Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits.

Area of Science:

• Quantum Information Theory
• Quantum Many-Body Systems
• Quantum Computing

Background:

• The separability problem in quantum mechanics aims to distinguish between entangled and separable quantum states.
• Bound entangled states pose a challenge as they are entangled but cannot be distilled via local operations and classical communication.
• Existing criteria like the Peres-Horodecki criterion (or PPT criterion) fail to detect all forms of bound entanglement.

Purpose of the Study:

• To develop an efficient method for solving the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT).
• To classify bipartite qutrit states into separable, free entangled, and bound entangled categories.
• To determine the relative volumes of these state classes and compare the effectiveness of entanglement detection criteria.

Main Methods:

• Utilizing a geometrical representation of quantum states in Euclidean space.
• Developing novel classification methods for Bell diagonal PPT states.
• Analyzing the efficacy of various criteria in detecting bound entanglement.

Main Results:

• An efficient classification method for separable and bound entangled Bell diagonal PPT states was developed with 95% success.
• The study determined the precise volumes of separable ([Formula: see text]) and bound entangled ([Formula: see text]) states within the analyzed set.
• A comparison of entanglement detection criteria revealed that no single criterion can identify all bound entangled states.

Conclusions:

• The developed geometrical and classification methods provide an efficient solution to the separability problem for a significant class of quantum states.
• The findings quantify the prevalence of separable, free entangled, and bound entangled states, offering insights into their relative abundance.
• The study highlights the limitations of current entanglement detection criteria, emphasizing the need for more comprehensive methods to identify bound entanglement.