Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

Area of Science:

• Quantum Information Science
• Quantum Foundations
• Quantum Computing

Background:

• Understanding the resources required for quantum correlations is crucial in quantum information science.
• Bipartite quantum systems and their correlations are fundamental to quantum mechanics.
• Quantifying the minimum Hilbert space dimension is key to characterizing quantum correlations.

Purpose of the Study:

• To identify an easily computable lower bound for the Hilbert space dimension required to generate a given two-party quantum correlation.
• To assess the tightness of this bound on known quantum correlations.
• To explore the bound's utility in ruling out finite-dimensional quantum representations and detecting non-convexity.

Main Methods:

• Developing a novel mathematical framework to establish a lower bound on Hilbert space dimension.
• Testing the bound's efficacy against established quantum correlations.
• Analyzing the bound's properties, including multiplicativity under product correlations.

Main Results:

• An easy-to-compute lower bound on the smallest Hilbert space dimension for generating two-party quantum correlations was identified.
• The bound was demonstrated to be tight for many common correlations.
• The bound successfully ruled out certain correlations from having finite-dimensional quantum representations.
• The bound was shown to be multiplicative for product correlations and capable of detecting non-convexity.

Conclusions:

• The developed lower bound provides a practical tool for resource quantification in quantum information.
• This method advances the understanding of the relationship between quantum correlations and system dimensionality.
• The findings have implications for the study of quantum complexity and the limitations of quantum representations.