Quantum information theory and Fourier multipliers on quantum groups

• URL: http://arxiv.org/abs/2008.12019v13
• Date: Mon, 8 Mar 2021 15:41:41 GMT
• Title: Quantum information theory and Fourier multipliers on quantum groups
• Authors: C\'edric Arhancet
• Abstract summary: We compute the exact values of the minimum output entropy and the completely bounded minimal entropy of quantum channels acting on matrix algebras. Our results use a new and precise description of bounded Fourier multipliers from $mathrmL1(mathbbG)$ into $mathrmLp(mathbbG)$ for $1 p leq infty$ where $mathbbG$ is a co-amenable locally compact quantum group.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of very large classes of quantum channels acting on matrix algebras $\mathrm{M}_n$. Our new and simple approach relies on the theory of locally compact quantum groups and our results use a new and precise description of bounded Fourier multipliers from $\mathrm{L}^1(\mathbb{G})$ into $\mathrm{L}^p(\mathbb{G})$ for $1 < p \leq \infty$ where $\mathbb{G}$ is a co-amenable locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. Indeed, our approach even allows to use convolution operators on quantum hypergroups. This enable us to connect equally the topic of computation of entropies and capacities to subfactor planar algebras. We also give a upper bound of the classical capacity of each considered quantum channel which is already sharp in the commutative case. Quite surprisingly, we observe by direct computations that some Fourier multipliers identifies to direct sums of classical examples of quantum channels (as dephasing channel or depolarizing channels). Indeed, we show that the study of unital qubit channels can be seen as a part of the theory of Fourier multipliers on the von Neumann algebra of the quaternion group $\mathbb{Q}_8$. Unexpectedly, we also connect ergodic actions of (quantum) groups to this topic of computation, allowing some transference to other channels. Finally, we investigate entangling breaking and $\mathrm{PPT}$ Fourier multipliers and we characterize conditional expectations which are entangling breaking.

Related papers

• Hybrid Oscillator-Qubit Quantum Processors: Simulating Fermions, Bosons, and Gauge Fields [31.51988323782987]
  We develop a hybrid oscillator-qubit processor framework for quantum simulation of strongly correlated fermions and bosons. This framework gives exact decompositions of particle interactions as well as approximate methods based on the Baker-Campbell Hausdorff formulas. While our work focusses on an implementation in superconducting hardware, our framework can also be used in trapped ion, and neutral atom hardware.
  arXiv  Detail & Related papers  (2024-09-05T17:58:20Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
  We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model. We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
  arXiv  Detail & Related papers  (2024-05-14T15:28:37Z)
• Tensor cumulants for statistical inference on invariant distributions [49.80012009682584]
  We show that PCA becomes computationally hard at a critical value of the signal's magnitude. We define a new set of objects, which provide an explicit, near-orthogonal basis for invariants of a given degree. It also lets us analyze a new problem of distinguishing between different ensembles.
  arXiv  Detail & Related papers  (2024-04-29T14:33:24Z)
• Entanglement-assisted classical capacities of some channels acting as radial multipliers on fermion algebras [0.0]
  We investigate a new class of unital quantum computation channels on $mathrmM_2k$. We identify the matrix algebra $mathrmM_2k$ with a finite-dimensional fermion algebra. Our calculations yield exact values applicable to the operators of the fermionic Ornstein-Uhlenbeck semigroup.
  arXiv  Detail & Related papers  (2024-02-23T16:58:31Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Primitive Quantum Gates for Dihedral Gauge Theories [0.0]
  We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$ serves as an approximation to $U(1)timesbbZ$ lattice gauge theory.
  arXiv  Detail & Related papers  (2021-08-30T15:16:47Z)
• Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations [3.8117315001626966]
  partitioning of the Hilbert space greatly reduces the number of variational operators. A single-term representation suffices to reach required accuracy for various molecules tested. The number of controlled-NOT gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35.
  arXiv  Detail & Related papers  (2020-06-19T16:28:26Z)
• 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)
• Quantum Fourier Analysis [1.776439648597615]
  Quantum Fourier analysis is a new subject that combines an algebra with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information.
  arXiv  Detail & Related papers  (2020-02-10T00:25:53Z)

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.