1 · The essentials
When two independent light inputs are mixed by a beam splitter, how much uncertainty must remain in the output? This work answers that question using a quantity called the entropy photon number, which expresses uncertainty on a photon-number scale. The result says that the output’s value is at least the weighted average of the two input values, with weights set by how the beam splitter mixes them. It applies to any finite number of light modes, provided each input has finite average energy. The two inputs must be independent, but the modes within either input may be entangled with one another. Product thermal inputs reach the bound exactly, even when the two inputs have different amounts of uncertainty. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine mixing two ingredients whose labels give a measure of uncertainty rather than weight or volume. The result supplies a minimum uncertainty score for the mixture, based on the mixing proportions. This is only an image: the actual score is entropy photon number, and the ingredients are independent bosonic inputs.
Why it matters
The manuscript describes the result as resolving the entropy photon-number conjecture in the finite-energy setting. It also reports two consequences: an exact minimum output entropy for repeated identical thermal attenuators and an exact classical capacity region for a specified two-receiver bosonic broadcast channel. These connect the mixing inequality to limits on output uncertainty and communication.
2 · Going further
The precise problem concerns beam-splitter mixing of two independent bosonic inputs, each with finite mean energy, in any finite number of modes. The entropy photon-number inequality places a lower bound on the output’s entropy photon number: it cannot be smaller than the transmissivity-weighted average of the corresponding input quantities. Independence is required between the inputs, not between the modes inside each input; arbitrary entanglement within either input is allowed.
The repository summary identifies product thermal inputs as equality cases, including when their entropies differ. Thus, the lower bound can be attained and is not restricted to equal-entropy inputs. The abstract describes the result as resolving the conjecture under the finite-energy hypothesis. It does not describe the argument or provide intermediate steps, so no proof mechanism can be extracted from the supplied text.
As consequences, the manuscript reports the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, allowing finite-energy inputs entangled across modes. It also reports the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint. Only the main manuscript is supplied here; no companion manuscripts or their relationship to this result are described.
Useful notions
- Entropy as a measure of uncertainty
- Bosonic modes and photon number
- Beam-splitter mixing and transmissivity
- Independence versus entanglement
- Thermal inputs and thermal attenuators
What was known before / what is new
Before
The abstract identifies the entropy photon-number statement as a conjecture. The supplied text does not give its history, earlier partial results, or previously available bounds.
New
The main result establishes the inequality for independent finite-energy inputs in every finite number of modes, while allowing arbitrary entanglement within each input. The repository summary states that product thermal inputs attain equality even at unequal input entropies. The manuscript also reports exact results for the thermal-attenuator minimum output entropy problem and the specified bosonic broadcast-channel capacity region.
Limits and assumptions
The inequality assumes independent inputs, finite mean energy, and a finite number of modes; entanglement is allowed within either input. The thermal-attenuator consequence concerns finite tensor powers of an identical attenuator and finite-energy inputs at fixed input entropy. The broadcast-channel consequence concerns a degraded two-receiver pure-loss channel under a mean photon constraint. The main result has a Lean formalization, but the supplied material does not specify its detailed scope or establish that it covers the reported consequences or equality cases. No remaining open problems are explicitly listed, and settings outside these hypotheses are not addressed.
Mini glossary
- Entropy photon number —
- The entropy-based quantity on a photon-number scale whose input and output values are compared by the inequality.
- Transmissivity —
- The beam-splitter mixing parameter that sets the weights in the inequality.
- Finite mean energy —
- The requirement that each input’s average energy be finite.
- Entanglement across modes —
- Quantum dependence among modes, which the result permits within either input while requiring the two inputs to be independent.
- Equality case —
- An allowed choice of inputs for which the output reaches the stated lower bound exactly.
Keywords : Entropy photon-number inequality · Beam-splitter mixing · Bosonic channels
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The entropy photon-number inequality
24/09/2026We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.