Two unbounded classicalization gaps with a qubit

Zhen-Peng Xu, Satoya Imai and Otfried Gühne, Fate of multiparticle entanglement when one particle becomes classical, arXiv:2206.12834v2, Phys. Rev. A 107, L040401, Section V.B, page 5:

Inspired by those two examples, an interesting question arises whether there exist entanglement measures E and states ρ_ABC such that both Δ_E(ρ_ABC) and Δ̃_E(ρ_ABC) − Δ_E(ρ_ABC) can be arbitrarily large in the sense that they are not limited by the size of C, even if C is only a qubit. We leave this question for further research.

The answer is Proved for the fixed tripartite negativity of Equation (4), with no dimension-dependent rescaling. The source and conventions are in the Library note; #19638 is the preregistration.

Statement echo

| Source clause | Formal counterpart | Scope | | --- | --- | --- | | Equation (4), sum of three bipartite negativities | tripartiteNegativity | Partial transposes on C, A and B; positive negativity $(|\rho^{T_Y}|_1-1)/2$. | | Equation (1), $\sum_i p_i\sigma_i\otimes\tau_i$ | branch, classicalize | Unnormalized branches $p_i\sigma_i=\operatorname{tr}_C(\rho m_i)$, including zero-probability outcomes. | | Equation (2), minimum entanglement loss | delta | Infimum over every finite POVM and every finite register dimension; attainment is proved for the constructed family. | | Equation (9), discard C and prepare its zero flag | deltaTilde | The prepared flag is the zero basis state of a qubit. | | Both gaps arbitrarily large on the same state | claim, result | Every real threshold admits one positive semidefinite trace-one state on $\operatorname{Fin}(a)\times\operatorname{Fin}(b)\times\operatorname{Fin}(2)$ exceeding it in both gaps. |

The density and POVM conditions appear literally in the public proposition and the infimum. The proofs use the frozen IsDensity and IsPOVM predicates with those same defining conditions. The source prints a sum of negative eigenvalues without an absolute value; its examples use the standard positive negativity adopted here.

A single qubit can control two distinct losses: the unavoidable loss when its quantum information is replaced by any measurement-and-prepare register, and the further loss caused by discarding that register. The source's flower-state example has zero unavoidable loss and unbounded further loss; Observation 6 gives unbounded unavoidable loss. The settlement supplies one family with both losses unbounded simultaneously.

The literature screen in #19638 covers arXiv v1 and v2, Imai's dissertation Section 12.5.2, and the related works arXiv:2501.04526, 2407.13348 and 2402.06677. The recorded screen finds no settlement in those works. Christandl–Winter, quant-ph/0501090, is not read in full. The bounded search result is not-found-in-searched-scope; exhaustive priority is ASSUMED-UNVERIFIED.

Let $|\Phi_d\rangle=d^{-1/2}\sum_{r=0}^{d-1}|rr\rangle$ and $|W\rangle=(|001\rangle+|010\rangle+|100\rangle)/\sqrt3$. Regroup $\rho_d=|\Phi_d\rangle\langle\Phi_d|\otimes|W\rangle\langle W|$ onto $A=A'A_0$, $B=B'B_0$ and C. Thus A and B have dimension $2d$ and C has dimension two.

Rank-one refinement of a qubit POVM writes each refined vector as $v_i=\alpha_i|0\rangle+\beta_i|1\rangle$, with $\sum_i|\alpha_i|^2=\sum_i|\beta_i|^2=1$. Its normalized W branch has coefficient matrix $K_i=\left(\begin{smallmatrix}\overline{\beta_i}&\overline{\alpha_i}\\overline{\alpha_i}&0\end{smallmatrix}\right)/\sqrt3$, giving partial-transpose trace norm $(4|\alpha_i|^2+|\beta_i|^2)/3$. Summing yields $5/3$. Tensor multiplicativity and the trace-norm triangle inequality bound the two A and B output cuts by $5d/3$ each. The output-register cut is positive under partial transpose and has unit trace, hence contributes zero negativity. Computational-basis measurement with orthogonal qubit flags attains the bound.

The compiled private declarations Family.family_initial, Family.family_all_channel_bound, Family.family_attains and Family.family_discarded_value establish respectively

$$E(\rho_d)=d-1+\frac{(2d+1)\sqrt2}{3},\qquad \max_{\Phi_C}E(\Phi_C(\rho_d))=\frac{5d}{3}-1,\qquad E(\rho_{AB}\otimes|0\rangle\langle0|)=\frac{d(2+\sqrt5)}3-1.$$

The private Family.finite_family_data identifies the infimum with the attained optimum and proves both gap identities:

$$\Delta_E(\rho_d)=\frac{2d(\sqrt2-1)+\sqrt2}{3},\qquad \widetilde\Delta_E(\rho_d)-\Delta_E(\rho_d)=\frac{d(3-\sqrt5)}3.$$

Both slopes are positive. unbounded_linear_gaps chooses d for every real threshold, and result proves the public claim.

A restriction to a chosen POVM, a fixed outcome count or a bounded register dimension would not settle the stated infimum. The upper bound quantifies all finite outcome types and register dimensions. A dimension-dependent rescaling of E, growth of C, or different states for the two gaps would not answer the question; E is fixed, C is exactly a qubit, and both inequalities use the same state. The public theorem has no hypotheses.

The canonical mathematical source is XuImaiGuhneSimultaneousUnboundedness.lean. Its sole public theorem is result : claim. Its public definitions have complete formulas in the corresponding Blueprint. The admission basis is open-problem-resolution (#19638; Proved); there is no atom.

The four-slot escape audit for D5.S3.Quantum.Classicalization.XuImaiGuhneSimultaneousUnboundedness.result is unfinished under #19716. The dependent threshold-indexed family needs compiled source-reconstruction, equivalence, variation, sensitivity and observational-dependence evidence. No declared_validated registration is claimed.

Numerical experiment

The experiment is the pinned experiment directory. Command: OPENBLAS_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 python3 /Users/auric/trureturing-experiments/docs/reports/xu-imai-guhne-2022-classicalization-simultaneous-unboundedness/check.py; exit 0, RESULT PASS. Script SHA-256: 977d6aef0d5b5991e2237a1cdd3030f0888d5eecbb758fe1a2119e41c324ea7e.

The tested scope is $d=2,3,4$, tolerance $10^{-9}$, 400 random channels per d with 2–4 effects and register dimension 1–3. These are numerical checks, not a proof of the all-channel bound.

| d | $E(\rho_d)$ | Attained output E | Largest sampled output E | $\Delta_E$ | $\widetilde\Delta_E-\Delta_E$ | | --- | --- | --- | --- | --- | --- | | 2 | 3.357022604 | 2.333333333 | 2.095592731 | 1.023689 | 0.509288 | | 3 | 5.299831646 | 4.000000000 | 3.675282957 | 1.299832 | 0.763932 | | 4 | 7.242640687 | 5.666666667 | 5.129618788 | 1.575974 | 1.018576 |

Tier 1 published question; settlement Proved. The scope is finite-dimensional tripartite negativity.

What the settlement shows

| Target item | Status and evidence kind | Evidence | | --- | --- | --- | | One fixed tripartite negativity measure suffices; no dimension-dependent rescaling | Proved; kernel-checked | result uses the single definition tripartiteNegativity at every threshold. | | Both gaps grow linearly with d while C stays exactly a qubit | Proved; kernel-checked | Private Family.finite_family_data gives the two identities above for every $d\ge2$, and unbounded_linear_gaps establishes simultaneous unboundedness. | | The optimal record can be a qubit with two orthogonal flags | Proved; kernel-checked | Private WChannel.computational_povm, WChannel.computational_state, Family.family_attains and Family.finite_family_data use two computational-basis flags on Fin 2 and prove attainment. | | Output-register cut contributes zero negativity; amplification acts through A and B cuts | Proved; paper argument from kernel-checked identities | Private WChannel.output_normC_eq gives raw trace norm 3; Amplification.amplify_normC multiplies it by d, and multiplication by $(3d)^{-1}$ gives trace norm 1 and negativity $(1-1)/2=0$. Amplification.amplify_normA and Amplification.amplify_normB give the additional factor d on the other normalized cuts. | | Experiment readings | Computed; floating-point experiment | The pinned experiment entry tests precisely $d=2,3,4$ and 400 sampled channels per d. | | Exhaustive literature priority | Open; incomplete literature screen | The scope and unread work are stated in Gap. | | Logarithmic negativity and other measures | Open | No theorem in this module establishes these extensions. |

The decisive mechanism is the POVM-independent branch-norm sum followed by entanglement amplification in A and B. It supplies the simultaneous example left open by Section V.B; it does not alter the source's flower-state or Bell-pair results. No extension to infinite-dimensional registers or to other entanglement measures is claimed.

Literature completeness and worldwide priority are ASSUMED-UNVERIFIED. The numerical experiment is approximate and finite. The four-slot escape audit remains unfinished under #19716; it is separate from the kernel proof of the settlement.