Введение
Понятие в теории информации
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that
where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem),
He showed that for any such functions the sum of the Shannon entropies is non negative,
A tighter bound,
was conjectured by Hirschman proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space.
В квантовой механике, теории информации и анализе Фурье, энтропическая неопределенность, или неопределенность Хиршмана, определяется как сумма временной и спектральной энтропий Шеннона. Оказывается, принцип неопределенности Гейзенберга можно выразить как нижнюю границу для суммы этих энтропий. Это более строгое утверждение, чем обычная формулировка принципа неопределенности в терминах произведения стандартных отклонений. В 1957 году Хиршман рассматривал функцию f и её преобразование Фурье g таким образом, что где символ "≈" указывает на сходимость в L2, и нормализовал их так, что (в соответствии с теоремой Планшереля),
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that
where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem),
He showed that for any such functions the sum of the Shannon entropies is non negative,
A tighter bound,
was conjectured by Hirschman proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space.
Он показал, что для любых таких функций сумма энтропий Шеннона неотрицательна,
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that
where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem),
He showed that for any such functions the sum of the Shannon entropies is non negative,
A tighter bound,
was conjectured by Hirschman proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space.
Более точная граница,
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that
where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem),
He showed that for any such functions the sum of the Shannon entropies is non negative,
A tighter bound,
was conjectured by Hirschman proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space.
была предложена Хиршманом, доказана в 1975 году У. Бекнером и в том же году интерпретирована Бялыницким-Бирулой и Мицельским как обобщенный принцип квантовомеханической неопределенности. Равенство достигается в случае гауссовского распределения. Следует, однако, отметить, что вышеуказанная функция энтропической неопределенности существенно отличается от квантовой энтропии фон Неймана, представленной в фазовом пространстве.
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that
where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem),
He showed that for any such functions the sum of the Shannon entropies is non negative,
A tighter bound,
was conjectured by Hirschman proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space.
Схема доказательства
Доказательство этого строгого неравенства опирается на так называемую (q, p)-норму преобразования Фурье. (Установление этой нормы – наиболее сложная часть доказательства.) На основе этой нормы можно установить нижнюю оценку для суммы (дифференциальных) энтропий Реньи, где , которые являются обобщением энтропий Шеннона. Для простоты мы рассматриваем это неравенство только в одном измерении; обобщение на многомерный случай прямолинейно и представлено в опубликованных работах.