Semiclassical Geometric Tensor in Multiparameter Quantum Information

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.