No-go for quantum seals

We introduce the concept of a quantum “seal” and investigate its feasibility. We define a seal as a state provided by Alice to Bob along with a set of instructions which Bob can follow in order to “break the seal” and read the classical message stored inside. We define two success criteria for a seal: the probability Bob can successfully read the message if he follows Alice’s instructions without any further input must be high, and if Alice asks for the state back from Bob, the probability Alice can tell if Bob broke the seal without permission must be high. We prove that within the constraints of quantum mechanics these two criteria are mutually exclusive. Finally, we derive upper and lower bounds on the achievability of a probabilistic seal for various message lengths, and show that for a 1-bit message our bound is tight.