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Semiclassical Geometric Tensor in Multiparameter Quantum Information.

Satoya Imai1,2,3,4, Jing Yang5,6, Luca Pezzè3,4

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Summary
This summary is machine-generated.

This study introduces the semiclassical geometric tensor (SCGT) to geometrically frame the quantum-classical Fisher information gap. The SCGT sharpens quantum information bounds and extends the Berry phase concept.

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Area of Science:

  • Quantum Information Theory
  • Quantum Geometry
  • Mathematical Physics

Background:

  • The quantum and classical Fisher information matrices (QFIM and CFIM) quantify distinguishability in quantum and classical systems, respectively.
  • A fundamental gap exists between QFIM and CFIM, representing an intrinsic quantum obstruction in information processing.

Purpose of the Study:

  • To develop a geometrical framework for understanding the quantum-classical Fisher information gap.
  • To introduce and analyze the semiclassical geometric tensor (SCGT) and its relation to the quantum geometric tensor (QGT).

Main Methods:

  • Introduction of the semiclassical geometric tensor (SCGT) as a novel geometrical tool.
  • Relating SCGT to the quantum geometric tensor (QGT).
  • Proving a matrix inequality between QGT and SCGT.

Main Results:

  • The SCGT provides a sharpened inequality between QFIM and CFIM, offering novel multiparameter information bounds.
  • The real part of SCGT equals CFIM plus a non-negative term quantifying quantum obstruction.
  • A natural extension of the Berry phase to the semiclassical setting is motivated.

Conclusions:

  • The SCGT offers a powerful geometrical perspective on quantum information obstructions.
  • This framework enhances our understanding of quantum distinguishability and its bounds.
  • The study paves the way for new developments in quantum metrology and quantum phase generalization.