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 apriori 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 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 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 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.
Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers m and l. We encode any tensor product of m many states emitted by the quantum stochastic source by breaking the tensor product into m many blocks where each block has length l, and considering sequences of m many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call \lq\lq special block codes\rq\rq\ in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for m=1. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.
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 α-sandwiched Rényi divergence (α-SRD). We first investigate the limit α→0 to determine whether this recovers the well-known 0-Rényi relative divergence. We then prove various new results for entropic quantities derived from the α-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 α-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 α-SRD.
Suppose there are two correlated sources, one producing classical data and the other producing mixed states with commuting density operators. We combine the two sources into one and call it a classical-quantum source. In this paper, we consider the problem of distributed compression of the classical-quantum source. This is a special case of the classical-quantum Slepian-Wolf problem addressed by Devetak and Winter [1]. Suppose Alice (resp. Bob) receives classical (resp. quantum) data from the classical-quantum source. Alice and Bob, who cannot communicate with each other, want to send their received data to Charlie at the minimum communication cost. We consider two scenarios, the blind and visible scenarios, for Bob's quantum data compression. We formulate our problem in terms of classical information theory and then using some results in classical information theory, we show partial results on the achievable rate region for asymptotically lossless distributed compression of the classical-quantum source in the blind and visible scenarios.