Local Additivity Revisited

• URL: http://arxiv.org/abs/2111.11385v4
• Date: Wed, 1 Mar 2023 22:52:52 GMT
• Title: Local Additivity Revisited
• Authors: Mary Beth Ruskai and Jon T. Yard
• Abstract summary: We make a number of simplifications in Gour and Friedland's proof of local additivity of minimum output entropy of a quantum channel. We use a different approach to reduce the general case to that of a square positive definite matrix. We extend this result to the maximum relative entropy with respect to a fixed reference state.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We make a number of simplifications in Gour and Friedland's proof of local additivity of minimum output entropy of a quantum channel. We follow them in reframing the question as one about entanglement entropy of bipartite states associated with a $d_B \times d_E $ matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information.

Related papers

• One-Shot Min-Entropy Calculation And Its Application To Quantum Cryptography [21.823963925581868]
  We develop a one-shot lower bound calculation technique for the min-entropy of a classical-quantum state. It gives an alternative tight finite-data analysis for the well-known BB84 quantum key distribution protocol. It provides a security proof for a novel source-independent continuous-variable quantum random number generation protocol.
  arXiv  Detail & Related papers  (2024-06-21T15:11:26Z)
• The Limits of Pure Exploration in POMDPs: When the Observation Entropy is Enough [40.82741665804367]
  We study a simple approach of maximizing the entropy over observations in place true latent states. We show how knowledge of the latter can be exploited to compute a regularization of the observation entropy to improve principled performance.
  arXiv  Detail & Related papers  (2024-06-18T17:00:13Z)
• Sum rule for the pseudo-Rényi entropy [0.07366405857677226]
  We establish an operator sum rule that pertains to the reduced transition matrix and reduced density matrices corresponding to the superposition states of $|phirangle$ and $|psirangle$. We provide proof of the operator sum rule and verify its validity in both finite-dimensional systems and quantum field theory.
  arXiv  Detail & Related papers  (2023-08-09T23:53:35Z)
• Local Intrinsic Dimensional Entropy [29.519376857728325]
  Most entropy measures depend on the spread of the probability distribution over the sample space $mathcalX|$. In this work, we question the role of cardinality and distribution spread in defining entropy measures for continuous spaces. We find that the average value of the local intrinsic dimension of a distribution, denoted as ID-Entropy, can serve as a robust entropy measure for continuous spaces.
  arXiv  Detail & Related papers  (2023-04-05T04:36:07Z)
• Asymptotic Equipartition Theorems in von Neumann algebras [24.1712628013996]
  We show that the smooth max entropy of i.i.d. states on a von Neumann algebra has an rate given by the quantum relative entropy. Our AEP not only applies to states, but also to quantum channels with appropriate restrictions.
  arXiv  Detail & Related papers  (2022-12-30T13:42:35Z)
• Integral formula for quantum relative entropy implies data processing inequality [0.0]
  We prove the monotonicity of quantum relative entropy under trace-preserving positive linear maps. For a simple application of such monotonicities, we consider any divergence' that is non-increasing under quantum measurements. An argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states.
  arXiv  Detail & Related papers  (2022-08-25T16:32:02Z)
• 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)
• Entropy and relative entropy from information-theoretic principles [24.74754293747645]
  We find that every relative entropy must lie between the R'enyi divergences of order $0$ and $infty$. Our main result is a one-to-one correspondence between entropies and relative entropies.
  arXiv  Detail & Related papers  (2020-06-19T14:50:44Z)
• Theory of Ergodic Quantum Processes [0.0]
  We consider general ergodic sequences of quantum channels with arbitrary correlations and non-negligible decoherence. We compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy can be computed exactly. Other physical implications of our results are that most Floquet phases of matter are metastable and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned.
  arXiv  Detail & Related papers  (2020-04-29T18:00:03Z)

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.