Simultaneous transmission of classical and quantum information under channel uncertainty and jamming attacks

Abstract

We derive universal codes for simultaneous transmission of classical messages and entanglement through quantum channels, possibly under attack of a malignant third party. These codes are robust to different kinds of channel uncertainty. To construct such universal codes, we invoke and generalize properties of random codes for classical and quantum message transmission through quantum channels. We show these codes to be optimal by giving a multi-letter characterization of regions corresponding to capacity of compound quantum channels for simultaneously transmitting and generating entanglement with classical messages. Also, we give dichotomy statements in which we characterize the capacity of arbitrarily varying quantum channels for simultaneous transmission of classical messages and entanglement. These include cases where the malignant jammer present in the arbitrarily varying channel model is classical (chooses channel states of product form) and fully quantum (is capable of general attacks not necessarily of product form). S(ρ) := −tr(ρ log ρ).

Universal superposition codes: Capacity regions of compound quantum broadcast channel with confidential messages

“…This results from time sharing arguments applied on the entropic quantities appearing in (7). For a short proof of a similar statement, see [16].…”

“…Proof. Properties 1-3 result directly from Lemma 14 [16]. Properties 4-6 and (17), result from applying the same concatenation arguments as in the proof of Lemma 14 [16], on inequalities (4)- (7) from [17].…”

“…Relaxing the assumption of the perfectly known channel, requires coding strategies that work for all channels belonging to a set of possibly infinite cardinality and are hence, significantly more sophisticated. A case in point is the coding strategy established by the authors of the current paper in [16] to derive capacity results for simultaneous transmission of classical (public) messages and quantum information over the quantum channel, given that those developed for the perfectly known channel in [21] did not provide the structure needed to deal with channel uncertainty. The compound model consists of an indexed set of channels {W s } s∈S .…”

Universal random codes: capacity regions of the compound quantum multiple-access channel with one classical and one quantum sender

“…A clever combination of singleterminal random codes results in a random code for the QMAC which has sufficient performance on the average. In quantum information theory, this technique was used for simultaneous classical and quantum coding over a single sender quantum channel and can be generalized to channels with uncertainty also in this case [15]. By extracting powerful universal random coding results for classical message and entanglement transmission from the literature ( [9], [34]), we were able to make successful use of this strategy also in case of channel uncertainty.…”

“…Remark 2 In [11,12], actually a continuous random code distributed according to the Haar measure on the unitary group on the encoding subspace was constructed. For the finite random code in Proposition 2 see [15].…”