We introduce a new operational technique for deriving chain rules for general information theoretic quantities. This technique is very different from the popular (and in some cases, fairly involved) methods like SDP formulation and operator algebra or norm interpolation. Instead, our framework considers a simple information transmission task and obtains lower and upper bounds for it. The lower bounds are obtained by leveraging a successive cancellation encoding and decoding technique. Pitting the upper and lower bounds against each other gives us the desired chain rule. As a demonstration of this technique, we derive a chain rule for the smooth-Hypothesis testing mutual information.

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., (2005). 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.

We introduce a new operational technique for deriving a chain rule for general information theoretic quantities. This technique is very different from the popular (and in some cases fairly involved) methods like SDP formulation and operator algebra or norm interpolation. Instead, our framework considers a simple information transmission task and obtains lower and upper bounds for it. The lower bounds are obtained leveraging a successive cancellation encoding and decoding technique. Pitting the upper and lower bounds against each other gives us the desired chain rule. As a demonstration of this technique we derive the chain rule for the smooth-Hypothesis testing mutual information.The full version of this paper can be found online, see [CK].

We investigate the problem of producing local pure states by performing local noisy operations assisted by one-way classical communication on a given bipartite mixed state in the one-shot setting. We consider the following two scenarios:1)Scenario I: A party, say Alice, is provided with a single copy of some quantum state ρAon system A. The task for Alice is to extract pure qubit states using only noisy operations on A. We call this task purity concentration.2)Scenario II: Two parties, Alice and Bob possess the A and B sub-systems, respectively, of a given bipartite quantum state ρAB. They are allowed to perform any local noisy operations and communicate via a one-way dephasing (i.e., classical) channel. The task for them is to design a protocol using these resources such that together they can extract pure local qubit states from the shared state ρAB. We call this task local purity distillation.

This paper is eligible for the Jack Reil Wolf ISIT Student Paper Award.We prove new one shot achievability results for measurement compression of quantum instruments with side information at the receiver. Unlike previous one shot results for this problem, our one shot bounds are nearly optimal and do not need catalytic randomness. In fact, we state a more general problem called centralised multi link measurement compression with quantum side information and provide one shot achievability results for it. As a simple corollary, we obtain one shot measurement compression results for quantum instruments with side information that we mentioned earlier. All our one shot results lead to the standard results for this problem in the asymptotic iid setting. We prove our achievability bounds by first proving a novel sequential classical quantum multipartite covering lemma, which should be of independent interest.

We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k > 3\Delta$ and returns a random proper $k$-coloring of $G$. The distribution of the coloring is perfectly uniform over the set of all proper $k$-colorings; the expected running time of the algorithm is ${poly}(k,n)=\widetilde{O}(n\Delta^2\cdot \log(k))$. This improves upon a result of Huber [ Proceedings of the $30$th ACM Symposium on Theory of Computing (STOC), 1998, pp. 31--40], who obtained a polynomial time perfect sampling algorithm for $k>\Delta^2+2\Delta$. Prior to our work, no algorithm with expected running time ${poly}(k,n)$ was known to guarantee perfectly sampling with a subquadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Huber's) is based on the coupling from the past method. Inspired by the bounding chain approach, pioneered independently by Huber (STOC 1998) and Häggström and Nelander [ Scand. J. Stat., 26 (1999), pp. 395--411], we employ a novel bounding chain to derive our result for the graph coloring problem.

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 on$A_{1}A_{2}R$, where$A_{1}, A_{2}$are treated as the two ‘sender’ systems and$R$is the reference system. We apply independent Haar random unitaries in tensor product on$A_{1}$and$A_{2}$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 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 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. [22]. 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-shot setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid 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].

We present a randomized algorithm that takes as input an undirected n-vertex graph G with maximum degree Δ and an integer k > 3Δ, and returns a random proper k-coloring of G. The distribution of the coloring is perfectly uniform over the set of all proper k-colorings; the expected running time of the algorithm is poly(k,n)=O(nΔ2· log(k)). This improves upon a result of Huber (STOC 1998) who obtained a polynomial time perfect sampling algorithm for k>Δ2+2Δ. Prior to our work, no algorithm with expected running time poly(k,n) was known to guarantee perfectly sampling with sub-quadratic number of colors in general.