[Paper Review] Limit on Time-Energy Uncertainty with Multipartite Entanglement

Manabendra Nath Bera, R. Prabhu+3 more|arXiv (Cornell University)|Mar 4, 2013

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper establishes a fundamental lower bound on the time-energy uncertainty relation in quantum systems, showing it is directly constrained by the geometric measure of multipartite entanglement. Using geometric quantum uncertainty relations based on Fubini-Study, Hilbert-Schmidt, and Bures metrics, the authors derive that the product of the time-averaged energy fluctuation and evolution time is bounded below by the multipartite entanglement of the target state, valid for both pure and mixed states.

ABSTRACT

We establish a relation between the geometric time-energy uncertainty and multipartite entanglement. In particular, we show that the time-energy uncertainty relation is bounded below by the geometric measure of multipartite entanglement for an arbitrary quantum evolution of any multipartite system. The product of the time-averaged speed of the quantum evolution and the time interval of the evolution is bounded below by the multipartite entanglement of the target state. This relation holds for pure as well as for mixed states. We provide examples of physical systems for which the bound reaches close to saturation.

Motivation & Objective

• To investigate whether multipartite entanglement imposes a fundamental lower bound on quantum evolution time.
• To establish a quantitative connection between geometric quantum uncertainty and multipartite entanglement in many-body quantum systems.
• To generalize the time-energy uncertainty relation to include entanglement as a physical constraint, beyond Planck’s constant.
• To validate the bound using physical models such as the Heisenberg spin chain and various quantum metrics.

Proposed method

• Derives the geometric quantum uncertainty relation (GQUR) using the Fubini-Study metric for pure states, linking the total path length in projective Hilbert space to energy fluctuations.
• Applies the Hilbert-Schmidt metric to extend the GQUR to mixed states, showing the bound remains independent of the specific metric used.
• Introduces the Bures metric to define a distance-based measure of multipartite entanglement, particularly for genuine multipartite entanglement.
• Uses time-independent and time-dependent Hamiltonians to derive the uncertainty relation: τΔH ≥ ħ𝐺(ℰ), where 𝐺 is the geometric measure of entanglement.
• Applies the relation to physical systems like the Heisenberg spin chain to demonstrate saturation of the bound.
• Employs fidelity-based measures (Uhlmann fidelity) and trace distance to quantify entanglement and relate it to the geometric uncertainty.

Experimental results

Research questions

• RQ1 Can multipartite entanglement serve as a fundamental lower bound on the time-energy uncertainty relation in quantum dynamics?
• RQ2 How does the geometric measure of multipartite entanglement constrain the minimum evolution time of a quantum state?
• RQ3 Does the time-energy uncertainty relation remain bounded by entanglement when generalized to mixed quantum states?
• RQ4 To what extent can physical systems like the Heisenberg spin chain saturate the derived uncertainty bound?
• RQ5 Is the bound robust across different Riemannian metrics (Fubini-Study, Hilbert-Schmidt, Bures)?

Key findings

• The time-energy uncertainty relation is bounded below by the geometric measure of multipartite entanglement, with the relation τΔH ≥ ħ𝐺(ℰ) holding for both pure and mixed states.
• The bound is independent of the choice of Riemannian metric, as demonstrated for Fubini-Study, Hilbert-Schmidt, and Bures metrics.
• For the Heisenberg spin chain model, the bound is shown to be close to saturation, indicating physical realizability of the theoretical limit.
• The geometric measure of entanglement 𝐺(ℰ) quantifies the minimum time required for a quantum state to evolve to a target entangled state, given a fixed energy fluctuation.
• The relation τΔH ≥ ħ𝐺(ℰ) generalizes the standard time-energy uncertainty principle by incorporating entanglement as a physical constraint.
• The use of Uhlmann fidelity in the Bures metric formulation enables a robust, contractive measure of entanglement that supports the uncertainty bound.