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arXiv:1403.2543 · IEEE Transactions on Information Theory

Second-Order Asymptotics for Source Coding, Dense Coding, and Pure-State Entanglement Conversions

Nilanjana Datta, Felix Leditzky

Published Nov 4, 2014• 53 citations
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Abstract

We introduce two variants of the information spectrum relative entropy defined by Tomamichel and Hayashi, which have the particular advantage of satisfying the data-processing inequality, i.e., monotonicity under quantum operations. This property allows us to obtain one-shot bounds for various information-processing tasks in terms of these quantities. Moreover, these relative entropies have a second-order asymptotic expansion, which in turn yields tight second-order asymptotics for optimal rates of these tasks in the independent and identically distributed setting. The tasks studied in this paper are fixed-length quantum source coding, noisy dense coding, entanglement concentration, pure-state entanglement dilution, and transmission of information through a classical-quantum channel. In the latter case, we retrieve the second-order asymptotics obtained by Tomamichel and Tan. Our results also yield the known second-order asymptotics of fixed-length classical source coding derived by Hayashi. The second-order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols-a quantity which, in the case of entanglement dilution, was studied by Harrow and Lo. We prove how the discrepancy between the optimal rates of these two processes in the second-order implies the irreversibility of entanglement concentration established by Kumagai and Hayashi. In addition, the spectral divergence rates of the information spectrum approach (ISA) can be retrieved from our relative entropies in the asymptotic limit. This enables us to directly obtain the more general results of the ISA from our one-shot bounds.

DOI
10.1109/tit.2014.2366994

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We introduce two variants of the information spectrum relative entropy defined by Tomamichel and Hayashi, which have the particular advantage of satisfying the data-processing inequality, i.e., monotonicity under quantum operations. This property allows us to obtain one-shot bounds for various information-processing tasks in terms of these quantities. Moreover, these relative entropies have a second-order asymptotic expansion, which in turn yields tight second-order asymptotics for optimal rates of these tasks in the independent and identically distributed setting. The tasks studied in this paper are fixed-length quantum source coding, noisy dense coding, entanglement concentration, pure-state entanglement dilution, and transmission of information through a classical-quantum channel. In the latter case, we retrieve the second-order asymptotics obtained by Tomamichel and Tan. Our results also yield the known second-order asymptotics of fixed-length classical source coding derived by Hayashi. The second-order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols-a quantity which, in the case of entanglement dilution, was studied by Harrow and Lo. We prove how the discrepancy between the optimal rates of these two processes in the second-order implies the irreversibility of entanglement concentration established by Kumagai and Hayashi. In addition, the spectral divergence rates of the information spectrum approach (ISA) can be retrieved from our relative entropies in the asymptotic limit. This enables us to directly obtain the more general results of the ISA from our one-shot bounds.
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the ααα-sandwiched Rényi divergence (ααα-SRD). We first investigate the limit α→0α\to 0α→0 to determine whether this recovers the well-known 000-Rényi relative divergence. We then prove various new results for entropic quantities derived from the ααα-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the ααα-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the ααα-SRD.
The simplest example of a quantum information source with memory is a mixed source, which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second-order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of the second-order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han. To prove the achievability part of our main result, we introduce universal quantum source codes achieving the second-order asymptotic rates. These are obtained by an extension of Hayashi's construction of their classical counterparts.
The simplest example of a quantum information source with memory is a mixed source, which emits signals entirely from one of two memoryless quantum sources with given a prioria~prioria priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second-order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of the second-order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han. To prove the achievability part of our main result, we introduce universal quantum source codes achieving the second-order asymptotic rates. These are obtained by an extension of Hayashi’s construction of their classical counterparts.
We consider two fundamental tasks in quantum information theory, data compression with quantum side information, as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. These characterizations-in contrast to earlier results-enable us to derive tight second-order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree (up to logarithmic terms) with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our technique also naturally yields bounds on operational quantities for finite block lengths.
The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from either one of two memoryless quantum sources with given a priori probabilities. For such a source, we derive the second order asymptotic rates for fixedlength (visible) source coding. This provides the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [21]. Moreover, the known optimal asymptotic rate and strong converse rate for fixed-length source coding for an arbitrary sequence of mixed sources (derived in the classical case by Han [11] and in the quantum case in [4]) are directly obtainable from our results.
We consider asymptotic convertibility of an arbitrary sequence of bipartite pure states into another by local operations and classical communication (LOCC). We adopt an information-spectrum approach to address cases where each element of the sequences is not necessarily a tensor power of a bipartite pure state. We derive necessary and sufficient conditions for the LOCC convertibility of one sequence to another in terms of spectral entropy rates of entanglement of the sequences. Based on these results, we also provide simple proofs for previously known results on the optimal rates of entanglement concentration and dilution of general sequences of bipartite pure states.