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arXiv:1103.1135 · New Journal of Physics

The apex of the family tree of protocols: optimal rates and resource inequalities

Nilanjana Datta, Min-Hsiu Hsieh

Published Sep 30, 2011• 57 citations
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Abstract

We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the fully quantum Slepian–Wolf (FQSW) protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in quantum information theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic independent and identically distributed scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previous bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.

DOI
10.1088/1367-2630/13/9/093042

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We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the fully quantum Slepian–Wolf (FQSW) protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in quantum information theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic independent and identically distributed scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previous bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.
We present the unification of many previously disparate results in noisy quantum Shannon theory and the unification of all of noiseless quantum Shannon theory. More specifically we deal here with bipartite, unidirectional, and memoryless quantum Shannon theory. We find all the optimal protocols and quantify the relationship between the resources used, both for the one-shot and for the ensemble case, for what is arguably the most fundamental task in quantum information theory: sharing entangled states between a sender and a receiver. We find that all of these protocols are derived from our one-shot superdense coding protocol and relate nicely to each other. We then move on to noisy quantum information theory and give a simple, direct proof of the "mother" protocol, or rather her generalization to the Fully Quantum Slepian-Wolf protocol(FQSW). FQSW simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation, and state transfer from the sender to the receiver. As a result, in addition to her other "children," the mother protocol generates the state merging primitive of Horodecki, Oppenheim, and Winter as well as a new class of distributed compression protocols for correlated quantum sources, which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called "father" protocol, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.
We give a simple, direct proof of the ‘mother’ protocol of quantum information theory. In this new formulation, it is easy to see that the mother, or rather her generalization to the fully quantum Slepian–Wolf protocol, simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation and state transfer from the sender to the receiver. As a result, in addition to her other ‘children’, the mother protocol generates the state-merging primitive of Horodecki, Oppenheim and Winter, a fully quantum reverse Shannon theorem, and a new class of distributed compression protocols for correlated quantum sources which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called ‘father’ protocol whose children provide the quantum capacity and the entanglement-assisted capacity of a quantum channel, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.
In quantum state redistribution as introduced by Luo and Devetak and Devetak and Yard, there are four systems of interest: the A system held by Alice; the B system held by Bob; the C system that is to be transmitted from Alice to Bob; and the R system that holds a purification of the state in the ABC registers. We give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one-shot setting. Our bounds are in terms of the smooth conditional minand max-entropy, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work of Oppenheim: it decomposes the quantum state redistribution task into two simpler coherent state merging tasks by introducing a coherent relay. In the independent and identical (i.i.d.) asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information I(C; R|B), and our bounds for the total cost converge to the conditional entropy H(C|B). This yields an alternative proof of optimality of these rates for quantum state redistribution in the i.i.d. asymptotic limit. In particular, we obtain a strong converse for quantum state redistribution, which even holds when allowing for feedback.
In the last few years, there has been a great interest in extending the information-theoretic scenario for the non-asymptotic or one-shot case, i.e., where the channel is used only once. We provide the one-shot rate region for the distributed source-coding (Slepian-Wolf) and the multiple-access channel. Our results are based on defining a novel one-shot typical set based on smooth entropies that yields the one-shot achievable rate regions while leveraging the results from the asymptotic analysis. Our results are asymptotically optimal, i.e., for the distributed source coding they yield the same rate region as the Slepian-Wolf in the limit of unlimited independent and identically distributed (i.i.d.) copies. Similarly for the multiple-access channel the asymptotic analysis of our approach yields the rate region which is equal to the rate region of the memoryless multiple-access channel in the limit of large number of channel uses.
A missing piece in quantum information theory, with very few exceptions, has been to provide the random coding exponents for quantum information-processing protocols. We remedy the situation by providing these exponents for a variety of protocols including those at the top of the family tree of protocols. Our line of attack is to provide an exponential bound on the decoupling error for a restricted class of completely positive maps where a key term in the exponent is in terms of a Rényi α-information-theoretic quantity for any α∈\in∈ (1,2]. Among the protocols covered are fully quantum Slepian-Wolf, quantum state merging, quantum state redistribution, quantum/classical communication across channels with side information at the transmitter with or without entanglement assistance, and quantum communication across broadcast channels.
We study the problem of state redistribution both in the classical (shared randomness assisted) and quantum (entanglement assisted) one-shot settings and provide new upper bounds on the communication required. Our classical bounds are in terms of smooth-min and max relative entropies and the quantum bounds are in terms of max relative entropy and Renyi relative entropy of order 12\frac{1}{2}21​. We also consider a special case of this problem in the classical setting, previously studied by Braverman and Rao (2011). We show that their upper bound is optimal. In addition we provide an alternate protocol achieving a priory different looking upper bound. However, we show that our upper bound is essentially the same as their upper bound and hence also optimal. For the quantum case, we show that our achievability result is upper bounded by the achievability result obtained in Berta, Christandl, Touchette (2016).