The entropy power inequality with quantum conditioning

Palma, Giacomo De

doi:10.1088/1751-8121/aafff4

Cited by 8 publications

(6 citation statements)

References 44 publications

(99 reference statements)

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“…The main idea of the proof of ( 35) is perturbing the state with the quantum heat semigroup. The same idea has been crucial in the proofs of several quantum versions of the Entropy Power Inequality [43][44][45][46][47][48][49][50][51][52][53][54], of which (35) can be considered a generalization. Let Ḡ achieve the maximum in (35) (if the maximum is not achieved, the result can be obtained with a limiting argument).…”

Section: Generalized Strong Subadditivitymentioning

confidence: 96%

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

Palma

Hackl

2022

SciPost Phys.

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We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models.

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Section: Generalized Strong Subadditivitymentioning

confidence: 96%

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

Palma

Hackl

2022

SciPost Phys.

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“…In passing, it should be noted that the form of the Rényi EP expressed in ( 13 ) is not universally accepted version. In a number of works, it is defined merely as an exponent of RE, see, e.g., [ 75 , 76 ]. Our motivation for the form ( 13 ) is twofold: first, it has a clear interpretation in terms of variances of Gaussian distributions and, second, it leads to simpler formulas, cf.…”

Section: Rényi Entropy Based Estimation Theory and Rényi Entropy Pmentioning

confidence: 99%

“…In a number of works it is defined merely as an exponent of RE, see, e.g. [69,70]. Our motivation for the form ( 13) is twofold: first, it has a clear interpretation in terms of variances of Gaussian distributions and second, it leads to simpler formulas, cf.…”

Section: Rényi's Entropy Power and Generalized Isoperimetric Inequalitymentioning

confidence: 99%

From Rényi Entropy Power to Information Scan of Quantum States

Jizba

Dunningham

Prokš

2021

Entropy

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In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed.

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“…Here, the history of quantum EPI, which is a quantum analog of classical EPI, is briefly summarized. The quantum EPI on bosonic Gaussian systems was first derived by König and Smith [10], and it was generalized to bosonic Gaussian [11,37] and discrete systems [38] as well as conditional cases [39][40][41][42]. While the classical capacity is additive, quantum channel capacities are generally nonadditive; hence, quantum EPIs have the potential power of finding (tight upper bounds) realistic capacities on quantum channels.…”

Section: Introductionmentioning

confidence: 99%

Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems

Jeong¹,

Lee²

2022

Preprint

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In this study, the quantum Rényi entropy power inequality of order p > 1 and power κ is suggested as a quantum analog of the classical Rényi-p entropy power inequality. To derive this inequality, we first exploit the Wehrl-p entropy power inequality on bosonic Gaussian systems via the mixing operation of quantum convolution, which is a generalized beamsplitter operation. This observation directly provides a quantum Rényi-p entropy power inequality over a quasi-probability distribution for D-mode bosonic Gaussian regimes. The proposed inequality is expected to be useful for the nontrivial computing of quantum channel capacities, particularly universal upper bounds on bosonic Gaussian quantum channels.

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The entropy power inequality with quantum conditioning

Cited by 8 publications

References 44 publications

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

From Rényi Entropy Power to Information Scan of Quantum States

Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems

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