Historical research guide · 0654 · 2019
Fundamental Limitations to Local Energy Extraction in Quantum Systems
Álvaro M. Alhambra · Georgios Styliaris · Nayeli A. Rodríguez-Briones · Jamie Sikora · Eduardo Martín-Martínez
In one minute
Álvaro Alhambra, Georgios Styliaris, Nayeli Rodríguez-Briones, Jamie Sikora and Eduardo Martín-Martínez ask a question anyone designing a vacuum-energy device eventually meets: when a system is strongly coupled and its low-energy states are entangled, how much energy can you draw out by touching only one side of it? They give the complete answer. Using semidefinite programming — an optimisation method borrowed from quantum information and applied here for the first time to energy extraction — they reduce the whole question to one operator inequality. Satisfy it and no local operation of any kind lowers the system’s energy; fail it and the same calculation hands you the best operation and the exact energy it yields. They then make the result physical, bounding the ground-state population and the temperature below which this local passivity switches on, and showing it survives to arbitrarily large baths whenever spatial correlations fall off fast enough. The closing move matters most: let the two sides exchange classical messages and the door opens again.
Connection
Chapter 6 is about getting usable energy out of a strongly coupled quantum system, and this is the paper that draws the exact boundary of the purely local version of that move — and then names the ingredient, classical communication between the two regions, that puts the energy back on the table. Chapter 2’s picture of a correlated, entangled ground state is precisely what makes the theorem bite.
From the source
There is a necessary and sufficient condition for it to be impossible to extract any energy from a bipartite quantum system by acting on one side alone with the most general local operation. It takes the form of a single operator inequality on a matrix the size of that subsystem, built from the state and the global Hamiltonian, and it can be checked directly.Theorem 1, Eq. 7
The maximum energy extractable by a local operation is a semidefinite program, so it is not merely bounded but computable — and when the system is not passive the same program returns the optimal extracting map. This is the first use of semidefinite programming in the context of energy extraction and passivity.Setting, Eq. 1; Discussion, second paragraph
For a Hamiltonian whose ground state is non-degenerate and of full Schmidt rank, the authors give an explicit upper bound on the ground-state population above which local extraction becomes impossible, written in terms of the ground-state entanglement and the gap to the first excited state — and a corresponding finite threshold temperature strictly above absolute zero.Theorem 2, Eq. 8; threshold temperature, Eq. 9
The result extends to the thermodynamic limit: for a thermal state on a lattice whose correlations cluster, a shielding region of finite thickness is enough, so the passivity of the patch you act on is set by its neighbourhood rather than by the whole bath, and numerical examples show the threshold temperature converging as the system grows.Theorem 3, Eqs. 12 to 14; Fig. 2
The effect is essentially quantum. In a fully classical setting — an incoherent state with product energy eigenstates — this passivity can arise only under a strong restriction on the support of the state or a highly degenerate Hamiltonian, and no thermal state of full support with any non-degeneracy obeys it. Frustration and an entangled ground state are what make it real; the bound also weakens as the region you act on grows.Classical CP-local passivity, Eqs. 16 to 21; frustration bound, Eq. 11
The authors name the way past their own no-go: allow classical communication between the two regions and the setting becomes quantum energy teleportation, and they propose their necessary and sufficient conditions as a design tool for better teleportation-based protocols, already applied in quantum field theory and in algorithmic cooling.Discussion, final paragraph
The way in
https://doi.org/10.1103/PhysRevLett.123.190601How to cite it
Álvaro M. Alhambra, Georgios Styliaris, Nayeli A. Rodríguez-Briones, Jamie Sikora, Eduardo Martín-Martínez (2019) Fundamental Limitations to Local Energy Extraction in Quantum Systems doi:10.1103/PhysRevLett.123.190601