We prove fully smooth one shot multipartite covering, aka convex split, results as well as fully smooth multipartite decoupling results for quantum states. Fully smooth one shot results for these problems were not known earlier, though the works of Cheng, Gao and Berta (arXiv:2304.12056) for convex split, and Colomer and Winter (arXiv:2304.12114) for decoupling, had made substantial progress by introducing a technique called telescoping cum mean zero decomposition of quantum states. We show that the telescoping cum mean zero decomposition technique can in fact be simplified and further extended in order to prove fully smooth decoupling and convex split results. Our techniques allow us to prove the first fully smooth one shot inner bounds for various fundamental network quantum information theory problems like e.g. the generalised Slepian Wolf problem of Anshu, Jain and Warsi (arXiv:1703.09961). We can also prove for the first time the natural polyhedral inner bound for sending quantum information over a quantum multiple access channel with limited entanglement assistance, first conjectured in Chakraborty, Nema and Sen (arXiv::2102.02187).
We prove fully smooth one shot multipartite covering, aka convex split, results as well as fully smooth multipartite decoupling results for quantum states. Fully smooth one shot results for these problems were not known earlier, though the works of Cheng, Gao and Berta (arXiv:2304.12056) for convex split, and Colomer and Winter (arXiv:2304.12114) for decoupling, had made substantial progress by introducing a technique called telescoping cum mean zero decomposition of quantum states. We show that the telescoping cum mean zero decomposition technique can in fact be simplified and further extended in order to prove fully smooth decoupling and convex split results. Our techniques allow us to prove the first fully smooth one shot inner bounds for various fundamental network quantum information theory problems like e.g. the generalised Slepian Wolf problem of Anshu, Jain and Warsi (arXiv:1703.09961). We can also prove for the first time the natural polyhedral inner bound for sending quantum information over a quantum multiple access channel with limited entanglement assistance, first conjectured in Chakraborty, Nema and Sen (arXiv::2102.02187).
We provide a powerful machinery to prove fully smooth one shot multipartite covering, aka convex split, type results for quantum states. In the important case of smooth multipartite convex split for classical quantum states, aka smooth multipartite soft covering, our machinery works even when certain marginals of these states do not satisfy pairwise independence. The recent paper (arXiv:2410.17893) gave the first proof of fully smooth multipartite convex split by simplifying and extending a technique called telescoping, developed originally for convex split by (arXiv:2304.12056). However, that work as well as all earlier works on convex split assumed pairwise or even more independence amongst suitable marginals of the quantum states. We develop our machinery by leveraging known results from (arXiv:1806.07278) involving tilting and augmentation smoothing of quantum states, combined with a novel observation that a natural quantum operation `flattening' quantum states actually preserves the fidelity. This machinery is powerful enough to lead to non pairwise independent results as mentioned above. As an application of our soft covering lemma without pairwise independence, we prove the `natural' one shot inner bounds for sending private classical information over a quantum wiretap interference channel, even when the classical encoders at the input lose pairwise independence in their encoding strategies to a certain extent. This result was unknown earlier even in the classical setting.
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. A full version of this paper is accessible at [5].
Abstract We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old (Dupuis et al. in Commun Math Phys 328:251–284, 2014) and new methods (Dupuis in IEEE Trans Inf Theory 69:7784–7792, 2023), we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, which are additionally optimal for entanglement of assistance and state merging.
In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say onA1A2R, whereA1,A2are treated as the two ‘sender’ systems andRis the reference system. We apply independent Haar random unitaries in tensor product onA1andA2and then send the resulting systems through a quantum channel. We want the channel outputBto be almost in tensor with the untouched referenceR. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over ak-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
In this work, we prove a novel one-shot multi-sender decoupling theorem generalising Dupuis result. We start off with a multipartite quantum state, say on A1 A2 R, where A1, A2 are treated as the two sender systems and R is the reference system. We apply independent Haar random unitaries in tensor product on A1 and A2 and then send the resulting systems through a quantum channel. We want the channel output B to be almost in tensor with the untouched reference R. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a k-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard, Dein the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.