=I(V;Y)+I(X_V;Y\mid V)$. Thus, detection and estimation contributions multiply in the volume domain and add in the bit domain, and one sensing bit halves the posterior effective measure. We further prove that posterior-preserving cascades attain the direct-inference limit, while arbitrary intermediate compression incurs the exact information loss $I(Ξ;Y\mid Z)$. Numerical results for a single-target presence-range model illustrate the information composition, posterior entropy-volume contraction, and cascade-interface loss.