bibkey: starke2025swapping authors: Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero year: 2025 title: “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 claim: “The improved entanglement-swapping upper-bound conjecture, Eq. (59), bounds the average post-measurement l1 entanglement by the product of the input l1 entanglements divided by d-1.” strata_touched:

• D5/S3/Quantum/Entanglement/QuditSwappingProductBoundRefutation license: citation-only triage: anchor

Entanglement swapping for partially entangled qudits

D. S. Starke, M. L. W. Basso, L. C. Céleri and J. Maziero, arXiv:2508.00813v2 (quant-ph). The source is the section “The qutrit case and a conjecture”, p. 8, Eq. (59):

Therefore, we conjecture that the improved upper bound has the form $⟨E_{l_1}(|{\hat{\varphi }}_{pq}^{AB}⟩)⟩\le \frac{E_{l_1}(|\xi {⟩}_{AC})E_{l_1}(|\eta {⟩}_{C^{\prime }B})}{d-1}.$

The inputs (p. 4, Eqs. (19)–(20)) are $|\xi {⟩}_{AC}={\sum }_j{c}_j|jj{⟩}_{AC}$ and $|\eta {⟩}_{C^{\prime }B}={\sum }_k{d}_k|kk{⟩}_{C^{\prime }B}$, with ${\sum }_j|c_j{|}^2={\sum }_k|d_k{|}^2=1$. The paper uses the unscaled entanglement measure (p. 4, Eq. (24)):

$E_{l_1}(|\xi {⟩}_{AC})={\sum }_{j\ne k}|c_j{c}_k|$.

The Bell measurement gives (p. 5, Eq. (30)):

$|{\varphi }_{pq}^{AB}⟩=\frac{1}{\sqrt{d}}{\sum }_{k=0}^{d-1}{c}_{p\oplus k}{d}_k{\bar{\omega }}^{qk}|p\oplus k,k⟩$.

Here $\omega =\exp (2\pi i/d)$, and $p\oplus k$ is addition modulo $d$. The probability (p. 5, Eq. (31)) is:

${\Vert |{\varphi }_{pq}^{AB}⟩\Vert }^2=\frac{1}{d}{\sum }_{k=0}^{d-1}|c_{p\oplus k}{|}^2|d_k{|}^2=\mathrm{Pr}\left({\Phi }_{pq}^{C{C}^{\prime }}\right)$.

The average (p. 5, Eq. (41)) is:

$⟨E_{l_1}(|{\hat{\varphi }}_{pq}^{AB}⟩)⟩={\sum }_{p,q=0}^{d-1}\mathrm{Pr}({\Phi }_{pq}^{C{C}^{\prime }}\left)E_{l_1}\right(|{\hat{\varphi }}_{pq}^{AB}⟩)$.

The Lean encoding renames the paper’s second vector $d_k$ to $b_k$ and uses ZMod d indices. Each post-measurement state is in Schmidt form ${\sum }_k{\alpha }_k|\sigma (k),k⟩$, with the bijection $\sigma (k)=p\oplus k$; its entanglement is the ordered off-diagonal sum ${\sum }_{j\ne k}|{\alpha }_j{\alpha }_k|$. Normalization divides the state by its Hilbert norm. Outcomes of probability zero contribute zero.

For $d=4$ and $c=b=(7/10,1/10,7/10,1/10)$, both inputs are normalized and have entanglement $39/25$. The average is $723/625$, exceeding the proposed bound $(39/25{)}^2/3=507/625$.

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DOI: https://doi.org/10.48550/arXiv.2508.00813

Source: https://arxiv.org/abs/2508.00813v2

Conjecture: https://arxiv.org/html/2508.00813v2#S4.SS2, Eq. (59), and https://arxiv.org/pdf/2508.00813v2, p. 8.