Private information via the Unruh effect

In a relativistic theory of quantum information, the possible presence of horizons is a complicating feature placing restrictions on the transmission and retrieval of information. We consider two inertial participants communicating via a noiseless qubit channel in the presence of a uniformly accelerated eavesdropper. Owing to the Unruh effect, the eavesdropper’s view of any encoded information is noisy, a feature the two inertial participants can exploit to achieve perfectly secure quantum communication. We show that the associated private quantum capacity is equal to the entanglement-assisted quantum capacity for the channel to the eavesdropper’s environment, which we evaluate for all accelerations. Q uantum information theory for the most part assumes that the senders, receivers and eavesdroppers involved in a protocol share an inertial frame. For many of the applications envisioned in the field this is a good approximation and sometimes, as in the case of quantum key distribution, even a rigorously justifiable simplification. To the extent that quantum information theory attempts to identify fundamental rules governing information processing imposed by the laws of physics , however, neglecting relativity is ultimately unacceptable. Luckily, much of the formalism of quantum information remains valid in relativistic settings and the effect of changing reference frames can usually be modeled as the introduction of noise. Thus, there has been a significant amount of work done to calculate how entanglement degrades under a boost or acceleration [1], [2], [3], [4], [5], [6] and how basic quantum information theoretic protocols like teleportation, which were designed for inertial participants, break down under acceleration [7]. The natural next step is to design communications protocols specifically adapted to relativistic situations and, possibly, take advantage of uniquely relativistic features to accomplish otherwise impossible tasks. Kent has demonstrated, for example, that secure bit commitment is possible using a protocol exploiting relativistic causality constraints even though it is known to be impossible otherwise [8]. In this article, we consider a scenario in which two inertial participants communicate via a noiseless, bosonic, dual-rail qubit channel in the presence of a uniformly accelerated eavesdropper. In this context, the eavesdropper’s Unruh noise becomes a resource which the inertial participants can potentially exploit to encrypt their communications. The private quantum capacity is the optimal rate at which a sender (Alice) can send qubits to a receiver (Bob) while keeping them private from an eavesdropper (Eve). It had not previously been studied because in most settings it is redundant to require privacy in quantum communication: if the eavesdropper is modeled as being part of the environment of the communications channel then quantum communication is automatically private. This was in fact the insight behind † School of Computer Science, McGill University, Montréal, Canada Devetak’s proof of the quantum capacity theorem [9]. On the other hand, if Eve is assumed to have unrestricted access to the states while they are in transit from Alice to Bob, then unconditional privacy is impossible without secret keys in a nonrelativistic model because Eve and Bob are effectively interchangeable. This symmetry is broken, however, if Eve is assumed to be accelerating. The private quantum capacity problem therefore provides an example of a question which is poorly motivated in non-relativistic settings but very natural when relativity is taken into account. Because of structural features of the Unruh effect, this private quantum capacity is exactly zero if Alice is restricted to isometric encodings. However, for general encodings it is given by the same formula as the entanglement-assisted quantum capacity [10] of the channel to the eavesdropper’s environment, despite the absence of any operational connection between the two problems. Of course, it is also possible to define a private classical capacity for this setting, which we study for the purposes of comparison with its quantum version.