High-Dimensional Entanglement in States with Positive Partial Transposition.

Area of Science:

• Quantum Information Theory
• Entanglement Theory
• Quantum Communication

Background:

• High-dimensional entanglement is crucial for quantum communication, surpassing low-dimensional system limitations.
• Positive partial transpose (PPT) states are typically considered weakly entangled and cannot be distilled into pure entangled states.

Purpose of the Study:

• To investigate the possibility of high Schmidt numbers in PPT states.
• To construct explicit families of PPT states with high Schmidt numbers and analyze their properties.
• To address conjectures regarding PPT states and their Schmidt number behavior.

Main Methods:

• Explicit construction of PPT states with linear scaling in Schmidt number.
• Probabilistic analysis of random PPT states.
• Connecting Schmidt number to entangled sub-block matrices.
• Proving bounds on Schmidt numbers for specific PPT state classes.

Main Results:

• First explicit construction of PPT states exhibiting linear scaling of Schmidt number with local dimension.
• Demonstration that random PPT states typically share this linear scaling property.
• Proof of a conjecture by Chen et al. on PPT states with arbitrarily large Schmidt number increase upon partial transposition.
• Establishment of a link between Schmidt number and entangled sub-block matrices.
• Proof that certain PPT states (transposition-invariant or absolutely PPT) cannot achieve maximal Schmidt number.

Conclusions:

• PPT states can exhibit genuine high-dimensional entanglement, challenging the notion of them being only weakly entangled.
• The findings provide a new resource for quantum communication and deepen the understanding of entanglement theory.
• New insights into the fundamental properties and limitations of entanglement in quantum states were revealed.