The Second Law of Quantum Complexity and the Entanglement Wormhole

TL;DR

This paper explores the relationship between quantum complexity, entropy, and wormhole geometry, proposing a second law of quantum complexity and linking it to the growth of Einstein-Rosen bridges in holography.

Contribution

It introduces the Second Law of Quantum Complexity and connects quantum complexity growth to wormhole interior expansion in AdS/CFT correspondence.

Findings

Quantum complexity tends to increase towards a maximum value.

The growth of wormhole volume correlates with quantum complexity.

A classical analogy helps relate entropy to quantum complexity.

Abstract

This work is originally a Cambridge Part III essay paper. Quantum complexity arises as an alternative measure to the Fubini metric between two quantum states. Given two states and a set of allowed gates, it is defined as the least complex unitary operator capable of transforming one state into the other. Starting with K qubits evolving through a k-local Hamiltonian, it is possible to draw an analogy between the quantum system and an auxiliary classical system. Using the definition of complexity to define a metric for the classical system, it is possible to relate its entropy with the quantum complexity of the K qubits, defining the Second Law of Quantum Complexity. The law states that, if it is not already saturated, the quantum complexity of a system will increase with an overwhelming probability towards its maximum value. In the context of AdS/CFT duality and the ER=EPR conjecture, the growth of the volume of the Einstein Rosen bridge interior is proportional to the quantum complexity of the instantaneous state of the conformal field theory. Therefore, the interior of the wormhole connecting two entangled CFT will grow as a natural consequence of the complexification of the boundary state.