Limit Distribution Theory for Quantum Divergences

• URL: http://arxiv.org/abs/2311.13694v3
• Date: Mon, 14 Oct 2024 18:23:34 GMT
• Title: Limit Distribution Theory for Quantum Divergences
• Authors: Sreejith Sreekumar, Mario Berta,
• Abstract summary: We show that a limit distribution theory which characterizes the fluctuations of the estimation error is still premature. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting distribution is a normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain performance guarantees for a multi-hypothesis testing problem.
• Score: 8.11839312231511
• License:
• Abstract: Estimation of quantum relative entropy and its R\'{e}nyi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fr\'{e}chet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.

Related papers

• Reducing the sampling complexity of energy estimation in quantum many-body systems using empirical variance information [45.18582668677648]
  We consider the problem of estimating the energy of a quantum state preparation for a given Hamiltonian in Pauli decomposition. We construct an adaptive estimator using the state's actual variance.
  arXiv  Detail & Related papers  (2025-02-03T19:00:01Z)
• Converse bounds for quantum hypothesis exclusion: A divergence-radius approach [7.812210699650153]
  We apply strong converse results for asymmetric binary hypothesis testing to distinguishing an arbitrary dummy'' hypothesis from each of the concerned candidates. This leads to the desired upper bounds in terms of divergence via a geometrically inspired argument.
  arXiv  Detail & Related papers  (2025-01-16T18:01:19Z)
• Generalized quantum asymptotic equipartition [11.59751616011475]
  We prove that all operationally relevant divergences converge to the quantum relative entropy between two sets of quantum states. In particular, both the smoothed min-relative entropy between two sequential processes of quantum channels can be lower bounded by the sum of the regularized minimum output channel divergences. We apply our generalized AEP to quantum resource theories and provide improved and efficient bounds for entanglement distillation, magic state distillation, and the entanglement cost of quantum states and channels.
  arXiv  Detail & Related papers  (2024-11-06T16:33:16Z)
• Quantum tomography of helicity states for general scattering processes [55.2480439325792]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• Efficiency of estimators for locally asymptotically normal quantum statistical models [0.0]
  We establish a representation theorem for locallyally normal quantum statistical models. We study the efficiency of quantum estimators such as quantum regular estimators and quantum minimax estimators.
  arXiv  Detail & Related papers  (2022-09-02T06:10:05Z)
• Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition [62.997667081978825]
  We analyze the role of boundary conditions in the statistics of topological defects. We show that for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate.
  arXiv  Detail & Related papers  (2022-07-08T09:55:05Z)
• Maximum entropy quantum state distributions [58.720142291102135]
  We go beyond traditional thermodynamics and condition on the full distribution of the conserved quantities. The result are quantum state distributions whose deviations from thermal states' get more pronounced in the limit of wide input distributions.
  arXiv  Detail & Related papers  (2022-03-23T17:42:34Z)
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification [56.61325554836984]
  The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. We derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance.
  arXiv  Detail & Related papers  (2021-11-01T16:35:41Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• The variance of relative surprisal as single-shot quantifier [0.0]
  We show that (relative) surprisal gives sufficient conditions for approximate state-transitions between pairs of quantum states in single-shot setting. We further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy.
  arXiv  Detail & Related papers  (2020-09-17T16:06:54Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.