A four-dimensional counterexample to the improved swapping bound

Abstract

The improved product bound for entanglement swapping conjectured by Starke, Basso, Celeri and Maziero fails for two identical partially entangled ququarts: the average is 723/625, whereas the proposed bound is 507/625.

Definition 1.1 (Bell phase).

$$\forall d:\mathbb{N},\mathrm{omega}\left(d\right)=\exp \left((\frac{2\cdot \pi }{(d:\mathbb{R})}:\mathbb{C})\cdot i\right)$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.omega (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

The generalized Bell basis uses omega = exp(2 pi i / d). The imaginary unit is i; the natural dimension is cast to a real number in the quotient.

Definition 1.2 (Unnormalized conditional state).

$$\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall c:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall b:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall p:\mathrm{ZMod}\left(d\right),\forall q:\mathrm{ZMod}\left(d\right),\forall x:(\mathrm{ZMod}\left(d\right)\times \mathrm{ZMod}\left(d\right)),\mathrm{phi}\left(d,c,b,p,q,x\right)=\frac{1}{(\sqrt{(d:\mathbb{R})}:\mathbb{C})}\cdot (\sum _{k\in \mathrm{ZMod}\left(d\right)}c\left(p+k\right)\cdot b\left(k\right)\cdot \mathrm{star}{\left(\mathrm{omega}\left(d\right)\right)}^{\mathrm{val}\left(q\right)\cdot \mathrm{val}\left(k\right)}\cdot \mathrm{ite}\left(x=(p+k,k),1,0\right))$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.phi (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Page 5, Eq. (30): \ket{\phi_{pq}^{AB}} = \frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}c_{p\oplus k} d_k \bar{\omega}^{qk}|p\oplus k,k\rangle. The paper’s d_k is renamed b_k. Indices are ZMod d, p plus k is addition modulo d, and val selects the representative in 0,…,d-1 for the natural exponent. The ket is its computational-basis delta function: ite(P,a,b) is a if P holds and b otherwise. All real scalars in complex arithmetic are cast to C. Translation k to p+k is a bijection, so the post-measurement state is in Schmidt form with sigma(k)=p+k.

Definition 1.3 (Hilbert norm).

$$\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall v:((\mathrm{ZMod}\left(d\right)\times \mathrm{ZMod}\left(d\right))\to \mathbb{C}),\mathrm{stateNorm}\left(d,v\right)=\sqrt{\sum _{x\in (\mathrm{ZMod}\left(d\right)\times \mathrm{ZMod}\left(d\right))}{\Vert v\left(x\right)\Vert }^2}$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.stateNorm (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

In the orthonormal computational basis ZMod d times ZMod d, the Hilbert norm is the square root of the sum of squared coordinate moduli. This is the norm used to normalize each conditional state.

Definition 1.4 (Bell-outcome probability).

$$\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall c:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall b:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall p:\mathrm{ZMod}\left(d\right),\forall q:\mathrm{ZMod}\left(d\right),\mathrm{prob}\left(d,c,b,p,q\right)=\mathrm{stateNorm}{\left(d,\mathrm{phi}\left(d,c,b,p,q\right)\right)}^2$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.prob (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Page 5, Eq. (31): \left\Vert\ket{\phi_{pq}^{AB}}\right\Vert^2 = \frac{1}{d}\sum_{k=0}^{d-1}|c_{p\oplus k}|^2 |d_k|^2 = \Pr\big(\Phi_{pq}^{CC'}\big). The definition is the squared Hilbert norm of the conditional state, with d_k renamed b_k.

Definition 1.5 (Entanglement in Schmidt form).

$$\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall a:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),El1\left(d,a\right)=\sum _{j\in \mathrm{ZMod}\left(d\right)}\sum _{k\in \mathrm{ZMod}\left(d\right)}\mathrm{ite}\left(j=k,0,\Vert a\left(j\right)\cdot a\left(k\right)\Vert \right)$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.El1 (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Page 4, Eq. (24): E_{l_1}(|\xi\rangle_{AC}) = \sum_{j\ne k}|c_j c_k|. For a normalized state in Schmidt form sum_k a_k |sigma(k),k>, with sigma a bijection, this is the sum over ordered pairs of distinct indices of the modulus of a_j a_k. No division by d-1 is included in El1.

Definition 1.6 (Average post-measurement entanglement).

$$\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall c:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall b:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),averageEl1\left(d,c,b\right)=\sum _{p\in \mathrm{ZMod}\left(d\right)}\sum _{q\in \mathrm{ZMod}\left(d\right)}\mathrm{ite}\left(\mathrm{prob}\left(d,c,b,p,q\right)=0,0,\mathrm{prob}\left(d,c,b,p,q\right)\cdot El1\left(d,(k:\mathrm{ZMod}\left(d\right)\mapsto \frac{\mathrm{phi}\left(d,c,b,p,q,(p+k,k)\right)}{(\mathrm{stateNorm}\left(d,\mathrm{phi}\left(d,c,b,p,q\right)\right):\mathbb{C})})\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.averageEl1 (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Page 5, Eq. (41): \big\langle E_{l_1}\big(|\hat{\phi}_{pq}^{AB} \rangle\big) \big\rangle = \sum_{p,q = 0}^{d-1} \Pr\big(\Phi_{pq}^{CC'}\big) E_{l_1}\big( |\hat{\phi}_{pq}^{AB} \rangle \big). Each conditional state is divided by its Hilbert norm. Its Schmidt coefficients are the coordinates at (p+k,k); zero-probability outcomes contribute zero. The sum includes all d squared Bell outcomes.

Definition 1.7 (The improved upper-bound conjecture).

$$claim\iff (\forall d:\mathbb{N},[\mathrm{NeZero}\left(d\right)],\forall c:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),\forall b:(\mathrm{ZMod}\left(d\right)\to \mathbb{C}),(2\le d)\Rightarrow \left((\sum _{j\in \mathrm{ZMod}\left(d\right)}{\Vert c\left(j\right)\Vert }^2=1)\Rightarrow \left((\sum _{k\in \mathrm{ZMod}\left(d\right)}{\Vert b\left(k\right)\Vert }^2=1)\Rightarrow averageEl1\left(d,c,b\right)\le \frac{El1\left(d,c\right)\cdot El1\left(d,b\right)}{(d:\mathbb{R})-1}\right)\right))$$

Formalization. D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.claim (✓ std3).

Citation. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Page 8, Eq. (59), verbatim (the source formula is quoted in TeX): “Therefore, we conjecture that the improved upper bound has the form \big\langle E_{l_1}\big(|\hat{\phi}_{pq}^{AB} \rangle\big) \big\rangle \le \frac{E_{l_1}(|\xi\rangle_{AC}) E_{l_1}(|\eta\rangle_{C'B})}{d-1}.” The quantified encoding ranges over every dimension d at least 2 and every pair c,b of normalized complex coefficient vectors. NeZero d supplies the finite ZMod d index type and follows from d at least 2. The paper’s second vector d_k is b_k. The denominator is real d minus 1.

Theorem 1.8 (The bound fails in dimension four).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.result (✓ std3). ∎

Resolves. Problems/starke-2025-qudit-swapping-product-bound-refutation (refuted) by D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.result.

Source. Repository-derived.

Acknowledgement. Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Commentary.

Take d=4 and c=b=(7/10,1/10,7/10,1/10). Both sums of squared moduli are 1 and both El1 values are 39/25. For arbitrary positive dimension, the phase has modulus 1 and the weighted entanglement of an outcome is (1/d) times the ordered off-diagonal sum of |c_(p+j)c_(p+k)b_j b_k|. In the zero-probability case, every supported coefficient vanishes; otherwise the squared normalizing factor cancels the probability. Summing over q removes the factor 1/d. Evaluating the resulting correlation sum at these inputs gives 723/625, strictly greater than (39/25)^2/3=507/625.

References

• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.El1
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.averageEl1
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.omega
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.phi
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.prob
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation.stateNorm