Layer Cake Representations for Quantum Divergences

• URL: http://arxiv.org/abs/2507.07065v2
• Date: Thu, 10 Jul 2025 17:53:52 GMT
• Title: Layer Cake Representations for Quantum Divergences
• Authors: Po-Chieh Liu, Christoph Hirche, Hao-Chung Cheng,
• Abstract summary: We propose a new approach to define suitable quantum extensions of classical divergences.<n>The resulting quantum R'enyi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations.
• Score: 8.397730500554047
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum R\'enyi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the R\'enyi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.

Related papers

• Grassmann Variational Monte Carlo with neural wave functions [45.935798913942904]
  We formalize the framework introduced by Pfau et al.citepfau2024accurate in terms of Grassmann geometry of the Hilbert space.<n>We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.
  arXiv  Detail & Related papers  (2025-07-14T13:53:13Z)
• Generalization Bounds for Quantum Learning via Rényi Divergences [45.45698347373077]
  This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms.<n>Our primary contribution is the derivation of these bounds in terms of quantum and classical R'enyi divergences.
  arXiv  Detail & Related papers  (2025-05-16T09:21:31Z)
• Covariant non-perturbative pointer variables for quantum fields [44.99833362998488]
  We derive and renormalize the integro-differential equation that governs the detector pointer-variable dynamics.<n>Our formal solution, expressed in terms of Green's functions, allows for the covariant, and causal analysis of induced observables on the field.
  arXiv  Detail & Related papers  (2025-02-03T11:53:31Z)
• Some properties and applications of the new quantum $f$-divergences [13.231546105751013]
  Recently, a new definition for quantum $f$-divergences was introduced based on an integral representation.<n>In this work, we investigate alternative ways of expressing these quantum $f$-divergences.
  arXiv  Detail & Related papers  (2025-01-07T14:06:56Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics.<n>It allows to model situations where the order of observations does not matter.<n>It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Quantum Bayesian Inference in Quasiprobability Representations [0.0]
  Bayes' rule plays a crucial piece of logical inference in information and physical sciences alike. quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.
  arXiv  Detail & Related papers  (2023-01-05T08:16:50Z)
• Finitely Repeated Adversarial Quantum Hypothesis Testing [22.102728605081534]
  We formulate a passive quantum detector based on a quantum hypothesis testing framework under the setting of finite sample size. Under the assumption that the attacker adopts separable optimal strategies, we derive that the worst-case average error bound converges to zero exponentially. We adopt our formulations upon a case study of detection with quantum radars.
  arXiv  Detail & Related papers  (2022-12-02T17:08:17Z)
• 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)
• Holomorphic representation of quantum computations [0.0]
  We study bosonic quantum computations using the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which gives a canonical description of bosonic quantum computing.
  arXiv  Detail & Related papers  (2021-10-29T23:26:51Z)
• Quantum indistinguishability through exchangeable desirable gambles [69.62715388742298]
  Two particles are identical if all their intrinsic properties, such as spin and charge, are the same. Quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles. We show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems.
  arXiv  Detail & Related papers  (2021-05-10T13:11:59Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• 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)
• 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.