Quantum Energy Teleportation

Quantum energy teleportation (QET) is a correlation-assisted protocol for local energy extraction. Alice performs a local measurement and injects energy into a correlated ground or low-energy state. She sends the classical outcome to Bob through an ordinary causal channel. Bob then chooses a local operation that can lower the system’s energy near him, transferring that decrease to his apparatus. The energy Alice injected does not jump to Bob, and neither information nor usable energy travels outside the light cone.

Required background. Passivity, work, and information fixes the ground-state energy reference; measurement-induced energy, noise, and backreaction supplies the instrument account; and quantum energy inequalities constrain specified negative-energy averages in continuum models.

Helpful background. Vacuum entanglement harvesting distinguishes pre-existing correlations from excitations exchanged during an interaction.

The three-step protocol

Let $H\ge0$ be the regulated system Hamiltonian with ground state $\rho_0$ and $\operatorname{tr}(H\rho_0)=0$. Alice’s localized instrument has Kraus operators $M_a$ satisfying $\sum_aM_a^\dagger M_a=I$. Define the unnormalized conditional states

$$\widetilde\rho_a=M_a\rho_0M_a^\dagger, \qquad p_a=\operatorname{tr}\widetilde\rho_a.$$

The operational sequence is:

1. Alice measures in region $A$, stores $a$, and prepares $\widetilde\rho_a$.
2. A classical system carries $a$ from $A$ to $B$ no faster than light.
3. After receiving $a$, Bob applies a local unitary $U_a^B$ and couples the resulting system-energy decrease into a battery or work storage device.

The mean energy injected into the regulated system by Alice and extracted from it by Bob are

$$E_A=\sum_a\operatorname{tr}(H\widetilde\rho_a),$$ $$E_B=\sum_a\operatorname{tr}\!\left[ H\left(\widetilde\rho_a-U_a^B\widetilde\rho_aU_a^{B\dagger}\right) \right].$$

A successful QET stroke has $E_B>0$. The final system energy is

$$E_{\rm sys}^{f}=E_A-E_B\ge0,$$

so $E_B\le E_A$ in a ground-state protocol with a nonnegative total Hamiltonian. The idealized system account is not yet the apparatus account. Alice’s measurement device supplies $E_A$ to the system; Bob’s battery receives at most $E_B$; communication, record storage, conditional control, and reset have their own costs. For the delivery stroke,

$$W_{\rm delivery}^{\rm out} =E_B-E_A-W_{\rm message}-W_{\rm reset} \le0$$

when those auxiliary costs are nonnegative. The system still stores $E_A-E_B$. A later return to the ground state can recover at most that residual energy, so an ideal complete cycle has $W_{\rm cycle}^{\rm out}\le-W_{\rm message}-W_{\rm reset}\le0$ unless some additional resource is consumed. QET is a protocol for delivering locally extractable energy, not a perpetual-motion engine.

This measurement–message–operation structure is explicit in the original relativistic field proposal Hotta 2008, §§ II–IV and the spin-chain construction Hotta 2009, §§ 2–4.

Why classical feed-forward matters

Before the message arrives, Bob does not know $a$. In a relativistic theory, Alice’s nonselective local operation cannot change the expectation of a spacelike-separated Bob observable $O_B$. Local commutativity gives $[M_a,O_B]=0$, and therefore

$$\begin{aligned} \sum_a\operatorname{tr}(M_a\rho_0M_a^\dagger O_B) &=\operatorname{tr}\!\left( \rho_0\sum_aM_a^\dagger O_BM_a\right)\\ &=\operatorname{tr}(\rho_0O_B). \end{aligned}$$

Conditional states at $B$ can depend on $a$, but that dependence is inaccessible without the classical record. This is the same distinction between an ensemble and its labeled components that underlies no-signaling.

Correlations are the resource that makes the label useful. They need not be summarized by one universal entanglement measure, but the relevant conditional correlation must couple to Bob’s allowed operation. Removing cross-correlations while preserving the local marginals is a strong control in a separated model. Any signal that remains must be traced to direct interaction, propagation, a changed marginal, or another resource rather than to the claimed QET mechanism.

Exact two-qubit benchmark

The smallest analytically transparent model uses two qubits and natural units $\hbar=1$:

$$H=H_A+H_B+V,$$ $$\begin{aligned} H_A&=h\sigma_A^z+\frac{h^2}{\sqrt{h^2+k^2}},\\ H_B&=h\sigma_B^z+\frac{h^2}{\sqrt{h^2+k^2}},\\ V&=2k\sigma_A^x\sigma_B^x +\frac{2k^2}{\sqrt{h^2+k^2}}. \end{aligned}$$

The constants make the ground-state expectations of $H_A$, $H_B$, and $V$ separately zero, and the lowest eigenvalue of $H$ is zero. Alice measures $\sigma_A^x$ with

$$P_A(\alpha)=\frac{I+\alpha\sigma_A^x}{2}, \qquad \alpha=\pm1.$$

The mean injection is

$$E_A=\frac{h^2}{\sqrt{h^2+k^2}}.$$

After receiving $\alpha$, Bob applies

$$U_B(\alpha)=I\cos\theta-i\alpha\sigma_B^y\sin\theta,$$

where

$$\cos2\theta=\frac{h^2+2k^2} {\sqrt{(h^2+2k^2)^2+h^2k^2}}, \qquad \sin2\theta=\frac{hk} {\sqrt{(h^2+2k^2)^2+h^2k^2}}.$$

Optimizing over $\theta$ gives

$$E_B= \frac{\sqrt{(h^2+2k^2)^2+h^2k^2}-(h^2+2k^2)} {\sqrt{h^2+k^2}}>0.$$

These formulas are derived in Hotta 2011, § 3, Eqs. (5)–(14), Open PDF, based on the primary spin-chain protocol.

Set $h=k=1$. Then

$$\begin{aligned} E_A&=\frac{1}{\sqrt2}=0.7071068,\\ \theta&=\frac12\arctan\frac13=0.1608753,\\ E_B&=\frac{\sqrt{10}-3}{\sqrt2}=0.1147476,\\ E_A-E_B&=0.5923591. \end{aligned}$$

Thus Bob extracts about $16.23\%$ of the energy Alice injected into the two-qubit system. If Bob applies an outcome-independent unitary $W_B$, the averaged postmeasurement state cannot yield positive extraction in this model: its energy change reduces to the negative of a ground-state expectation of $W_B^\dagger H W_B$, which is nonpositive because $H\ge0$ Hotta 2011, § 3, Eq. (13). That is a local-passivity control, not a claim that every correlated state enables QET.

The two-qubit Hamiltonian contains a direct $A$–$B$ interaction and is nonrelativistic. It is an energy-accounting benchmark, not a demonstration of spacelike separation.

Relativistic timing and local energy

Let the centers of $A$ and $B$ be separated by distance $L$. If Alice’s measurement occurs near $t_A$, Bob’s outcome-dependent operation must satisfy

$$t_B-t_A\ge \frac{L}{c}$$

in Minkowski space. A relativistic QET geometry must additionally show that Bob’s output is not ordinary absorption of the positive-energy excitation created by Alice. At the time of $U_a^B$, that excitation must be absent from Bob’s interaction region—because it was directed elsewhere or because it has already passed without being captured. This separation cannot be manufactured by sending the classical record superluminally.

Bob’s local operation can leave the renormalized field energy near $B$ negative relative to the reference state while transferring positive energy to his apparatus. The compensating positive energy remains elsewhere, and the complete evolution must satisfy the applicable QEI or averaged-energy restrictions. A negative local density is not a negative total Hamiltonian.

The decisive causal checks are therefore distinct:

• No message: replace $U_a^B$ by one operation independent of $a$ and optimize again.
• No initial correlation: in a separated model, replace the state by one with the same local marginals but vanishing $A$–$B$ correlations, without introducing a direct coupling signal.
• No Alice injection: include the measurement device and verify that a claimed $E_B$ is not funded by an omitted change of apparatus or interaction energy.
• Timing control: compare $t_B$ with $L/c$ and with the full spacetime support of Alice-generated energy; verify that the excitation does not overlap Bob’s extraction operation.

From a lattice model to a continuum field

A regulated lattice protocol does not automatically represent the same operator sector as its continuum candidate. The comparison must hold fixed the measurement channel, charge sector, local energy definition, and coarse graining while the lattice spacing and volume are varied.

A concrete example is the massive Thirring/sine–Gordon analysis of Ikeda 2026. In the continuum, a weak trigonometric binary POVM has a leading signal governed by a neutral conserved-current correlator. The conventional lattice protocol instead accesses charged sectors because a lattice $U(1)$ selection rule removes that neutral contribution from Bob’s subsystem. A separately constructed lattice neutral-current protocol reproduces the coarse-grained current correlator, with extracted energy quadratic in measurement strength. This is a model-specific matching result, and it shows why agreement of Hamiltonian names or critical exponents is insufficient.

For a field-chain calculation, report at least

$$(a,L_{\rm box},L,\lambda,\epsilon_{\mathrm{meas}}, E_A,E_B,W_{\rm message},W_{\rm reset})$$

together with local-energy operators and sector projectors. Run the no-message and no-correlation controls at every lattice spacing. A stable $E_B$ without stable operator matching is not a continuum QET result.

What has been demonstrated

This evidence summary is current through 26 August 2026. The NMR experiment of Rodríguez-Briones et al. 2023 observed activation of a strongly locally passive bipartite state and implemented a finite-system QET protocol. Ikeda 2023 executed an encoded QET Hamiltonian on superconducting quantum hardware. These are genuine finite-device demonstrations, but neither is a direct laboratory extraction of energy from two spatially separated regions of a continuum relativistic field. The 2026 lattice–continuum work is a theoretical sector-matching result, not an experimental continuum realization.

The chapter orientation, claim-validity table, and failure controls compare these evidence levels with passivity and energy inequalities.

Common pitfalls

Saying that Alice’s energy is transported instantaneously. Alice injects energy locally. Bob later extracts energy locally using a causally delivered record and pre-existing correlations; the final positive and negative energy distribution obeys ordinary dynamics.

Equating correlation with a signal. The unlabeled state at Bob cannot reveal Alice’s outcome. The classical label is an indispensable causal input to the conditioned operation.

Ignoring the interaction term. In lattice and few-body models, local energy near $B$ often includes Hamiltonian terms crossing a boundary. Dropping $V$ can reverse the energy conclusion.

Promoting hardware simulation to continuum observation. An encoded two-qubit Hamiltonian, a condensed-matter effective system, and a relativistic field experiment answer different questions.

Exercises

1. Reproduce the two-qubit numbers

For $h=k=2\ \mathrm{meV}$, compute $E_A$, $E_B$, $E_B/E_A$, and $\theta$.

Solution

Both energies scale linearly when $h$ and $k$ are scaled together. Therefore

$$E_A=\frac{2}{\sqrt2}=1.414214\ \mathrm{meV},$$ $$E_B=2\frac{\sqrt{10}-3}{\sqrt2} =0.229495\ \mathrm{meV}.$$

The dimensionless ratio remains $\sqrt{10}-3=0.162278$, and $\theta=\tfrac12\arctan(1/3)=0.160875$ radians.

2. Prove no-signaling for the nonselective instrument

Assume $[M_a,O_B]=0$ for every $a$. Show that Alice’s trace-preserving local instrument cannot change Bob’s expectation of $O_B$ before the outcome is sent.

Solution

Cyclicity of the trace, locality, and completeness give

$$\begin{aligned} \langle O_B\rangle' &=\sum_a\operatorname{tr}(M_a\rho M_a^\dagger O_B)\\ &=\operatorname{tr}\!\left(\rho O_B \sum_aM_a^\dagger M_a\right)\\ &=\operatorname{tr}(\rho O_B). \end{aligned}$$

The individual conditional expectations may differ. Bob cannot sort them into those subensembles until the classical label arrives.

3. Check a proposed relativistic schedule

Alice and Bob are $30\ \mathrm{km}$ apart. Alice measures at $t_A=0$, and Bob proposes to act at $50\ \mu\mathrm s$ after receiving her one-bit outcome. Is this a relativistic QET schedule?

Solution

The light-travel time is

$$\frac{L}{c}\approx \frac{3.0\times10^4\ \mathrm m} {3.0\times10^8\ \mathrm{m\,s^{-1}}} =100\ \mu\mathrm s.$$

At $50\ \mu\mathrm s$, Bob is spacelike separated from Alice’s measurement and cannot possess the outcome through a causal channel. The schedule is invalid. Acting at or after $100\ \mu\mathrm s$ satisfies the message constraint, but one must still verify from the field dynamics that Alice’s injected excitation does not overlap Bob’s operation and supply the output by ordinary absorption.

4. Close the energy account

In a finite model, $E_A=5$, $E_B=1.2$, messaging costs $0.05$, and resetting the outcome record costs $0.2$, all in the same units. What are the final system energy and the net output of the declared delivery stroke? What changes if all residual system energy can later be recovered?

Solution

The system ends with energy $E_A-E_B=3.8$ above its ground state. Alice’s apparatus supplied 5, Bob’s received 1.2, and the two auxiliary steps consume 0.25. The declared delivery stroke has net external output

$$1.2-5-0.05-0.2=-4.05.$$

If a later operation recovers the full residual 3.8, the ideal complete-cycle output becomes $-4.05+3.8=-0.25$, precisely the message and reset overhead. The protocol has delivered 1.2 locally to Bob, but it has not generated net energy.

References

• Hotta, Masahiro. “Quantum Measurement Information as a Key to Energy Extraction from Local Vacuums.” Physical Review D 78 (2008): 045006. DOI.
• Hotta, Masahiro. “Quantum Energy Teleportation in Spin Chain Systems.” Journal of the Physical Society of Japan 78 (2009): 034001. DOI.
• Hotta, Masahiro. “Quantum Energy Teleportation: An Introductory Review.” arXiv:1101.3954 (2011). Open PDF.
• Ikeda, Kazuki. “Demonstration of Quantum Energy Teleportation on Superconducting Quantum Hardware.” Physical Review Applied 20 (2023): 024051. DOI.
• Ikeda, Kazuki. “Quantum Energy Teleportation across a Lattice and the Continuum.” Physical Review D 114 (2026): 045016. DOI.
• Rodríguez-Briones, Nayeli A., Hemant Katiyar, Eduardo Martín-Martínez, and Raymond Laflamme. “Experimental Activation of Strong Local Passive States with Quantum Information.” Physical Review Letters 130 (2023): 110801. DOI.