Lecture on Quantum Phase Transitions and Quantum Information Theory

Introduction to Quantum Phase Transitions

• Quantum phase transitions are driven by quantum fluctuations at zero temperature.
• Classifications:
  • First Order Quantum Phase Transition:
    • Energy level crossings at a critical point (gc).
  • Second Order Quantum Phase Transition:
    • Avoided energy level crossings at gc.
• Verified in architectures such as cold atoms and quantum dots.

Quantum Analog of the Ising Model

• Describes quantum many-body systems using Pauli spin matrices and external magnetic field.
• At zero temperature, described by a complex ground state wave function.
• Emergence of phases such as superconductivity and magnetism.
• Analyzed using quantum information theoretic measures.

Notions of Entanglement and Quantum Correlations

• Entanglement:
  • Introduced in 1935 by Einstein, Podolsky, and Rosen.
  • Cannot describe properties of entangled particles independently.
  • Used in applications like quantum communication, cryptography, and teleportation.
• Quantum Discord:
  • Introduced to quantify quantum correlations beyond entanglement.
  • Measures difference between total correlations and classical correlations.
• Quantum Coherence:
  • Measures coherence using quantum Jensen-Shannon divergence.

XY Model Analysis

• Describes a lattice of spin-1/2 particles with interaction parameter γ and magnetic field h.
• Second-order phase transition at hc = 1.
• Studies long-range quantum correlations between first site and its neighbors.

Wigner Function Approach

• Represents quantum states in phase space using Wigner function.
• Wigner function can detect entanglement and transitions in quantum systems.
• Analyzed using phase operators and displacement operator.
• Demonstrated for complex systems like the XXZ model.

Non-Equilibrium Phase Transitions

• Studied using the thermodynamic approach due to the connection between information and thermodynamics.
• Utilizes stochastic thermodynamics to treat non-equilibrium settings.

LMG Model (Lipkin-Meshkov-Glick Model)

• Collective spin model representing infinite connectivity limit of the Ising model.
• Second-order phase transition studied through various scenarios of magnetic field quenches.
• Quantified using Loschmidt echo and work probability distribution.

Summary of Findings

• Quantum information theoretic measures like entanglement, quantum discord, and quantum coherence can detect critical properties of quantum systems.
• Wigner function successfully detects various quantum phase transitions including topological phase transitions.
• Future work aims to apply these analyses to quantum thermal machines and study energetic advantage in quantum computation.

Research Outputs

• Results were published in international journals and presented at conferences in Morocco, Italy, Austria, Portugal, and Japan.

Future Directions

• Study how critical phenomena can enhance the efficiency of quantum thermal machines.
• Explore the energetic advantage of quantum computation and tensor networks.

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