We derive the optimal exponent of the error probability of the quantum fixed-length pure state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al. (PRL, 81, 1714 (1998))'s universal code. In the direct part, a group representation theoretical type method is essential. In the converse part, Nielsen and Kempe (PRL, 86, 5184 (2001))'s lemma is essential.
We examine dense coding with an arbitrary pure entangled state sharing between the sender and the receiver. Upper bounds on the average success probability in approximate dense coding and on the probability of conclusive results in unambiguous dense coding are derived. We also construct the optimal protocol which saturates the upper bound in each case.
A complementarity relation is established between the capacity of multiport classical information transmission via quantum states and multiparty quantum correlation measures for three-qubit pure states. The multiparty quantum correlation measures considered are the generalized geometric measure, the tangle, and the discord monogamy score. The complementarity relation is revealed by the identification of a one-parameter family of pure three-qubit states, which we call the maximally dense coding capable family of states. These states have the maximal multiport dense coding capacity among all thr…
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 i.i.d. setting. The tasks studied in this paper are fixed-length quantum source…
In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponentia…
The question of the measurements on the Bell states by making use of mode change (from mixed to pure) of one qubit is considered. Such a mode change cannot be taken advantage of for superluminal communication in teleportation, and it may define constraints on the size of the gates.
We construct an optimal quantum universal variable-length code that achieves the admissible minimum rate, i.e., our code is used for any probability distribution of quantum states. Its probability of exceeding the admissible minimum rate exponentially goes to 0. Our code is optimal in the sense of its exponent. In addition, its average error asymptotically tends to 0.
How to uses shared entanglement and forward classical communication to remotely prepare an arbitrary (mixed or pure) state has been fascinating quantum information scientists. A constructive scheme has been given by Berry for remotely preparing a general pure state with a pure entangled state and finite classical communication. Based on this scheme, for high-dimensional systems it is possible to use a coding of the target state to optimize the classical communication cost. Unfortunately, for low-dimensional systems such as a pure qubit the coding method is inapplicable. Because qubit plays a c…
Continuous variable dense coding under realistic non-ideal scenarios
Harish-Chandra Research Institute（哈里希-钱德拉研究所）；Homi Bhabha National Institute（霍米·巴哈国家研究所）；Dipartimento di Fisica “Aldo Pontremoli,” Università degli Studi di Milano（米兰大学阿尔多·蓬特罗莫利物理系）
In this paper we consider the notion of quantum entanglement from the perspective of the logos categorical approach [26, 27]. Firstly, we will argue that the widespread distinctions, on the one hand, between pure states and mixed states, and on the other, between separable states and entangled states, are completely superfluous when considering the orthodox mathematical formalism of QM. We will then argue that the introduction of these distinctions within the theory of quanta is due to another two completely unjustified metaphysical presuppositions, namely, the idea that there is a "collapse" …
A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest o…
We describe a method to non-obliviously communicate a 2l-qubit quantum state by physically transmitting l+o(l) qubits of communication, and by consuming l ebits of entanglement and some shared random bits. In the non-oblivious scenario, the sender has a classical description of the state to be communicated. Our method can be used to communicate states that are pure or entangled with the sender's system; l+o(l) and 3l+o(l) shared random bits are sufficient respectively.
In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over c-q channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an …
The more than thirty years old issue of the (classical) information capacity of quantum communication channels was dramatically clarified during the last years, when a number of direct quantum coding theorems was discovered. The present paper gives a self contained treatment of the subject, following as much in parallel as possible with classical information theory and, on the other side, stressing profound differences of the quantum case. An emphasis is made on recent results, such as general quantum coding theorems including cases of infinite (possibly continuous) alphabets and constrained i…
We investigate the effect of noisy channels in a classical information transfer through a multipartite state which acts as a substrate for the distributed quantum dense coding protocol between several senders and two receivers. The situation is qualitatively different from the case with one or more senders and a single receiver. We obtain an upper bound on the multipartite capacity which is tightened in case of the covariant noisy channel. We also establish a relation between the genuine multipartite entanglement of the shared state and the capacity of distributed dense coding using that state…
We derive a general limit on the fidelity of a quantum channel conveying an ensemble of pure states. Unlike previous results, this limit applies to arbitrary coding and decoding schemes, including nonunitary decoding. This establishes the converse of the quantum noiseless coding theorem for all such schemes.
We develop a package of numerical simulations implemented in MATLAB to solve complex many-body quantum systems. We focus on widely used examples that include the calculation of the magnetization dynamics for the closed and open Ising model, dynamical quantum phase transition in cavity QED arrays, Markovian dynamics for interacting two-level systems, and the non-Markovian dynamics of the pure-dephasing spin-boson model. These examples will be useful for undergraduate and graduate students with a medium or high background in MATLAB, and also for researches interested in numerical studies applied…
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…
We prove a theorem for coding mixed-state quantum signals. For a class of coding schemes, the von Neumann entropy $S$ of the density operator describing an ensemble of mixed quantum signal states is shown to be equal to the number of spin-$1/2$ systems necessary to represent the signal faithfully. This generalizes previous works on coding pure quantum signal states and is analogous to the Shannon's noiseless coding theorem of classical information theory. We also discuss an example of a more general class of coding schemes which {\em beat} the limit set by our theorem.
We show that the classical capacity of quantum states, as quantified by its ability to perform dense coding, respects an exclusion principle, for arbitrary pure or mixed three-party states in any dimension. This states that no two bipartite states which are reduced states of a common tripartite quantum state can have simultaneous quantum advantage in dense coding. The exclusion principle is robust against noise. Such principle also holds for arbitrary number of parties. This exclusion principle is independent of the content and distribution of entanglement in the multipartite state. We also fi…
We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters alpha>1. It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Renyi divergence of states, and hence the Rényi capacities of channels. Our exponent is in terms of the Renyi capacities corresponding to a version of the Renyi divergences that has been introduced recently in [Müller-Lennert, Dupuis, S…
There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant compositi…
In [20] it was argued against the orthodox definition of quantum entanglement in terms of pure and separable states. In this paper we attempt to discuss how the logos categorical approach to quantum mechanics [18, 19] is able to provide an objective formal account of the notion of entanglement --completely independent of both purity and separability-- in terms of the potential coding of intensive relations and effective relations. We will show how our novel redefinition allows us to provide an anschaulich content to the notion of entanglement, erasing in this way the "spookiness" still present…
In the low-energy high-energy-efficiency regime of classical optical communications---relevant to deep-space optical channels---there is a big gap between reliable communication rates achievable via conventional optical receivers and the ultimate (Holevo) capacity. Achieving the Holevo capacity requires not only optimal codes but also receivers that make collective measurements on long (modulated) codeword waveforms, and it is impossible to implement these collective measurements via symbol-by-symbol detection along with classical postprocessing. Here, we apply our recent results on the classi…
In this paper we obtain a lower bound of exponent of average probability of error for classical quantum multiple access channel, which implies that for all rate pairs in the capacity region is achievable by a code with exponential probability of error. Thus we re-obtain the direct coding theorem.
