No abstract

Decoupling theorems are an important tool in quantum information theory where they are used as building blocks in a host of information transmission protocols. A decoupling theorem takes a bipartite quantum state shared between a system and a reference, applies some local operation on the system, and then, if suitable conditions are met, proves that the resulting state is close to a product state between the output system and the untouched reference. The theorem is said to be non-catalytic if it does not require an additional input of a quantum state, in tensor with the given input state, in order to perform the decoupling. Dupuis [Dup10] proved an important non-catalytic decoupling theorem where the operation on the system was a Haar random unitary followed by a fixed superoperator, unifying many decoupling results proved earlier. He also showed a concentration result for his decoupling theorem viz. with probability exponentially close to one a Haar random unitary gives rise to a state close to a product state.In this paper we give a new concentration result for non-catalytic decoupling by showing that, for suitably large t, a unitary chosen uniformly at random from an approximate t-design gives rise to a state close to a product state with probability exponentially close to one. A unitary t-design is a finite set of unitaries with the property that the first t-moments of the matrix entries have the same expectation under the uniform distribution on the finite set as under the Haar measure over the full unitary group. Our concentration, though exponential, is less than that of Dupuis. However for many important applications it uses less random bits than Dupuis. In particular, we prove that approximate |A 1 |-designs decouple a quantum system in the Fully Quantum Slepian Wolf (FQSW) theorem wherein the fixed superoperator traces out the subsystem A 2 from a system A 1 ⊗ A 2 . This immediately leads to a saving in the number of random bits toefficient constructions of approximate |A 1 |-designs exist. Furthermore, this result implies that approximate unitary |A 1 |-designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. Previous works using unitary designs [SDTR13, NHMW17] did not obtain exponentially high concentration.

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard et al. [YDH05]. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

No abstract

We provide the first inner bounds for sending private classical information over a quantum multiple access channel. We do so by using three powerful information theoretic techniques: rate splitting, quantum simultaneous decoding for multiple access channels, and a novel smoothed distributed covering lemma for classical quantum channels. Our inner bounds are given in the one shot setting and accordingly the three techniques used are all very recent ones specifically designed to work in this setting. The last technique is new to this work and is our main technical advancement. For the asymptotic iid setting, our one shot inner bounds lead to the natural quantum analogue of the best classical inner bounds for this problem.