Quantum Private Broadcasting

TL;DR

This paper explores quantum private broadcasting, presenting three methods to securely broadcast quantum messages to multiple recipients with optimized key lengths, including novel symmetric unitary t-designs.

Contribution

It introduces a new concept of symmetric unitary t-designs and demonstrates improved key length bounds for quantum private broadcasting.

Findings

Key length can be reduced to logarithmic in the number of recipients using unitary t-designs.

Symmetric unitary t-designs can further optimize key length based on the dimension of the symmetric subspace.

Three solutions for t-recipient quantum private broadcasting are compared in terms of key efficiency.

Abstract

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 new concept that we define of symmetric unitary $t$-designs, that may be of independent interest.