slug: orthey-philip-kondra-streltsov-2026-four-copy-concentration-bound
bibkey: ortheyphilipkondrastreltsov2026optimality
doi: 10.1103/bz3j-9njg
url: https://arxiv.org/abs/2509.09423v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/PureState/FourCopyConcentrationRefutation.result

Direct four-copy universal entanglement concentration beats the bound of Eq. (8)

Problem

A. C. Orthey, A. Philip, T. V. Kondra and A. Streltsov, Optimality of universal conclusive entanglement concentration protocols, Phys. Rev. A 113, 062404 (2026), arXiv:2509.09423v2, study LOCC protocols that turn copies of an unknown two-qubit pure state into the Bell state ϕ+.

• Universality. Definition 1: an n-to-1 LOCC map E1 is a universal conclusive protocol if ⟨ϕ+|E1(ρ⊗n)|ϕ+⟩ = Tr E1(ρ⊗n) for every input in the set, here every pure state |ψ⟩ = c1|00⟩ + c2|01⟩ + c3|10⟩ + c4|11⟩ (Eq. (1)).
• The bound. Theorem 3 bounds the four-copy success probability of concatenated protocols based on two-qubit operations by B(c) = 2|c2⁴c3⁴| + 2|c1⁴c4⁴| − 4Re[c1²c4²(c2*)²(c3*)²] (Eq. (8)).
• The conjecture. §VIII: “protocols that process four copies directly without intermediate Schmidt-state conversion warrant investigation. We conjecture that such protocols cannot surpass the success probability bound in Eq. (8)”.

The verbatim statements are in the literature note.

Issue #17404 fixes the reading: for every universal conclusive LOCC map on four copies and every normalized ψ, Tr E1(ψ⊗4) ≤ B(c).

The conjecture is refuted.

Motivation

Universal protocols must work without knowing the Schmidt basis of the input, and the paper quantifies the cost of universality by comparing Eq. (8) with Vidal’s state-dependent optimum. The conjecture would make the two-stage protocol of Kálmán et al. optimal among all universal four-copy protocols.

Gap

Issue #17404 records the literature check before any Lean:

• arXiv v1 and v2 both state the conjecture, and the journal version is listed;
• OpenAlex and INSPIRE list no citing work;
• no settlement was found on Zenodo, Crossref or in the repository Naixu-Guo/quantum-open-problems;
• the repository had no entry for the paper.

The mechanism used here is Hayashi–Matsumoto’s universal concentration (quant-ph/0209030), which the paper does not cite.

not-found-in-searched-scope.

Route

1. The protocol. Each party applies the same local success operator K = |0⟩⟨u| + |1⟩⟨v| to its four qubits, and the parties keep the output when both succeed. Here:
   • u and w are the products of two singlets on positions (1, 2), (3, 4) and on positions (1, 3), (2, 4);
   • v = (2w − u)/√3, so K K† = 1 and K†K is a projection.
2. Universality. For every coefficient matrix C, (C ⊗ C) maps the singlet to det C times the singlet. So C^{⊗4} fixes the span of u and w up to the factor (det C)², and the output coefficient matrix is K C^{⊗4} Kᵀ = (det C)² I.
3. The success probability is 2|det C|⁴ for every ψ.
4. The counterexample. At ψ0 = (|00⟩ + |01⟩ − |10⟩ + |11⟩)/2, a maximally entangled state, det C = 1/2. The success probability is 1/8, while B = 0.

Falsifier

The refutation would not settle the conjecture if:

• the protocol were not LOCC;
• it were not universal for every pure two-qubit state;
• it used an intermediate Schmidt-state conversion;
• the module’s B differed from Eq. (8);
• the success probability were computed for a different branch.

The module states the conjecture for every pair of local contractions K_A, K_B whose product success operator is universal, and each such pair completes to local instruments. It gives the exact output (det C)² times the Bell array for every C, and evaluates the literal Eq. (8) at ψ0.

Evidence

The canonical source is D5/S3/Quantum/PureState/FourCopyConcentrationRefutation.lean (155 lines, generality: I, utility: kind=certified-instance; basis=refutes).

• It defines Q, normalized, amp4, output, universal, success, B and claim, and the witness u, v, K and C0, and proves result : ¬ claim.
• It reuses:
  • the frozen FiniteDimensional.QubitMatrix, QubitWitnesses.bellCoefficients and QutritThresholdSharing.normalization;
  • StatisticalStrengthMagicSquare.rationalSinglets, made public for this settlement, as the pairing vector on positions (1, 3), (2, 4), and through a coordinate permutation as u.
• It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.
• result is the designated refutation result and is exempt from the escape registration (CLAUDE.md §3.9); rationalSinglets is a definition.

Frozen state: Golden/Frozen/state/D5/S3/Quantum/PureState/FourCopyConcentrationRefutation.lean.json, statement sha256:f338a5562015cc769b3a09c11d969bf566f68053301d14ee9ec6ddfd7bdc4a2f.

Declaration statements:

• result: sha256:6b92e6162359d2efddbc8fd88ecde5f7c8c765fb537b0b3c551a4fa578b0fc2a;
• claim: sha256:6c6bd115986b2f367a38423db80b60df8dc840df25d07d702351c7faf97aa043.

Every other public declaration is a definition used by a public statement. The module’s 8 private theorems are bind-only helpers consumed on the proof path of result, and its compiler-generated declarations also lie on the proof path of result. The bind-only form is admitted under the open-problem-resolution basis of CLAUDE.md §3.2; no escape witness is claimed. The refutation is a certified instance with basis refutes. There is no digestion atom.

Triage

What the settlement shows

Refuted by result: a single universal one-round LOCC protocol on four copies has success probability 2|det C|⁴, which exceeds Eq. (8) at the maximally entangled input ψ0 (1/8 against 0).

Computed, not formalized (orchestrator, 15 October 2026; recorded in #17404):

• the identity K C^{⊗4} Kᵀ = (det C)² I holds exactly for symbolic C (SymPy);
• the maximal deviation over 2000 random normalized inputs is 1.7 × 10⁻¹⁶ (NumPy).

Argued, not formalized.

• With Δ = c1c4 − c2c3 and Σ = c1c4 + c2c3, Eq. (8) equals 2|Δ|²|Σ|². So the protocol beats Eq. (8) exactly when |Δ| > |Σ|, that is when Re(c1c4 · conj(c2c3)) < 0.
• The protocol is the Young-index (2, 2) sector of Hayashi–Matsumoto’s universal concentration. Their index (3, 1) sector yields a three-dimensional maximally entangled state, which LOCC converts into ϕ+ deterministically. By the Schur weights dim V_λ · s_λ(x, y) of the Schmidt weights x and y, this raises the success probability to 3xy(x² + xy + y²) + 2x²y², which is 11/16 at ψ0.
• At ψ0 the concatenated protocol of the paper succeeds with probability 0, since its Eq. (10) carries the factor of Eq. (8).

Open.

• The optimal success probability of universal four-copy concentration into ϕ+, and whether the combined Schur-sector protocol attains it.

Effect on the paper. The conjecture of §VIII is false: direct four-copy processing beats Eq. (8), and Schmidt-basis alignment is not an optimal strategy for universal operations. Theorem 3 is unaffected, since it concerns concatenated two-qubit protocols only.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty or priority. The statements in “Computed, not formalized” and “Argued, not formalized” are not formalized.