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Benjamin Schumacher - One of the best experts on this subject based on the ideXlab platform.

  • Information transmission through a noisy quantum channel
    Physical Review A, 1998
    Co-Authors: Michael A. Nielsen, Howard Barnum, Benjamin Schumacher
    Abstract:

    Noisy quantum channels may be used in many Information-carrying applications. We show that different applications may result in different channel capacities. Upper bounds on several of these capacities are proved. These bounds are based on the coherent Information, which plays a role in quantum Information Theory analogous to that played by the mutual Information in Classical Information Theory. Many new properties of the coherent Information and entanglement fidelity are proved. Two nonClassical features of the coherent Information are demonstrated: the failure of subadditivity, and the failure of the pipelining inequality. Both properties arise as a consequence of quantum entanglement, and give quantum Information new features not found in Classical Information Theory. The problem of a noisy quantum channel with a Classical observer measuring the environment is introduced, and bounds on the corresponding channel capacity proved. These bounds are always greater than for the unobserved channel. We conclude with a summary of open problems.

  • sending entanglement through noisy quantum channels
    Physical Review A, 1996
    Co-Authors: Benjamin Schumacher
    Abstract:

    This paper addresses some general questions of quantum Information Theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems {ital R} and {ital Q}, after which {ital Q} is subjected to a dynamical evolution given by the superoperator {ital E}{sup {ital Q}}. Two interesting quantities can be defined for this process: the entanglement fidelity {ital F}{sub {ital e}} and the entropy exchange {ital S}{sub {ital e}}. It turns out that neither of these quantities depends in any way on the system {ital R}, but only on the initial state and dynamical evolution of {ital Q}. {ital F}{sub {ital e}} and {ital S}{sub {ital e}} are related to various other fidelities and entropies and are connected by an inequality reminiscent of the Fano inequality of Classical Information Theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes. {copyright} {ital 1996 The American Physical Society.}

Masahide Sasaki - One of the best experts on this subject based on the ideXlab platform.

  • implementation of generalized quantum measurements superadditive quantum coding accessible Information extraction and Classical capacity limit
    Physical Review A, 2004
    Co-Authors: Masahiro Takeoka, Mikio Fujiwara, Jun Mizuno, Masahide Sasaki
    Abstract:

    Quantum-Information Theory predicts that when the transmission resource is doubled in quantum channels, the amount of Information transmitted can be increased more than twice by quantum-channel coding technique, whereas the increase is at most twice in Classical Information Theory. This remarkable feature, the superadditive quantum-coding gain, can be implemented by appropriate choices of code words and corresponding quantum decoding which requires a collective quantum measurement. Recently, an experimental demonstration was reported [M. Fujiwara et al., Phys. Rev. Lett. 90, 167906 (2003)]. The purpose of this paper is to describe our experiment in detail. Particularly, a design strategy of quantum-collective decoding in physical quantum circuits is emphasized. We also address the practical implication of the gain on communication performance by introducing the quantum-Classical hybrid coding scheme. We show how the superadditive quantum-coding gain, even in a small code length, can boost the communication performance of conventional coding techniques.

  • exceeding the Classical capacity limit in a quantum optical channel
    Physical Review Letters, 2003
    Co-Authors: Mikio Fujiwara, Masahiro Takeoka, Jun Mizuno, Masahide Sasaki
    Abstract:

    The amount of Information transmissible through a communications channel is determined by the noise characteristics of the channel and by the quantities of available transmission resources. In Classical Information Theory, the amount of transmissible Information can be increased twice at most when the transmission resource is doubled for fixed noise characteristics. In quantum Information Theory, however, the amount of Information transmitted can increase even more than twice. We present a proof-of-principle demonstration of this superadditivity of Classical capacity of a quantum channel by using the ternary symmetric states of a single photon, and by event selection from a weak coherent light source. We also show how the superadditive coding gain, even in a small code length, can boost the communication performance of the conventional coding technique.

Michael A. Nielsen - One of the best experts on this subject based on the ideXlab platform.

  • Information transmission through a noisy quantum channel
    Physical Review A, 1998
    Co-Authors: Michael A. Nielsen, Howard Barnum, Benjamin Schumacher
    Abstract:

    Noisy quantum channels may be used in many Information-carrying applications. We show that different applications may result in different channel capacities. Upper bounds on several of these capacities are proved. These bounds are based on the coherent Information, which plays a role in quantum Information Theory analogous to that played by the mutual Information in Classical Information Theory. Many new properties of the coherent Information and entanglement fidelity are proved. Two nonClassical features of the coherent Information are demonstrated: the failure of subadditivity, and the failure of the pipelining inequality. Both properties arise as a consequence of quantum entanglement, and give quantum Information new features not found in Classical Information Theory. The problem of a noisy quantum channel with a Classical observer measuring the environment is introduced, and bounds on the corresponding channel capacity proved. These bounds are always greater than for the unobserved channel. We conclude with a summary of open problems.

  • Quantum Information Theory
    Quantum Computation and Quantum Information, 2026
    Co-Authors: Michael A. Nielsen
    Abstract:

    Classical Information Theory is mostly concerned with the problem of sending Classical Information – letters in an alphabet, speech, strings of bits – over communications channels which operate in accordance with the laws of Classical physics. How does the picture change if we can build quantum-mechanical communications channels? Can we transmit Information more efficiently? Can we make use of quantum mechanics to transmit secret Information without being eavesdropped on? These are just two of the questions we may ask when communication channels are allowed to be quantum mechanical. This redefinition of what a channel is causes us to go back and re-examine the fundamental questions motivating Classical Information Theory, in the search for new answers. This chapter surveys what is known about quantum Information Theory, including some surprising and intriguing possibilities made possible by quantum communication channels. Quantum Information Theory is motivated by the study of communications channels, but it has a much wider domain of application, and it is a thought-provoking challenge to capture in a verbal nutshell the goals of the field. As described in Section 1.6, we can identify three fundamental goals uniting work on quantum Information Theory: to identify elementary classes of static resources in quantum mechanics (which we identify as types of ‘Information’); to identify elementary classes of dynamical processes in quantum mechanics (identified as types of ‘Information processing’); and to quantify resource tradeoffs incurred performing elementary dynamical processes.

Howard Barnum - One of the best experts on this subject based on the ideXlab platform.

  • Information transmission through a noisy quantum channel
    Physical Review A, 1998
    Co-Authors: Michael A. Nielsen, Howard Barnum, Benjamin Schumacher
    Abstract:

    Noisy quantum channels may be used in many Information-carrying applications. We show that different applications may result in different channel capacities. Upper bounds on several of these capacities are proved. These bounds are based on the coherent Information, which plays a role in quantum Information Theory analogous to that played by the mutual Information in Classical Information Theory. Many new properties of the coherent Information and entanglement fidelity are proved. Two nonClassical features of the coherent Information are demonstrated: the failure of subadditivity, and the failure of the pipelining inequality. Both properties arise as a consequence of quantum entanglement, and give quantum Information new features not found in Classical Information Theory. The problem of a noisy quantum channel with a Classical observer measuring the environment is introduced, and bounds on the corresponding channel capacity proved. These bounds are always greater than for the unobserved channel. We conclude with a summary of open problems.

Andrea Montanari - One of the best experts on this subject based on the ideXlab platform.

  • fundamental limits of weak recovery with applications to phase retrieval
    Conference on Learning Theory, 2018
    Co-Authors: Marco Mondelli, Andrea Montanari
    Abstract:

    In phase retrieval we want to recover an unknown signal \(x\in\mathbb C^d\) from \(n\) quadratic measurements of the form \(y_i = ||^2+w_i\) where \(a_i\in \mathbb C^d\) are known sensing vectors and \(w_i\) is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements \(n\) needed to produce an estimator \(\hat{x}(y)\) that is positively correlated with the signal \(x\)? We consider the case of Gaussian vectors \(a_i\). We prove that -- in the high-dimensional limit -- a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For \(n\le d-o(d)\) no estimator can do significantly better than random and achieve a strictly positive correlation. For \(n\ge d+o(d)\) a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on Classical Information-Theory arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper and lower bound generalize beyond phase retrieval to measurements \(y_i\) produced according to a generalized linear model. As a byproduct of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.

  • fundamental limits of weak recovery with applications to phase retrieval
    arXiv: Machine Learning, 2017
    Co-Authors: Marco Mondelli, Andrea Montanari
    Abstract:

    In phase retrieval we want to recover an unknown signal $\boldsymbol x\in\mathbb C^d$ from $n$ quadratic measurements of the form $y_i = |\langle{\boldsymbol a}_i,{\boldsymbol x}\rangle|^2+w_i$ where $\boldsymbol a_i\in \mathbb C^d$ are known sensing vectors and $w_i$ is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements $n$ needed to produce an estimator $\hat{\boldsymbol x}(\boldsymbol y)$ that is positively correlated with the signal $\boldsymbol x$? We consider the case of Gaussian vectors $\boldsymbol a_i$. We prove that - in the high-dimensional limit - a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For $n\le d-o(d)$ no estimator can do significantly better than random and achieve a strictly positive correlation. For $n\ge d+o(d)$ a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on Classical Information-Theory arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper and lower bound generalize beyond phase retrieval to measurements $y_i$ produced according to a generalized linear model. As a byproduct of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.