Admissible Subspaces for the Collective Haar Twirl: Schur-Weyl Classification and Heredity under Tensor-Factor Reduction

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 \(\mathcal H_d^{\otimes t}\) is generally unsuitable for \textsf{QPB}, identified \(\mathrm{Sym}^t(\mathcal H_d)\) 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 \(\mathcal S\subseteq\mathcal H_d^{\otimes t}\) on which the collective Haar twirl \(\mathcal T^{(t)}\) is a replacement channel. Precisely, every Schur-Weyl block weight must be independent of the encoded state, and in each active block \(V_λ\otimes W_λ\), 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 \(\mathcal T^{(t)}\). 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 \(s\leq(d+1)/2\). 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.