On a recent conjecture by Z. Van Herstraeten and N.J. Cerf for the quantum Wigner entropy

• URL: http://arxiv.org/abs/2303.10531v1
• Date: Sun, 19 Mar 2023 02:07:27 GMT
• Title: On a recent conjecture by Z. Van Herstraeten and N.J. Cerf for the quantum Wigner entropy
• Authors: Nuno Costa Dias and Jo\~ao Nuno Prata
• Abstract summary: Z. Van Herstraeten and N.J. Cerf claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant. We prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant. We also prove an analogous result for another conjecture stated by the same authors for the R'enyi entropy of positive Wigner functions.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We address a recent conjecture stated by Z. Van Herstraeten and N.J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the R\'enyi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb's inequalities.

Related papers

• 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)
• Wigner non-negative states that verify the Wigner entropy conjecture [0.0]
  We prove Wigner entropy conjecture for qubits formed by Fock states $|0rangle$ and $|1rangle$. We then consider general mixed states and derive a sufficient condition for Wigner non-negativity.
  arXiv  Detail & Related papers  (2024-06-13T15:49:55Z)
• The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs [36.135201624191026]
  We introduce three different parameterizations of finite discrete TEMs via Legendre functions of the negative tempered entropy function. Similar to the Hilbert geometry, the tempered Hilbert distance is characterized as a $t$-symmetrization of the oriented tempered Funk distance.
  arXiv  Detail & Related papers  (2023-11-22T15:24:29Z)
• Complex-valued Wigner entropy of a quantum state [0.0]
  We argue the merits of defining a complex-valued entropy associated with any Wigner function. It is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.
  arXiv  Detail & Related papers  (2023-10-30T06:30:03Z)
• Relative Entropy via Distribution of Observables [0.0]
  We obtain formulas for Petz-R'enyi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. All of the results presented here are valid in both finite and infinite dimensions.
  arXiv  Detail & Related papers  (2022-03-03T19:06:36Z)
• Quantum Wigner entropy [0.0]
  We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. We conjecture that it is lower bounded by $lnpi +1$ within the convex set of Wigner-positive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics.
  arXiv  Detail & Related papers  (2021-05-26T21:12:50Z)
• Uhlmann Fidelity and Fidelity Susceptibility for Integrable Spin Chains at Finite Temperature: Exact Results [68.8204255655161]
  We show that the proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of temperatures. The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates.
  arXiv  Detail & Related papers  (2021-05-11T14:08:02Z)
• Towards a functorial description of quantum relative entropy [0.0]
  Affine functor defines an affine functor in the special case where the relative entropy is finite. A recent non-commutative disintegration theorem provides a key ingredient in this proof.
  arXiv  Detail & Related papers  (2021-05-10T00:58:46Z)
• Exact thermal properties of free-fermionic spin chains [68.8204255655161]
  We focus on spin chain models that admit a description in terms of free fermions. Errors stemming from the ubiquitous approximation are identified in the neighborhood of the critical point at low temperatures.
  arXiv  Detail & Related papers  (2021-03-30T13:15:44Z)
• Catalytic Transformations of Pure Entangled States [62.997667081978825]
  Entanglement entropy is the von Neumann entropy of quantum entanglement of pure states. The relation between entanglement entropy and entanglement distillation has been known only for the setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving results an operational meaning also in entangled single-copy setup.
  arXiv  Detail & Related papers  (2021-02-22T16:05:01Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

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.