Historical research guide · 0877 · 2008
Quantum measurement information as a key to energy extraction from local vacuums
Masahiro Hotta
In one minute
Masahiro Hotta, at Tohoku University, shows how to draw energy out of a patch of vacuum that has no energy in it. The protocol has four steps. Alice measures the quantum field at her location. Measuring costs her energy — poking the vacuum locally always adds energy, never removes it — and what she puts in leaves as wavepackets travelling at the speed of light. Bob, standing further along, lets them go past without catching any of them. The field around him is now indistinguishable from vacuum: every measurement he could make there returns the vacuum’s answer. Then Alice tells him what her result was, over an ordinary classical channel, no faster than light. Bob applies the one local operation her answer selects, and his apparatus collects real positive energy out of a region that a moment earlier had none. What is left at Bob’s position is a dip below the vacuum level, which sets off after Alice’s wavepackets and stays loosely bound to them.
Connection
Chapter 6 asks what it would actually take to get energy out of the ground state, and this is the paper that answers it in a relativistic field rather than a lattice of spins: not a local pump, which the vacuum forbids, but a measurement here, a classical message, and a conditioned operation there. It gives chapter 2 a precise account of what a region below the ambient vacuum level is — a suppressed fluctuation, tethered to the surplus that paid for it — and chapter 13 the cleanest statement of measurement information itself as the thing being spent.
From the source
A local general measurement performed on the vacuum always costs energy: summing the vacuum expectation of the Hamiltonian over the measurement operators gives a strictly positive input, and that energy leaves Alice’s position as wavepackets propagating to spatial infinity at light speed. This is why nobody extracts zero-point energy at a single spot, and it is the input side of the protocol’s ledger.Section 4, phase 1 and the expression for the input energy
After Alice’s wavepackets have passed Bob, the state around him is a local vacuum state — Knight’s strictly localized state — in which every many-point function of the field in Bob’s vicinity equals the vacuum’s, and the expectation value of the energy operator windowed on his region is exactly zero. The region is not merely low in energy; it is indistinguishable from vacuum by any local measurement.Section 4, phase 2 and the discussion following Eq. 25
Bob’s move is a local unitary that displaces the field’s conjugate momentum by a function localized at his position, with an amplitude selected by Alice’s result. The windowed energy at Bob then becomes a quadratic in the coupling with a linear cross term built from a vacuum correlation between Alice’s measurement operator and Bob’s momentum operator; choosing the coupling to minimize it drives that energy strictly below zero, to minus the square of the cross term over twice the quadratic coefficient — and the matching positive energy is released from the field to Bob’s apparatus.Section 4, Eqs. 17 to 25
Nothing outruns light and nothing is free. The only thing that travels from Alice to Bob is a classical measurement result, which cannot arrive sooner than light; the energy Bob collects was paid in at Alice’s measurement. And the wavepackets Bob leaves behind, which sit below the vacuum level, cannot travel independently — a region below the vacuum level exists only while it is correlated with a region above it, so Bob’s packets chase Alice’s and form loosely bound states with them. Hotta notes that in more than one spatial dimension, where Alice’s packets need not be isotropic and Bob may sit where none of them pass, this binding becomes a distinctly non-trivial phenomenon.Section 4, closing discussion
The protocol is worked all the way through for an explicit two-valued general measurement built from cosine and sine of a Hermitian operator that smears the field momentum against a localized profile at Alice. The post-measurement states are superpositions of two coherent states; the energy Alice must supply is the integral of the squared gradient of her profile; and Bob’s yield has a closed form whose kernel falls off as the inverse square of the separation between the two profiles, so the extractable energy drops as Bob is placed further away.Section 5, Eqs. 26 to 37
Hotta names the next step and where it would be tested. The construction extends to the 3+1 dimensional electromagnetic field in the Coulomb gauge, with Alice’s measurement operator smearing the electric field against a three-dimensional vector profile and Bob’s unitary smearing the vector potential against a divergence-free one, as residual gauge symmetry requires — and his own assessment is that experimental checks of the protocol may be promising in quantum optics.Section 5, final paragraph
The way in
https://doi.org/10.1103/PhysRevD.78.045006How to cite it
Masahiro Hotta (2008) Quantum measurement information as a key to energy extraction from local vacuums doi:10.1103/PhysRevD.78.045006