we-solve (beta)

Phys-43 · physics · mathematics · Physics

Single-letter classical capacity of generalized amplitude damping

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every generalized amplitude-damping qubit channel $A_{\gamma ,\nu }$, $\gamma ,\nu \in [0,1]$, is its n-copy Holevo information exactly n times its one-copy value for every $n\geq 1$? Require that the one-copy optimizer is a phase-opposite equiprobable pair $\operatorname{sqrt}(1-p)|0>\pm \operatorname{sqrt}(p)|1>$, so its optimized output-entropy difference gives the unassisted classical capacity.

Short English statement

For every generalized amplitude-damping qubit channel $A_{\gamma ,\nu }$, $\gamma ,\nu \in [0,1]$, is its n-copy Holevo information exactly n times its one-copy value for every $n\geq 1$? Require that the one-copy optimizer is a phase-opposite equiprobable pair $\operatorname{sqrt}(1-p)|0>\pm \operatorname{sqrt}(p)|1>$, so its optimized output-entropy difference gives the unassisted classical capacity.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-43-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-43

Every generalized amplitude-damping qubit channel is claimed to have an explicit one-variable unassisted capacity formula achieved by binary pure states, with the listed universal partner-channel additivities. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Single-letter classical capacity of generalized amplitude damping

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Every generalized amplitude-damping qubit channel is claimed to have an explicit one-variable unassisted capacity formula achieved by binary pure states, with the listed universal partner-channel additivities.

Mathematical statement of the claimed result

Result 1.
For $\gamma,\nu\in[0,1]$, define the qubit channel \[ \mathcal A_{\gamma,\nu}\begin{pmatrix}1-q&z\\\overline z&q\end{pmatrix} =\begin{pmatrix}1-t&\sqrt{1-\gamma}z\\\sqrt{1-\gamma}\overline z&t\end{pmatrix}, \quad t=(1-\gamma)q+\gamma\nu. \] Put $h(t)=-t\log t-(1-t)\log(1-t)$, $g(u)=h((1+\sqrt{1-4u})/2)$, and $v_{\gamma,\nu}(p)=\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2$. The claimed Holevo-information formula in bits is \[ \chi(\mathcal A_{\gamma,\nu}^{\otimes n}) =n\chi(\mathcal A_{\gamma,\nu}) =\frac n{\log2}\max_{0\le p\le1} [h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))] \] for every $n\ge1$, so its unassisted classical capacity is single-letter. Independent products of the two equiprobable states $\sqrt{1-p}|0\rangle\pm\sqrt p|1\rangle$, at a maximizing $p$, attain this information.

Outline of the argument

A block-entropy inequality is reduced to log-determinant comparisons and contraction convexity. This controls correlated channel inputs, proves the one-use binary optimum, and extends the entropy and Holevo bounds to arbitrary finite-dimensional tensor partners.

Formal verification

For family 276, the documented theorem gives the attained scalar capacity formula and repeated-use additivity for the same amplitude-damping channel. Additivity with arbitrary different partner channels is beyond this entry. The listed Comparator files are AmplitudeDamping.lean. Named declarations include OAI.GAD.main. 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

Go to contribution →