Showing papers by "Andreas Winter published in 2002"
TL;DR: In this article, the authors present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma, which involves a development of explicit large deviation estimates to the case of random variables taking values in self-adjoint operators on a Hilbert space.
Abstract: We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum communication channels of Ahlswede's (1979, 1992) approach to classical channels. It involves a development of explicit large deviation estimates to the case of random variables taking values in self-adjoint operators on a Hilbert space. This theory is presented separately in an appendix, and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems.
520 citations
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TL;DR: A better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, and a review of a construction achieving the lower bound by using common randomness which is complement by showing the optimal use of the latter within a class of protocols.
Abstract: We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem.
After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it.
The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.
100 citations
TL;DR: This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.
Abstract: We study the visible compression of a source [script E] = {|phii>,pi} of pure quantum signal states or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy S([script E]) qubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade-off curve Q*(R), where Q*(R) qubits per signal is the optimal quantum rate for a given classical rate of R bits per signal. Our main result is an explicit characterization of this trade-off function by a simple formula in terms of only single-signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parametrized states. We also show that our result leads to a number of corollaries characterizing the trade-off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so-called remote state preparation problem. Finally, we develop a probability-free version of our main result which may be interpreted as an answer to the question: "How many classical bits does a qubit cost?" This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.
48 citations
TL;DR: In this article, the authors consider the problem of noisy dense coding in the context of approximate programmable quantum gates (SPQG) and show that such objects do not exist in the domain of usual quantum theory.
Abstract: We consider two apparently separated problems: in the first part of the article we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice properties that could have implications on the theory of universal computation. Unfortunately, as we prove, such objects do not exist in the domain of usual quantum theory. In the second part the problem of noisy dense coding (and generalizations) is addressed. We observe that the additivity problem for the classical capacity obtained is of apparently greater generality than for the usual quantum channel (completely positive maps): i.e., the latter occurs as a special case of the former, but, as we shall argue with the help of the nonexistence result of the first part, the former cannot be reduced to an instance of the latter. We conclude by suggesting that the additivity problem for the classical capacity of quantum channels, as posed until now, may conceptually not be in its appropriate generality.
21 citations
30 Jun 2002
TL;DR: In this paper, the authors bring together two subjects in the realm of quantum information theory that might at first glance seem far apart: the theory of universal computation in a quantum computer, and noise resistant coding of classical information in quantum channels.
Abstract: We bring together two subjects in the realm of quantum information theory that might at first glance seem far apart: the theory of universal computation in a quantum computer, and noise resistant coding of classical information in quantum channels.
4 citations
TL;DR: In this article, the authors studied the distillable entanglement, the number of Bell pairs distilled per copy, as a function of an error exponent, which represents the rate of decrease in failure probability as n tends to infinity.
Abstract: Consider entanglement concentration schemes that convert n identical copies of a pure state into a maximally entangled state of a desired size with success probability being close to one in the asymptotic limit. We give the distillable entanglement, the number of Bell pairs distilled per copy, as a function of an error exponent, which represents the rate of decrease in failure probability as n tends to infinity. The formula fills the gap between the least upper bound of distillable entanglement in probabilistic concentration, which is the well-known entropy of entanglement, and the maximum attained in deterministic concentration. The method of types in information theory enables the detailed analysis of the distillable entanglement in terms of the error rate. In addition to the probabilistic argument, we consider another type of entanglement concentration scheme, where the initial state is deterministically transformed into a (possibly mixed) final state whose fidelity to a maximally entangled state of a desired size converges to one in the asymptotic limit. We show that the same formula as in the probabilistic argument is valid for the argument on fidelity by replacing the success probability with the fidelity. Furthermore, we also discuss entanglement yield when optimal success probability or optimal fidelity converges to zero in the asymptotic limit (strong converse), and give the explicit formulae for those cases.
2 citations
TL;DR: In this paper, the Stinespring dilation theorem was used to give a formula for the channel capacity involving entanglement of formation, which can be used to show that additivity of the latter for some states can be inferred from the capacity for certain channels.
Abstract: The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a given bipartite state.
In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels.
We demonstrate this connection for a family of group--covariant channels, allowing us to calculate the entanglement cost for many states, including some where a strictly smaller upper bound on the distillable entanglement is known. Group symmetry is used for more sophisticated analysis, giving formulas valid for a class of channels. This is presented in a general framework, extending recent findings of Vidal, Dur and Cirac (e-print quant-ph/0112131).
We speculate on a general relation of superadditivity of the entanglement of formation, which would imply both the general additivity of this function under tensor products and of the Holevo capacity (with or without linear cost constraints).
2 citations
TL;DR: In this paper, the authors studied the trade-off between information gain and state disturbance for quantum sources and provided a simple formula in terms of only single signal, perfect fidelity encodings of the source.
Abstract: We study the visible compression of a source E of pure quantum signal states, or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy S(E) qubits per signal.
We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal is the optimal quantum rate for a given classical rate of R bits per signal.
Our main result is an explicit characterization of this trade--off function by a simple formula in terms of only single signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parameterized states. We also show that our result leads to a number of corollaries characterizing the trade--off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so--called remote state preparation problem. Finally, we develop a probability--free version of our main result which may be interpreted as an answer to the question: ``How many classical bits does a qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.
2 citations