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PreprintCC BY 4.0arXiv

Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information

  • Sunghyeon Jo
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Abstract

Shor conjectured that a von Neumann measurement attains the accessible information of every binary quantum ensemble. We prove the conjecture constructively in arbitrary finite dimension. For every finite-outcome positive operator-valued measure (POVM) M, we form an operator TM from the posterior label probabilities and show that its spectral projection-valued measure (PVM) ΠM satisfies IΠM(X:Y)≥ IM(X:Y); every rank-one refinement retains the inequality. Two applications of Jensen's operator inequality prove the comparison and yield an exact concave variational formula for the accessible information. The result is a special case of the general theorem of Fang, Fawzi, and Fawzi on measured f-divergences; the proof below isolates the binary argument and makes the replacement M↦ TM↦ΠM explicit.

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Cite

BibTeXRIS
BibTeX
@misc{jo2026shors,
  title = {{Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information}},
  author = {Sunghyeon Jo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19003},
  eprint = {2609.19003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19003},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information
AU  - Sunghyeon Jo
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.19003
UR  - https://arxiv.org/abs/2609.19003
AB  - Shor conjectured that a von Neumann measurement attains the accessible information of every binary quantum ensemble. We prove the conjecture constructively in arbitrary finite dimension. For every finite-outcome positive operator-valued measure (POVM) M, we form an operator T_M from the posterior label probabilities and show that its spectral projection-valued measure (PVM) Π_M satisfies I_Π_M(X:Y)≥ I_M(X:Y); every rank-one refinement retains the inequality. Two applications of Jensen's operator inequality prove the comparison and yield an exact concave variational formula for the accessible information. The result is a special case of the general theorem of Fang, Fawzi, and Fawzi on measured f-divergences; the proof below isolates the binary argument and makes the replacement M↦ T_M↦Π_M explicit.
ER  -