First Law of Holographic Complexity

Area of Science:

• Theoretical Physics
• Quantum Gravity
• Holographic Principle

Background:

• Holographic complexity quantifies quantum information in quantum field theories.
• Nielsen's geometric approach provides a method to calculate complexity.
• The complexity=action conjecture relates spacetime geometry to quantum complexity.

Purpose of the Study:

• To investigate the variation of holographic complexity between quantum states.
• To derive a fundamental law governing holographic complexity changes.
• To test the complexity=action conjecture in a specific physical scenario.

Main Methods:

• Applying Nielsen's geometric approach to calculate holographic complexity.
• Analyzing the complexity variation for perturbed anti-de Sitter vacuum states.
• Examining scalar field excitations in the context of the complexity=action conjecture.

Main Results:

• The variation of holographic complexity depends solely on the endpoint of the optimal trajectory.
• This finding is termed the 'first law of complexity'.
• In the examined case, gravitational contributions to complexity variation cancel out, leaving only boundary terms from the scalar field action.

Conclusions:

• The 'first law of complexity' offers a new perspective on holographic complexity.
• The null boundary of the Wheeler-DeWitt patch acts as a quantum circuit's endpoint.
• Scalar field actions play a crucial role in determining holographic complexity variations.