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.

The paper classifies subspaces $\mathcal S\subseteq(\mathbb C^d)^{\otimes t}$ on which collective Haar encryption $U^{\otimes t}$ erases all plaintext information, and asks whether privacy and finite-ensemble moment guarantees survive deletion of a tensor factor. This characterizes plaintext spaces supporting perfectly private collective encryption with local inverse-unitary decryption. In the Schur–Weyl decomposition $\bigoplus_\lambda V_\lambda\otimes W_\lambda$, admissibility requires input-independent block weights $c_\lambda$ and fixed multiplicity-space states $\omega_\lambda$, including the corresponding off-diagonal identities. An explicit purification-based normal form describes every admissible subspace: dimension $n$ is possible for prescribed active blocks and multiplicity states exactly when $n\operatorname{rank}\omega_\lambda\leq\dim V_\lambda$ in every active block. The characterization also follows from scalar compression of the twirl’s image algebra and privacy–correctability complementarity. Deletion is defined through $\operatorname{Red}_j(\mathcal S)=\operatorname{supp}(\operatorname{Tr}_j\Pi_{\mathcal S})$. Privacy on this entire reduced space and descent of every exact restricted design are distinct, logically independent properties; design descent is equivalent to inclusion of coefficient-function spaces. The symmetric tower has both properties. Exterior powers always preserve privacy, but preserve designs from level $s$ to $s-1$ exactly when $s\leq(d+1)/2$. Branching rules, Jucys–Murphy eigenspaces and Peter–Weyl theory give further criteria for canonical irreducible carriers. The generalized plaintext is an already prepared distributed state: local recovery need not give each recipient the logical message. The heredity results concern exact designs; they do not establish efficient ensemble constructions or quantitative approximate-heredity guarantees.