The Bogoliubov inner product in quantum statistics

  title={The Bogoliubov inner product in quantum statistics},
  author={D{\'e}nes Petz and G{\'a}bor Fejes T{\'o}th},
  journal={Letters in Mathematical Physics},
A natural Riemannian geometry is defined on the state space of a finite quantum system by means of the Bogoliubov scalar product which is infinitesimally induced by the (nonsymmetric) relative entropy functional. The basic geometrical quantities, including sectional curvatures, are computed for a two-level quantum system. It is found that the real density matrices form a totally geodesic submanifold and the von Neumann entropy is a monotone function of the scalar curvature. Furthermore, we… Expand
Geometry of canonical correlation on the state space of a quantum system
A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by theExpand
Quantum Statistical Manifolds
  • J. Naudts
  • Computer Science, Medicine
  • Entropy
  • 2018
Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection. Expand
2 8 M ay 2 01 8 Quantum Statistical Manifolds
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more generalExpand
Positive definite matrices of trace 1 describe the state space of a finite quantum system. This manifold can be endowed by the physically very relevant Bogoliubov-Kubo-Mori inner product as aExpand
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We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physicalExpand
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An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtainedExpand
For certain unitary representations of a Lie algebra we define the statistical manifold ℳ of states as the convex cone of for which the partition function Z=Trexp{-X} is finite. The Hessian of Ψ=logExpand
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In this paper, we study a family of quantum Fisher metrics based on a convex mixture of two well-known inner products, which covers the well-known symmetric logarithmic derivative, the rightExpand
Positive definite matrices of trace 1 describe the state space of a finite quantum system. This manifold can be endowed by the physically relevant Bogoliubov–Kubo–Mori inner product as a RiemannianExpand


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  • H. Yuen, M. Lax
  • Mathematics, Computer Science
  • IEEE Trans. Inf. Theory
  • 1973
It is shown that the optimal receiver measures the photon annihilation operator, which corresponds to optical heterodyning, demonstrating the possible optimality of nonselfadjoint operators and clearly indicates the importance of considering more general quantum measurements in quantum signal detection. Expand
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  • H. Yuen
  • Computer Science
  • IEEE Transactions on Information Theory
  • 1977
This volume contains advanced materials of queueing theory plus three application chapters, and gives a comprehensive account of the diffusion process approximation and heavy traffic theory as applied to queueing systems. Expand
Inequalities for the trace function
DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen,Expand