Koopmanian System Coupling Prevents Entanglement Between Two Qubits, Violates Conservation Laws

The fundamental nature of entanglement, a cornerstone of quantum mechanics, continues to challenge our understanding of how information connects physical systems, and researchers are now exploring whether this uniquely quantum phenomenon can be replicated, or even mediated, by classical systems. Chiara Marletto and Vlatko Vedral, both from the Clarendon Laboratory at the University of Oxford, investigate this question by proposing a method to connect two quantum bits, or qubits, through a classical system governed by Koopmanian dynamics. Their work demonstrates that while a combined quantum and classical system can exhibit complex behaviour, it fundamentally cannot generate entanglement between the qubits, revealing a key distinction between classical and quantum information processing. This finding clarifies the limits of semi-classical approximations and offers insights into the essential requirements for creating and sustaining quantum correlations.

Quantum Information and Emergent Conservation Laws

Current research builds upon a long-standing quest to reconcile quantum mechanics with fundamental conservation laws, a problem explored since the work of Witten. Investigations suggest a connection between quantum mechanics and the preservation of physical quantities, proposing that conservation laws emerge as a consequence of the quantum nature of information itself. This challenges conventional thinking and opens new avenues for understanding the foundations of physics, particularly in scenarios where conservation laws appear to be broken. Recent publications continue to refine this informational interpretation of conservation laws. The implications of this research extend to the quantization of gravity, suggesting that the structure of spacetime, and its quantum behaviour, may be linked to the preservation of information, offering a potential pathway towards a consistent theory of quantum gravity.

Koopmanian Dynamics Prevent Qubit Entanglement

The researchers develop a method for coupling a Koopmanian classical system to two quantum bits, demonstrating that the resulting dynamics can never lead to entanglement between the qubits. The team analyses classical dynamics within the framework of quantum mechanics, based on the Koopmanian approach, to investigate hybrid quantum-classical systems using a consistent joint formalism.

Classicality Requires Quantum System Coupling

The research builds upon arguments demonstrating that if a sector of the universe exhibits quantum behaviour and couples with another sector, the latter cannot be purely classical. The team defines classicality as the requirement that all operators representing physical variables of a system must commute, encoding position and momentum into degrees of freedom pertaining to two separate quantum systems. The system’s dynamics are determined by exponentiating the Hamiltonian, demonstrating consistency with Newtonian dynamics. Galilean transformations are implemented using a unitary transformation, resulting in transformations consistent with classical mechanics due to the absence of additional quantum phases. Investigating coupling this classical system to two quantum bits, the researchers demonstrate that this Koopmanian mediator cannot generate entanglement between the qubits, suggesting a fundamental limitation in utilising this approach for quantum information processing.

Entanglement Limited by Momentum and Position Coupling

The Hamiltonian is defined, and the system’s dynamics are determined by exponentiating it. Entanglement between the qubits is impossible, a fact unaffected by adding a potential to the Koopmanian system or by other individual couplings, due to the classical condition implemented in the model where observables from the two systems commute. To achieve entanglement, a Hamiltonian violating the classical condition is required, engaging at least two non-commuting degrees of freedom. This demonstrates that classical mediators cannot entangle quantum systems, a result proven using the Constructor Theory.

The Koopmanian treatment suggests a consistent coupling between quantum and classical sectors, utilizing Hilbert spaces and commuting degrees of freedom. However, this consistency is illusory, as semi-classical models lack back-reaction from the quantum system onto the classical system. Consider a conservation law applying to the composite system, where a quantity is conserved. Only unitaries commuting with this conserved quantity are allowed, resulting in trivial local evolutions. Inducing a transition requires a Hamiltonian violating the classicality condition.

Explaining static interactions between charges using the field as a mediator is problematic because the Hamiltonian is linear, necessary for reproducing classical behaviour. Witnesses ruling out semiclassical models do not require assuming deterministic evolution, relying instead on performing sufficient measurements. In summary, a fully classical system coupled to two qubits cannot generate entanglement without engaging non-commuting degrees of freedom, reinforcing the conclusion that hybrid semiclassical models are useful approximations but cannot provide a fundamental model for physical reality. Acknowledgments: This research was supported by the Gordon and Betty Moore Foundation, the Eutopia Foundation, and the John Templeton Foundation, as part of The Quantum Information Structure of Spacetime (QISS) Project. The opinions expressed are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.