Quantum private broadcasting

In private broadcasting, a single plaintext is broadcast to multiple recipients in an encrypted form, such that each recipient can decrypt locally. When the message is classical, a straightforward solution is to encrypt the plaintext with a single key shared among all parties, and to send to each recipient a copy of the ciphertext. Surprisingly, the analogous method is insufficient in the case where the message is quantum [i.e., in quantum private broadcasting (QPB)]. In this work, we give three solutions to t-recipient quantum private broadcasting (t-QPB) and compare them in terms of key lengths. The first method is the independent encryption with the quantum one-time pad, which requires a key linear in the number of recipients, t. We show that the key length can be decreased to be logarithmic in t by using unitary t-designs. Our main contribution is to show that this can be improved to a key length that is logarithmic in the dimension of the symmetric subspace, using a concept that we define of symmetric unitary t-designs, which may be of independent interest.