Admissible Subspaces for the Collective Haar Twirl: Schur-Weyl Classification and Heredity under…

Abstract

Quantum private broadcasting (\textsf{QPB}) asks whether a quantum message can be encrypted once, broadcast to several recipients, and decrypted locally while the joint ciphertext reveals no plaintext information. Broadbent, González-Guillén, and Schuknecht showed that the full tensor-power space ℋ_d^{⊗ t} is generally unsuitable for \textsf{QPB}, identified Sym^t(ℋ_d) as a natural plaintext space, introduced symmetric unitary t-designs, and asked which other subspaces support private broadcast. Using Schur-Weyl duality, we classify all subspaces 𝒮⊆ℋ_d^{⊗ t} on which the collective Haar twirl 𝒯^(t) is a replacement channel. Precisely, every Schur-Weyl block weight must be independent of the encoded state, and in each active block V_λ⊗ W_λ, the reduced state on the multiplicity space W_λ must be fixed. This yields explicit two-block and general multiblock constructions and realizes the classification as the Schur-Weyl form of the Knill-Laflamme conditions for a complementary channel of 𝒯^(t). We also introduce a canonical reduction under deletion of a recipient and characterize privacy heredity, design heredity, and \textsf{QPB}-heredity for exact restricted designs. The symmetric tower is fully \textsf{QPB}-hereditary, whereas the antisymmetric tower is fully privacy-hereditary and is design-hereditary from level s to s−1 exactly when s≤(d+1)/2. For canonical carriers in arbitrary Schur-Weyl blocks, hereditary reduction is governed by multiplicity-free branching along the Young graph and inclusion of the associated Peter-Weyl coefficient spaces.

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BibTeX RIS

BibTeX

@misc{ganguly2026admissible,
  title = {{Admissible Subspaces for the Collective Haar Twirl: Schur-Weyl Classification and Heredity under Tensor-Factor Reduction}},
  author = {Sayantan Ganguly},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33553},
  eprint = {2610.33553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33553},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Admissible Subspaces for the Collective Haar Twirl: Schur-Weyl Classification and Heredity under Tensor-Factor Reduction
AU  - Sayantan Ganguly
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33553
UR  - https://arxiv.org/abs/2610.33553
AB  - Quantum private broadcasting (\textsf{QPB}) asks whether a quantum message can be encrypted once, broadcast to several recipients, and decrypted locally while the joint ciphertext reveals no plaintext information. Broadbent, González-Guillén, and Schuknecht showed that the full tensor-power space ℋ_d^⊗ t is generally unsuitable for \textsf{QPB}, identified Sym^t(ℋ_d) as a natural plaintext space, introduced symmetric unitary t-designs, and asked which other subspaces support private broadcast. Using Schur-Weyl duality, we classify all subspaces 𝒮⊆ℋ_d^⊗ t on which the collective Haar twirl 𝒯^(t) is a replacement channel. Precisely, every Schur-Weyl block weight must be independent of the encoded state, and in each active block V_λ⊗ W_λ, the reduced state on the multiplicity space W_λ must be fixed. This yields explicit two-block and general multiblock constructions and realizes the classification as the Schur-Weyl form of the Knill-Laflamme conditions for a complementary channel of 𝒯^(t). We also introduce a canonical reduction under deletion of a recipient and characterize privacy heredity, design heredity, and \textsf{QPB}-heredity for exact restricted designs. The symmetric tower is fully \textsf{QPB}-hereditary, whereas the antisymmetric tower is fully privacy-hereditary and is design-hereditary from level s to s−1 exactly when s≤(d+1)/2. For canonical carriers in arbitrary Schur-Weyl blocks, hereditary reduction is governed by multiplicity-free branching along the Young graph and inclusion of the associated Peter-Weyl coefficient spaces.
ER  -

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