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Phys-55 · physics · mathematics · Physics

The entropy photon-number inequality for arbitrary finite-mode inputs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For independent finite-energy n-mode bosonic inputs $\rho _A$ and $\rho _B$, any finite $n\geq 1$ and $0\leq \eta \leq 1$, let $\rho _C$ be the n output modes of passive beam-splitter mixing $c_j=\operatorname{sqrt}(\eta )a_j+\operatorname{sqrt}(1-\eta )b_j$. Is $g^{-1}(S(\rho _C)/n)\geq \eta $ $g^{-1}(S(\rho _A)/n)+(1-\eta )g^{-1}(S(\rho _B)/n)$, where $g(x)=(x+1)\log (x+1)-x$ log x? Internal entanglement between modes within either input is permitted.

Short English statement

For independent finite-energy n-mode bosonic inputs $\rho _A$ and $\rho _B$, any finite $n\geq 1$ and $0\leq \eta \leq 1$, let $\rho _C$ be the n output modes of passive beam-splitter mixing $c_j=\operatorname{sqrt}(\eta )a_j+\operatorname{sqrt}(1-\eta )b_j$. Is $g^{-1}(S(\rho _C)/n)\geq \eta $ $g^{-1}(S(\rho _A)/n)+(1-\eta )g^{-1}(S(\rho _B)/n)$, where $g(x)=(x+1)\log (x+1)-x$ log x? Internal entanglement between modes within either input is permitted.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-55-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-55

Independent finite-energy bosonic inputs are claimed to satisfy the entropy photon-number inequality in any finite number of modes, allowing internal multimode entanglement. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The entropy photon-number inequality for arbitrary finite-mode inputs

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Independent finite-energy bosonic inputs are claimed to satisfy the entropy photon-number inequality in any finite number of modes, allowing internal multimode entanglement.

Mathematical statement of the claimed result

Result 1.
For finite $n\ge1$, take independent input states $\rho_A\otimes\rho_B$ on two collections of $n$ bosonic modes, each with finite total mean photon number. Internal correlations among the modes of either input are allowed. Let $\rho_C$ be the output on the modes $c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j$, $0\le\eta\le1$. Define $g(t)=(t+1)\log(t+1)-t\log t$ and von Neumann entropy $S$ in nats. The asserted inequality is \[ g^{-1}(S(\rho_C)/n)\ge \eta g^{-1}(S(\rho_A)/n)+(1-\eta)g^{-1}(S(\rho_B)/n). \]

Outline of the argument

Diagonal form comparisons are interpolated through a regularized entropy minimum. Its Hessian is controlled in a metric adapted to thermal relaxation, forcing the thermal extremizer and hence the inequality; fixed-entropy output and degraded broadcast-capacity consequences follow.

Formal verification

For family 273, the documented statement is the independent-input finite-energy multimode photon-number entropy inequality. Thermal-attenuator and broadcast-capacity consequences are not selected. The listed Comparator files are EntropyPhotonNumber.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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