Pith Number

pith:4FUBMAYR

pith:2026:4FUBMAYRDYAYWA4RGX75N6AV5E

not attested not anchored not stored refs pending

Counterexamples to additivity of minimum output $p$-R\'enyi entropy of quantum channels for $p>3/4$ and $0\leq p<1/4$

Benjamin Lovitz, Debbie Leung, Peixue Wu

arxiv:2607.15210 v1 · 2026-07-16 · quant-ph · math-ph · math.MP · math.PR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4FUBMAYRDYAYWA4RGX75N6AV5E}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-07-17T01:22:14.724284Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e1681603111e018b039135ffd6f815e9000e72acc3eab7e918a533a9e3c42ba2

Aliases

arxiv: 2607.15210 · arxiv_version: 2607.15210v1 · doi: 10.48550/arxiv.2607.15210 · pith_short_12: 4FUBMAYRDYAY · pith_short_16: 4FUBMAYRDYAYWA4R · pith_short_8: 4FUBMAYR

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/4FUBMAYRDYAYWA4RGX75N6AV5E \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: e1681603111e018b039135ffd6f815e9000e72acc3eab7e918a533a9e3c42ba2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d32cd2d95f3d3831f0f3bc142ffdb2b59ed8a0885242876cedc47d7c24fd116a",
    "cross_cats_sorted": [
      "math-ph",
      "math.MP",
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-07-16T17:08:47Z",
    "title_canon_sha256": "ce66c0a1a0f6ec03dc420adfe861d4e7773bfde757692ad89eff055f30c51f88"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2607.15210",
    "kind": "arxiv",
    "version": 1
  }
}