Sample Complexity of Quantum Mutual Information for Pure States

Abstract

We establish a fundamental limit on the number of experimental repetitions (samples) required to estimate the quantum mutual information I(A:B)I(A:B)—a key measure of total correlations—when the quantum system is in a pure bipartite state. By leveraging the Schmidt decomposition, we rely on the exact relation I(A:B)=2S(ρA)I(A:B)=2S(ρA), which connects this correlation measure to the local von Neumann entropy S(ρA)S(ρA). This allows us to directly import a recently proven lower bound by Wang for entropy estimation. Concretely, we show that achieving an estimate of I(A:B)I(A:B) with a prescribed accuracy ϵϵ demands a number of samples that scales as Ω~(d2/ϵ)Ω~(d2/ϵ)—where dd denotes the effective dimension of each subsystem and the Ω~Ω~ notation absorbs logarithmic corrections. This conclusion follows rigorously from Wang's results without introducing extra hypotheses. In addition, we briefly explore the implications for emergent spacetime models, pointing out that transferring this sample-complexity barrier to geometric reconstruction requires an extra condition of stability—namely, that small changes in geometry must translate into small changes in mutual information. Without such stability, the bound does not automatically apply to the geometric sector.