Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

Bardet, Ivan; Junge, Marius; LaRacuente, Nicholas; Rouzé, Cambyse; França, Daniel Stilck

doi:10.1109/tit.2021.3065452

Cited by 11 publications

(5 citation statements)

References 61 publications

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“…Let (E, ρ), where ρ represents the quantum density matrix and E is the quantum communication channel, whereby E : B ⊗ B → B forming a completely positive tracepreserving map, where B is C-algebra of bounded operators. This must satisfy the quantum Markov condition [69], that…”

Section: A Quantum Markov Chain Theorymentioning

confidence: 99%

Quantum-Inspired Machine Learning for 6G: Fundamentals, Security, Resource Allocations, Challenges, and Future Research Directions

Duong

Ansere

Narottama

et al. 2022

IEEE Open J. Veh. Technol.

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Quantum computing is envisaged as an evolving paradigm for solving computationally complex optimization problems with a large-number factorization and exhaustive search. Recently, there has been a proliferation growth of the size of multi-dimensional datasets, the input-output space dimensionality, and data structures. Hence, the conventional machine learning approaches in data training and processing face a huge limited computing capabilities to support the sixthgeneration (6G) networks with highly dynamic applications and services. Under this regard, the fast developing quantum computing with machine learning (QML) for 6G networks is investigated. QML algorithms can significantly enhanced the processing efficiency and exponentially computational speed-up for effective quantum data representation and superposition framework, highly capable of guaranteeing high data storage and secured communications. We present the state-of-the-art in quantum computing and provide comprehensive overviews through machine learning approaches for their applications in the 6G networks. Furthermore, we introduce quantum-inspired machine learning applications for 6G networks, considering their enabling technologies and potential challenges. Finally, some dominating research issues and future research directions for the quantum-inspired machine learning in 6G networks are elaborated.

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Section: A Quantum Markov Chain Theorymentioning

confidence: 99%

Quantum-Inspired Machine Learning for 6G: Fundamentals, Security, Resource Allocations, Challenges, and Future Research Directions

Duong

Ansere

Narottama

et al. 2022

IEEE Open J. Veh. Technol.

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“…3. The amalgamated norms obey a nice "transference principle" which allows to successfully transfer estimates about hypercontractivity of classical Markov semigroups to the quantum case [5]. 4.…”

Section: Remark 43mentioning

confidence: 99%

Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates

Bardet

Rouzé

2022

Ann. Henri Poincaré

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We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated$$\mathbb {L}_p$$ L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated $$\mathbb {L}_p$$ L p norms are not uniformly convex for $$1\le p \le 2$$ 1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

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“…3.2-3.4: Lemma 1 directly yields a generalization of a result of [20] for conditional expectations with respect to non-tracial states in Corollary 1. Moreover, using a noncommutative change of measure argument [4], we provide in Proposition 2 some first estimates of the strong and weak constants c and d in AT(c, d) in terms of the maximal and minimal eigenvalues of a common invariant state of the three conditional expectations involved.…”

Section: Weak Approximate Tensorization Of the Relative Entropymentioning

confidence: 99%

“…Note that, at infinite temperature, the conditional expectations are selfadjoint with respect to the Hilbert-Schmidt inner product, a property referred to as symmetric in [4,21]. Under this condition, in [20], a different extension of (SSA) was proposed.…”

Section: Introductionmentioning

confidence: 99%

Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

Bardet

Capel

Rouzé

2021

Ann. Henri Poincaré

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In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

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Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

Cited by 11 publications

References 61 publications

Quantum-Inspired Machine Learning for 6G: Fundamentals, Security, Resource Allocations, Challenges, and Future Research Directions

Quantum-Inspired Machine Learning for 6G: Fundamentals, Security, Resource Allocations, Challenges, and Future Research Directions

Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates

Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

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