[Paper Review] Using unitary operations to preserve quantum states in the presence of relaxation

Benjamin Recht, Yael Maguire+3 more|ArXiv.org|Oct 11, 2002

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper proposes a method to preserve specific quantum states in relaxing open quantum systems using unitary control Hamiltonians, shifting the system's fixed point to a desired non-equilibrium state. By applying tailored local unitary operations, the authors stabilize a highly entangled two-spin state with 0.355 ebits of entanglement of formation, even under Markovian relaxation, without requiring quantum error correction or ancilla qubits.

ABSTRACT

When a quantum system interacts with an external environment, it undergoes the loss of quantum correlation (decoherence) and the loss of energy (relaxation) and eventually all of the quantum information becomes classical. Here we show a general principle to use unitary operations to establish and preserve particular non-equilibrium states in arbitrary relaxing quantum systems. We elucidate these concepts with examples of state preservation in one-spin and two-spin entangled systems.

Motivation & Objective

• To address the challenge of preserving quantum coherence and entanglement in open quantum systems subject to relaxation and decoherence.
• To develop a control strategy that stabilizes specific non-equilibrium quantum states in relaxing semigroup dynamics, bypassing the need for quantum error correction.
• To demonstrate that unitary operations can shift the fixed point of a relaxing quantum system to a desired state, enabling long-term preservation.
• To provide a framework for stabilizing entangled states in multi-spin systems under Markovian dissipation using local Hamiltonian control.
• To explore the geometric structure of stabilizable states in higher-dimensional quantum systems, particularly for two-spin entangled states.

Proposed method

• Model the open quantum system using the Lindblad equation with a Hamiltonian part (A) and a dissipative part (B), represented in coherence vector form as dr/dt = (A + B)r + c.
• Identify relaxing semigroups where the system evolves toward a unique fixed point r_f = -(A + B)^(-1)c under Markovian dynamics.
• Apply a time-dependent control Hamiltonian H_c to modify the system's dynamics, shifting the fixed point to r_f^c = -(A_c + B)^(-1)c.
• Use local unitary operations on individual spins to engineer the control Hamiltonian H_c, enabling stabilization of a target entangled state.
• Solve the resulting system of 15 linear equations for the coherence vector to determine the fixed point under the controlled dynamics.
• Apply asymptotic analysis as J → ∞ to derive the limiting fixed point state ρ_e = ½|↑↑⟩⟨↑↑| + ½|ψ₂⟩⟨ψ₂|, where |ψ₂⟩ = (|↑↓⟩ + |↓↑⟩)/√2.

Experimental results

Research questions

• RQ1 Can unitary control Hamiltonians stabilize a desired non-equilibrium quantum state in a system undergoing relaxation?
• RQ2 What conditions allow the fixed point of a relaxing quantum semigroup to be shifted via unitary control?
• RQ3 Is it possible to preserve a specific entangled two-spin state with high entanglement of formation under Markovian dissipation?
• RQ4 How does the geometry of stabilizable states in higher-dimensional systems relate to the structure of the Lindblad generators?
• RQ5 Can local control operations on individual spins be used to stabilize a globally entangled state in a two-spin system?

Key findings

• Unitary control Hamiltonians can shift the fixed point of a relaxing quantum system to a desired state, enabling long-term preservation without quantum error correction.
• For a two-spin system with independent damping to the |↑↑⟩ state, the fixed point under control Hamiltonian H_c = (4√J/5)(X₁ + X₂) - J(Z₁ + Z₂) converges to ρ_e = ½|↑↑⟩⟨↑↑| + ½|ψ₂⟩⟨ψ₂| as J → ∞.
• The stabilized state ρ_e has an entanglement of formation of 0.355 ebits, demonstrating the preservation of significant quantum entanglement.
• The convergence rate to the fixed point is robust even for moderate J/γ ratios, indicating practical feasibility for small-scale systems.
• The method avoids the need for high-fidelity ancilla qubits or error thresholds, offering an alternative to quantum error correction for relaxation-dominated systems.
• The framework generalizes to arbitrary relaxing semigroups, showing that a large submanifold of quantum states can be stabilized using unitary control.