A certified lower bound on the quantum-capacity threshold of the depolarizing channel

Based on the provided text, here is a detailed technical summary of the paper "A certified lower bound on the quantum-capacity threshold of the depolarizing channel."

Problem Statement

The paper addresses the long-standing open problem of determining the noise threshold below which a qubit depolarizing channel retains positive quantum capacity. Specifically, it targets the qubit depolarizing channel defined as:
$$D_p(\rho) = (1-3p)\rho + p(X\rho X + Y\rho Y + Z\rho Z)$$
where $p$ is the per-Pauli error probability (total error rate $3p$).

For thirty years, the known lower bounds on this threshold have relied on floating-point evaluations of the coherent information ($I_c$) for specific input states. These numerical methods are insufficient for the regime where the coherent information margin is extremely small (order of $10^{-7}$ bits), as standard double-precision arithmetic cannot definitively distinguish positive capacity from zero. The paper seeks to provide the first proven, certified positivity result in this regime, eliminating reliance on floating-point approximations.

Methodology

The author employs a rigorous, computer-assisted proof strategy that reduces the verification of coherent information positivity to finite lists of big-integer comparisons. The methodology is structured into three "tiers" of verification:

1. State Representation (Tier 1):

   • The paper utilizes two specific 45-qubit input states: State R (the strongest public state from Agarwal et al. [6]) and State A (a new rank-two state discovered by the author).
   • Both states are encoded as 1,472-byte payloads containing little-endian complex128 coefficients. Crucially, the proof treats these bytes as exact dyadic rationals, not floating-point approximations.

2. Monotonicity and Semigroup Properties (Tier 2):

   • The author leverages the fact that the depolarizing channel family forms a semigroup under composition ($D_q \circ D_p = D_{p+q-4pq}$).
   • Using the data processing inequality, they prove that for a fixed input state, the coherent information $I_c(\rho, D_p^{\otimes n})$ is non-increasing on the interval $p \in [0, 1/4]$.
   • This allows a single certified positive point to establish positivity for the entire interval below it.

3. Certified Computation (Tier 3):

   • Block Decomposition: Exploiting the permutation symmetry of the input states (Dicke basis), the $2^{45}$-dimensional Hilbert space computation is collapsed into a sum of smaller blocks (up to 92-dimensional).
   • Exact Arithmetic: The coherent information is computed as a sum of entropies of these blocks. The author does not use eigensolvers. Instead, they use "untrusted hints" (a candidate eigenbasis and a center matrix) derived from floating-point calculations.
   • Fencing: These hints are rigorously "fenced" using three self-contained lemmas (Appendix A):
     • Inflated Gershgorin: To bound eigenvalues of the exact matrix based on the hint matrix.
     • Polar-Factor Perturbation: To bound the distance between the candidate basis and the true unitary.
     • Entropy Enclosure: To bound the von Neumann entropy using directed rational arithmetic for logarithms.
   • Verifier: A dependency-free Python verifier (using only int and Fraction) checks the certificates. It recomputes block matrices from the raw bytes, validates the hints against integer inequalities, and confirms the final sign of the coherent information.

Key Contributions

1. Certified Positivity Point: The paper establishes a certified lower bound on the depolarizing noise threshold at $p = 16239/250000 = 0.064956$ (total error rate $3p \approx 0.194868$). This exceeds the previous best numerical record of $0.064657$ by Agarwal et al. [6].
2. First Proven Ordering: The author proves that the positivity boundary of their new state A is strictly separated from the boundary of the previous best state R by more than $1/25000$. This is the first proven ordering of two competing code states in this regime.
3. Verification Framework: The paper introduces a framework where complex quantum information claims are reduced to big-integer comparisons. The entire proof chain is checkable by a short, dependency-free verifier without trusting the author, their code, or floating-point arithmetic.
4. Supplementary Artifact: The paper is accompanied by a public artifact containing the state payloads, four certificates (for the endpoints of the intervals), and the verifier code, allowing independent reproduction of the results.

Results

• State A: Certified to have positive coherent information at $p = 16239/250000$.
• State R: Certified to have negative coherent information at $p = 16229/250000$ (and positive at $4057/62500$).
• Separation: The zero-crossing (threshold) for State A is proven to be strictly greater than that of State R.
• Quantitative Bounds: The certified coherent information margin for State A at the threshold is approximately $8.4 \times 10^{-8}$ bits, a value far too small for standard numerical solvers to resolve with certainty.

Significance and Claims

The paper is modest in its scope, explicitly stating what it does not claim:

• It does not claim that State A is the optimal state for the depolarizing channel, nor that $0.064956$ is the true capacity threshold.
• It does not claim uniqueness of the zero-crossing for the coherent information functions.
• It does not provide a general capacity formula.

The primary significance lies in the methodology: demonstrating that superadditivity phenomena, which live in margins of $10^{-7}$ bits, can be rigorously certified using exact rational arithmetic and integer-based verification. The author argues that this "byte-level witness" approach, where floating-point results are merely hints fenced by rigorous integer checks, is the necessary path forward for verifying quantum capacity thresholds where floating-point evidence is insufficient. The result stands as the first proven positivity point for the depolarizing channel in this high-noise regime.