Bosonic quantum entropy equality holds only for identical gaussian inputs
Equality in the Bosonic Quantum Entropy Power Inequality
Information Theory
Summary
The bosonic quantum entropy power inequality relates the disorder (entropy) of light beams before and after they pass through a beam splitter. This paper shows that the inequality becomes an equality only when the incoming light beams are of a very special kind: Gaussian states with matching properties, except for their positions. The authors prove this by linking equality to the independence of output beams and applying advanced quantum information theory results. The work fully characterizes when this equality holds for any number of light modes and mixing ratios.
What this means in practice
- •For quantum communication engineers: Determine when beam splitter outputs preserve information optimally by recognizing input states that saturate entropy power inequalities.
- •For quantum optics experimentalists: Design experiments to generate or verify Gaussian input states by measuring entropy equality in beam splitter outputs.
A theory result. No direct application yet.
Authors
Yonglong Li
Abstract
The bosonic quantum entropy power inequality bounds the entropy of a beam-splitter output in terms of the input entropies. We determine its complete equality class among independent inputs of finite mean photon number: for any number of modes and any transmissivity strictly between zero and one, equality in either the linear or the exponential form holds if and only if the inputs are Gaussian with the same covariance matrix, allowing arbitrary displacements. The main step shows that equality for one output forces the two outputs to be independent: a thermal auxiliary converts equality into preservation of mutual information by a fixed noisy channel, and a two-outcome instrument shows that preservation of mutual information requires a product state with the reference. The quantum Darmois--Skitovich theorem then gives Gaussianity, while a central-limit argument reduces exponential equality to linear equality.