Historical research guide · 0662 · 2008
Quantum Energy Teleportation in Spin Chain Systems
Masahiro Hotta
In one minute
This is the paper that started quantum energy teleportation. Masahiro Hotta points out that ordinary teleportation moves a quantum state but not its energy — Bob still has to pay for the excitation locally — and then shows that energy itself can be moved using nothing but local operations and a phone call. His stage is a spin chain: many spins in a row, coupled to their neighbours, whose ground state is entangled across long distances. Alice measures one spin. Measuring costs her energy, which stays parked around her site, and she cannot take it back, because her measurement broke entanglement that only a non-local operation could rebuild. She tells Bob the one-bit result down an ordinary classical channel. Bob applies the single rotation that result selects, the energy around him drops below the ground-state level, and his device collects the difference as real, usable energy. Nothing travelled between them but a bit of information.
Connection
Chapter 6 asks what it would really take to draw energy out of a ground state, and this is where the answer was first written down: not a local pump, which the ground state forbids, but a measurement here, a classical message, and a conditioned operation there. It gives chapter 2 a concrete lattice picture of a region sitting below the ambient vacuum level — a fluctuation suppressed by interference, paid for elsewhere, with the system’s total energy still non-negative — and chapter 13 the claim that entanglement plus classical information is the resource being spent. The field-theory version of the same protocol is on this site at /library/stm-e7e45ffaae, and Hotta’s introductory review at /library/stm-5aa6b8b006.
From the source
Taking the zero of energy density to be the ground state’s own expectation value, regions below that level appear naturally. Local quantum fluctuation survives in the ground state, and by superposing eigenstates of the total Hamiltonian one can suppress that fluctuation more strongly in a small region than the ground state does, which puts the local energy density there below the reference level. The total energy of the system nevertheless stays non-negative — there is no state lower than the ground state.Introduction; Section 2, Ground-State Entanglement and Negative Energy Density
Alice’s local projective measurement on the ground state necessarily costs energy. The input is the sum over outcomes of the ground state’s expectation of the projected Hamiltonian, which is non-negative because the Hamiltonian is, and generically strictly positive because entanglement means the projected state is not proportional to the ground state. That energy stays localized: the expectation of every energy density term further than the interaction range from her site remains zero.Section 4, Eqs. 24 to 26
Alice cannot simply take her own deposit back. Whatever trace-preserving completely positive map she applies to her own spin, a strictly positive residual energy is left behind, because her measurement broke the entanglement between her spin and the rest of the chain and entanglement generation needs non-local operations. The energy is right in front of her and locally unreachable.Section 4, final paragraphs
The protocol itself has three steps: Alice measures a local spin observable and obtains a one-bit result; she announces it to Bob over a classical channel; Bob applies to his own spin a local unitary built from the identity and a Pauli operator, with a mixing angle fixed by two ground-state expectation values. The energy around Bob, which was exactly zero beforehand, then sits below the ground-state level, and local energy conservation requires that a positive energy equal to half of the difference between the root of the sum of squares of those two parameters and the first of them be released from the chain to Bob’s device.Section 5, Eqs. 27 to 33, with Eqs. 37 and 38 and Figures 1 to 4
Ground-state entanglement is the resource, and it sets the range. The parameter that carries Bob’s yield is a two-point correlation function between Alice’s measured observable and the time derivative of Bob’s: if the ground state is separable it factorises to zero and Bob gains nothing at all. Correlations decay over a characteristic length, so the protocol works best when the two sites are within that length of each other, and near-critical chains with long-range correlation — the transverse-field Ising chain among them — are the good candidates.Section 5, Eqs. 31 and 34 and the closing discussion
Nothing crosses the gap but a bit of classical information, and the books balance. The energy Bob collects was already sitting around Bob, hidden in the ground-state fluctuation; Alice’s result is the key that tells him which rotation extracts rather than adds; and Alice cannot afterwards withdraw more than her input minus Bob’s gain, since the total energy of the chain cannot go negative. What to watch: because what is transported is classical information rather than an excited physical entity, Hotta expects the dissipation rate in transport to be strongly suppressed — the loss figure a working implementation would need to measure.Abstract; Section 6, Conclusion
The way in
https://arxiv.org/abs/0803.0348How to cite it
Masahiro Hotta (2008) Quantum Energy Teleportation in Spin Chain Systems doi:10.1143/JPSJ.78.034001