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PreprintCC BY 4.0arXiv

No cardinality bound for squashed entanglement

  • Rabsan Galib Ahmed
  • Graeme Smith
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Abstract

Squashed entanglement is an additive bipartite entanglement measure. For a state ρAB, it is defined as the infimum of all conditional mutual information I(A:B| E) evaluated on extensions ρABE. It was unresolved whether there is a bound on the dimension of the conditioning system, E, needed for this optimization. We show that no such cardinality bound is possible. Explicitly, we find that the squashed entanglement for a partially dephased Bell pair on two qubits cannot be achieved for any finite dimensional extension E. A crucial ingredient of this proof is a direct sum step, which in a sense, symmetrizes two purifying systems while keeping the conditional mutual information unchanged and the coherence no worse.

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Cite

BibTeXRIS
BibTeX
@misc{ahmed2026cardinality,
  title = {{No cardinality bound for squashed entanglement}},
  author = {Rabsan Galib Ahmed and Graeme Smith},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14031},
  eprint = {2609.14031},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14031},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - No cardinality bound for squashed entanglement
AU  - Rabsan Galib Ahmed
AU  - Graeme Smith
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14031
UR  - https://arxiv.org/abs/2609.14031
AB  - Squashed entanglement is an additive bipartite entanglement measure. For a state ρ_AB, it is defined as the infimum of all conditional mutual information I(A:B| E) evaluated on extensions ρ_ABE. It was unresolved whether there is a bound on the dimension of the conditioning system, E, needed for this optimization. We show that no such cardinality bound is possible. Explicitly, we find that the squashed entanglement for a partially dephased Bell pair on two qubits cannot be achieved for any finite dimensional extension E. A crucial ingredient of this proof is a direct sum step, which in a sense, symmetrizes two purifying systems while keeping the conditional mutual information unchanged and the coherence no worse.
ER  -