Detecting the inseparability and distillability of continuous variable states in Fock space.

Area of Science:

• Quantum Information Theory
• Quantum Optics
• Mathematical Physics

Background:

• The partial transposition (PT) operation is crucial for identifying inseparability in mixed quantum states.
• Continuous variable states and Gaussian operators are fundamental in quantum information processing.
• Assessing the inseparability and distillability of quantum states is key to quantum communication and computation.

Purpose of the Study:

• To develop an explicit formula for the partial transposition (PT) operation on continuous variable states in Fock space.
• To establish necessary and sufficient conditions for the positivity of Gaussian operators.
• To derive criteria for the inseparability and distillability of multimode Gaussian states and their subspaces.

Main Methods:

• Derivation of an explicit formula for the partial transposition (PT) operation in Fock space.
• Analysis of the conditions for the positivity of Gaussian operators.
• Application of these conditions to establish inseparability and distillability criteria for Gaussian states.

Main Results:

• An explicit formula for the PT operation on continuous variable states is presented.
• A necessary and sufficient condition for the positivity of Gaussian operators is established.
• New criteria for the inseparability and distillability of multimode Gaussian states and states within subspaces of global Gaussian states are derived.

Conclusions:

• The developed PT formula provides a powerful tool for analyzing quantum state properties.
• The established conditions offer a rigorous framework for evaluating Gaussian operators.
• The derived criteria advance the understanding and manipulation of entanglement in complex quantum systems.