Teleported Ergotropy: Remote Work Activation

• Teleported ergotropy is a finite-temperature, LOCC-based protocol that uses shared topological codes and local operations to generate remote extractable work.
• The five-stage process involves local charging, syndrome measurement, classical communication, and conditional decoding to activate Bob’s battery.
• The protocol leverages information-to-work conversion while facing a thermodynamic horizon from quadratic channel-maintenance costs.

Teleported ergotropy is a finite-temperature, LOCC-based thermodynamic protocol in which a distant party, Bob, ends up with a battery state from which work can be extracted, even though no physical energy-carrying excitation is sent through the channel in the ordinary sense. In the formulation introduced in "Nonlocal Topological Maxwell Demon Teleporting Ergotropy via Surface-Code Quantum Error Correction" (Abd-Rabbou et al., 14 May 2026), the operative quantity is not bare internal energy propagating through space, but the availability of extractable work at Bob’s location, enabled by topological correlations, local charging operations, and information-guided decoding. The protocol uses a shared surface-code logical qubit maintained by active quantum error correction, classical syndrome communication from Alice to Bob, and local conditional operations on Bob’s battery. Within this framework, teleported ergotropy is presented as a nonlocal Maxwell demon at finite temperature, with quantum error correction functioning as a thermodynamic resource rather than only as a device for fault-tolerant computation (Abd-Rabbou et al., 14 May 2026).

1. Definition, ergotropy, and operational meaning

The foundation of the subject is the concept of ergotropy. For a system with Hamiltonian $H$ in state $\rho$, ergotropy is the maximum work extractable by a unitary process, namely by a cyclic, entropy-preserving control operation. It is defined by comparing the energy of $\rho$ with the minimum energy achievable by unitary rearrangement of its eigenvalues, that is, the corresponding passive state (Abd-Rabbou et al., 14 May 2026).

In the battery model of (Abd-Rabbou et al., 14 May 2026), Bob’s battery is a qubit with Hamiltonian

$H_B=\frac{\Delta E}{2}\sigma_B^z.$

After the protocol, Bob’s battery is a probabilistic mixture of the excited and ground states, with weights determined by the decoding success probability $P_{\mathrm{succ}}$. For $P_{\mathrm{succ}}\ge 1/2$, the passive state is obtained by swapping the larger population into the ground state, and the resulting ergotropy is

$\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$

or more generally

$$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$

Thus $P_{\mathrm{succ}}=1/2$ corresponds to a passive battery with zero ergotropy, while $P_{\mathrm{succ}}=1$ yields a fully charged battery with ergotropy $\rho$0 (Abd-Rabbou et al., 14 May 2026).

This operational meaning is central. Alice starts with a charged battery and spends energy $\rho$1. Bob, after receiving only classical information and using the shared code resource, can prepare his own battery in a state with ergotropy $\rho$2. The protocol is explicit that energy is not physically transported through the code as a propagating excitation. Rather, Alice uses a local interaction to encode a logical degree of freedom associated with the shared surface code, sends a classical syndrome record $\rho$3, and Bob uses that information to decode the logical state and conditionally charge his local battery. The syndrome string is described as an entropy rectifier: without it, Bob’s conditional operation is effectively random and no ergotropy is available (Abd-Rabbou et al., 14 May 2026).

This use of the term “teleported” is therefore structural and operational rather than literal in the standard quantum-information sense. The protocol has a preshared nonlocal resource, local operations, classical communication, and a conditional recovery operation, but what is reconstructed remotely is a work-bearing battery state rather than an arbitrary unknown quantum state (Abd-Rabbou et al., 14 May 2026).

2. Surface-code channel and five-stage protocol

The physical channel is a rotated surface code on a rectangular lattice

$\rho$4

where $\rho$5 is the code distance and $\rho$6 is the separation between Alice at $\rho$7 and Bob at $\rho$8. Physical qubits live on edges, with total number

$\rho$9

The code Hamiltonian is

$\rho$0

with star and plaquette stabilizers

$\rho$1

The code space encodes one logical qubit. A logical $\rho$2 operator is represented by a horizontal string

$\rho$3

joining Alice to Bob. By path independence on code states, any two such representatives differ by products of plaquettes and act identically in the code space; this path independence is what makes the logical action nonlocal while allowing Alice’s physical implementation to remain boundary-local (Abd-Rabbou et al., 14 May 2026).

The setup also contains two local batteries, $\rho$4 and $\rho$5. Alice’s battery has Hamiltonian

$\rho$6

and starts fully charged in $\rho$7. Bob’s battery is identical and starts empty in $\rho$8. The environment is a thermal bath at temperature $\rho$9, which induces anyon excitations in the surface code. External apparatus for syndrome measurement, erasure, and active error correction are assumed throughout, and they enter the thermodynamic bookkeeping because they consume work (Abd-Rabbou et al., 14 May 2026).

The protocol is organized into five stages.

In Stage 1, Alice locally couples her battery to the logical qubit through

$H_B=\frac{\Delta E}{2}\sigma_B^z.$0

with pulse area

$H_B=\frac{\Delta E}{2}\sigma_B^z.$1

The corresponding unitary is

$H_B=\frac{\Delta E}{2}\sigma_B^z.$2

Because

$H_B=\frac{\Delta E}{2}\sigma_B^z.$3

this operation does not change the stabilizer energy of the code. Alice’s battery flips from $H_B=\frac{\Delta E}{2}\sigma_B^z.$4 to $H_B=\frac{\Delta E}{2}\sigma_B^z.$5, with

$H_B=\frac{\Delta E}{2}\sigma_B^z.$6

The paper interprets the spent energy not as ordinary excitation energy of the code Hamiltonian, but as encoded in the logical degree of freedom that Bob later accesses via decoding (Abd-Rabbou et al., 14 May 2026).

In Stage 2, Alice measures $H_B=\frac{\Delta E}{2}\sigma_B^z.$7 boundary stabilizers on her side, $H_B=\frac{\Delta E}{2}\sigma_B^z.$8 of $H_B=\frac{\Delta E}{2}\sigma_B^z.$9-type and $P_{\mathrm{succ}}$0 of $P_{\mathrm{succ}}$1-type, obtaining a syndrome string

$P_{\mathrm{succ}}$2

In Stage 3, she sends $P_{\mathrm{succ}}$3 to Bob over a classical channel. This is the only communication between the two parties (Abd-Rabbou et al., 14 May 2026).

In Stage 4, Bob combines Alice’s record with the syndrome history from active monitoring of the code and uses minimum-weight perfect matching to infer a correction label

$P_{\mathrm{succ}}$4

In Stage 5, he conditionally applies

$P_{\mathrm{succ}}$5

thereby charging his battery whenever the decoded logical state indicates it (Abd-Rabbou et al., 14 May 2026).

The surface code therefore serves not as a wire carrying energy, but as a shared, topologically protected thermodynamic channel. The protocol is strictly LOCC in the sense that no quantum signal is sent during execution, but it is not resource-free, because the nonlocality is embodied in an already-established, actively maintained topological code (Abd-Rabbou et al., 14 May 2026).

3. Finite-temperature protection, thresholds, and information-to-work conversion

Temperature enters through thermal fluctuations that create anyon pairs with probability

$P_{\mathrm{succ}}$6

equivalently

$P_{\mathrm{succ}}$7

Increasing temperature therefore increases the physical error rate $P_{\mathrm{succ}}$8. The initial code is treated effectively as a known logical state $P_{\mathrm{succ}}$9 together with a thermal syndrome sector, while active monitoring is assumed to maintain the logical information against the bath. The paper emphasizes that without active monitoring, topological order in two dimensions is unstable at any finite temperature: anyons diffuse and create logical errors on a timescale independent of system length $P_{\mathrm{succ}}\ge 1/2$0, which destroys the stored ergotropy (Abd-Rabbou et al., 14 May 2026).

Below the topological threshold $P_{\mathrm{succ}}\ge 1/2$1, active monitoring suppresses logical errors exponentially with code distance. The asymptotic law quoted is

$P_{\mathrm{succ}}\ge 1/2$2

where $P_{\mathrm{succ}}\ge 1/2$3 is a geometry-dependent prefactor, $P_{\mathrm{succ}}\ge 1/2$4 is the number of syndrome rounds, and $P_{\mathrm{succ}}\ge 1/2$5 is the spacetime volume. Equivalently, at fixed $P_{\mathrm{succ}}\ge 1/2$6,

$P_{\mathrm{succ}}\ge 1/2$7

The quantity that is exponentially protected is therefore the decoding success probability, or equivalently the logical error rate, and through

$P_{\mathrm{succ}}\ge 1/2$8

the remotely recoverable ergotropy approaches $P_{\mathrm{succ}}\ge 1/2$9 exponentially in $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$0 when $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$1 (Abd-Rabbou et al., 14 May 2026).

Numerically, the threshold is reported as

$\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$2

For fixed separation $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$3, sub-threshold curves show $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$4 exponentially with $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$5, whereas above threshold, for $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$6,

$\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$7

so $\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$8 (Abd-Rabbou et al., 14 May 2026).

The demon interpretation is sharpened by an information-theoretic work bound. Using the Sagawa-Ueda result, the extracted ergotropy obeys

$\mathcal{E}_B=(2P_{\mathrm{succ}}-1)\,\Delta E,$9

where $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$0 is the mutual information between the relevant error variable and Alice’s syndrome string. Bob’s extractable work is thus limited by the information Alice sends. In the language of the paper, the syndrome record is not just bookkeeping; it is the operational fuel for the information-to-work conversion (Abd-Rabbou et al., 14 May 2026).

The same paper also studies a partial-information variant in which only a fraction $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$1 of Alice’s syndrome bits are transmitted. Numerically, it finds a critical information fraction

$$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$2

at $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$3, $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$4, $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$5, below which the decoding graph fails to percolate and Bob cannot distinguish logical sectors. Then $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$6 and $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$7. Near threshold, the observation

$$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$8

is presented as a percolation critical exponent associated with the $$\mathcal{E}_B=\max\!\bigl(0,(2P_{\mathrm{succ}}-1)\Delta E\bigr).$$9-dimensional decoding graph (Abd-Rabbou et al., 14 May 2026).

4. Thermodynamic accounting, second-law consistency, and horizon distance

The energy accounting is explicit. There are three work inputs: Alice’s battery contributes $P_{\mathrm{succ}}=1/2$0, erasure of classical memory costs $P_{\mathrm{succ}}=1/2$1, and active syndrome measurement throughout the channel costs $P_{\mathrm{succ}}=1/2$2. The balance is written as

$P_{\mathrm{succ}}=1/2$3

where $P_{\mathrm{succ}}=1/2$4 is dissipated heat (Abd-Rabbou et al., 14 May 2026).

The operational Landauer cost for erasing the syndrome register is estimated as

$P_{\mathrm{succ}}=1/2$5

The infrastructure cost is associated with maintaining a channel of length $P_{\mathrm{succ}}=1/2$6 and width $P_{\mathrm{succ}}=1/2$7 under repeated stabilizer measurement. Monitoring $P_{\mathrm{succ}}=1/2$8 stabilizers over $P_{\mathrm{succ}}=1/2$9 rounds at cost $P_{\mathrm{succ}}=1$0 per stabilizer-round gives

$P_{\mathrm{succ}}=1$1

Causality requires the classical signal from Alice to reach Bob before Bob can decode, so

$P_{\mathrm{succ}}=1$2

or approximately

$P_{\mathrm{succ}}=1$3

Substitution yields the quadratic cost law

$P_{\mathrm{succ}}=1$4

This quadratic infrastructure cost is presented as a consequence of the causal necessity of maintaining an $P_{\mathrm{succ}}=1$5 channel over a time proportional to $P_{\mathrm{succ}}=1$6, not as an artifact of inefficient engineering (Abd-Rabbou et al., 14 May 2026).

The net transferred work is defined as

$P_{\mathrm{succ}}=1$7

Since $P_{\mathrm{succ}}=1$8 is nearly distance-independent once the code is large enough and sub-threshold, while $P_{\mathrm{succ}}=1$9, the net transferred work must eventually become negative. This defines a thermodynamic horizon. Setting $\rho$00, the paper obtains

$\rho$01

Because $\rho$02 and $\rho$03 both scale linearly with $\rho$04, the authors state that $\rho$05 cancels, making $\rho$06 effectively independent of code distance once $\rho$07 is large enough to ensure $\rho$08. The scaling is summarized as

$\rho$09

This is the fundamental thermodynamic horizon on separation distance: beyond a certain range, the protocol cannot remain profitable because maintenance cost grows quadratically (Abd-Rabbou et al., 14 May 2026).

The second law is preserved through the same cost structure. Using the Sagawa-Ueda bound, the paper notes that

$\rho$10

so Bob’s ergotropy never exceeds the thermodynamic value of the information erased. With bath heat

$\rho$11

and Bob’s battery entropy increase

$\rho$12

the total entropy production is

$\rho$13

The chain of inequalities quoted in the main text is

$\rho$14

hence

$\rho$15

The Supplement gives the explicit lower bound

$\rho$16

Accordingly, the statement that the quadratic infrastructure cost strictly enforces the second law means that even under maximally favorable information-to-work conversion, unavoidable channel-maintenance cost makes the overall entropy production strictly positive (Abd-Rabbou et al., 14 May 2026).

5. Demon phase, thermal phase, and the meaning of nonlocality

The paper identifies a thermodynamic phase transition between a profitable demon phase and an unprofitable thermal phase. The operational order parameter is the sign of the net work,

$\rho$17

If

$\rho$18

the protocol is profitable and lies in the demon phase. If

$\rho$19

costs dominate and the system is in the thermal phase (Abd-Rabbou et al., 14 May 2026).

The control variable emphasized is the physical error rate $\rho$20, equivalently temperature $\rho$21, at fixed $\rho$22 and $\rho$23. The critical point $\rho$24 is defined implicitly by

$\rho$25

with

$\rho$26

Numerically, the paper reports

$\rho$27

so

$\rho$28

The distinction is explicit: the topological threshold $\rho$29 marks loss of asymptotic error-correcting power, whereas the thermodynamic threshold $\rho$30 marks loss of profitability. They are described as physically distinct because one is an information-theoretic boundary and the other a thermodynamic one (Abd-Rabbou et al., 14 May 2026).

The transition is described as continuous, with no latent heat and no discontinuity in $\rho$31, because $\rho$32 degrades smoothly with $\rho$33. The authors interpret it as analogous to a second-order transition connected to percolation in the spacetime decoding graph. They also qualify the claim: it is not a rigorously established equilibrium phase transition in the strict statistical-mechanical sense, but an operational, finite-size and asymptotic thermodynamic transition diagnosed through net work curves and supported by numerics (Abd-Rabbou et al., 14 May 2026).

The role of nonlocality is correspondingly narrow and precise. What is nonlocal is the logical structure of the code and the thermodynamic effect of shared topological information. Alice and Bob must preshare an extended surface-code resource spanning the separation $\rho$34. Alice’s physical charging operation is strictly local on her boundary, the only communication is the classical syndrome record $\rho$35, and Bob’s extraction operation is local on his own battery. Thus the protocol is strictly LOCC in execution, but not free of nonlocal resources. The most accurate interpretation given in the paper is topologically protected information-to-work conversion at a distance, or remote activation of extractable work using a shared encoded state (Abd-Rabbou et al., 14 May 2026).

6. Relation to adjacent ergotropy literature and scope of the concept

Teleported ergotropy, as formalized in (Abd-Rabbou et al., 14 May 2026), belongs to a broader line of work distinguishing energy transport from transport or activation of extractable work. "Correlations enable lossless ergotropy transport" (Simon et al., 2024) shows that ergotropy transport is fundamentally different from energy transport, and that under a strictly energy-conserving global unitary, correlations can make transport lossless or gainful. That work does not implement literal teleportation: there is direct global interaction, no Bell measurement, and no classical communication. Its relevance is conceptual and resource-theoretic: useful work can appear at the receiver without ordinary net energy flow through the channel because pre-existing correlations are consumed (Simon et al., 2024).

Two earlier correlation-based analyses sharpen the same point from different directions. "Quantum correlations and ergotropy" (Francica, 2022) develops same-energy reference-state comparisons and concludes that only discord correlations always give a non-negative contribution to work extraction, whereas total correlations, classical correlations, and even entanglement can reduce the extractable work in some mixed-state settings (Francica, 2022). "Ergotropy from quantum and classical correlations" (Touil et al., 2021) proves, for bipartite states with locally thermal marginals, that

$\rho$36

so extractable work can be encoded entirely in correlations even when each subsystem is individually passive (Touil et al., 2021). These results do not establish LOCC teleportation of work, but they supply the correlation-theoretic background for nonlocal work activation.

A different but related strand is the study of entanglement-enabled local ergotropy in many-body ground states. "Ergotropy and entanglement in critical spin chains" (Mula et al., 2022) shows that a subsystem of an entangled ground state can contain extractable work once isolated, and that the subsystem bound energy for half a free fermionic chain obeys

$\rho$37

This is not a teleportation protocol and does not involve LOCC, measurement, or feedback; its significance here is that entanglement alone can render a local reduced state active with respect to its local Hamiltonian (Mula et al., 2022).

Relative to those antecedents, (Abd-Rabbou et al., 14 May 2026) combines four themes that, according to the paper, had not previously been unified in that way: nonlocal generation of ergotropy, Maxwell-demon thermodynamics with spatially separated measurement and work extraction, finite-temperature operation using active topological protection, and the use of quantum error correction as a thermodynamic resource rather than solely an information-protection device (Abd-Rabbou et al., 14 May 2026).

The scope of the proposal remains bounded by explicit assumptions. It assumes a large preshared surface code spanning the separation between Alice and Bob, continuous syndrome monitoring, a measurement-apparatus cost model $\rho$38 per stabilizer-round, and Bob’s access to the necessary bulk syndrome history in addition to Alice’s boundary record. The finite-temperature model is simplified to an anyon-pair creation probability $\rho$39. Decoder complexity is identified as a natural extension rather than fully incorporated into the thermodynamic cost. The observed demon/thermal transition is operational rather than a full equilibrium proof. The interpretation of Alice’s spent energy as encoded in the logical qubit while $\rho$40 is explicitly a control-theoretic work-accounting perspective rather than ordinary storage in the code Hamiltonian (Abd-Rabbou et al., 14 May 2026).

Within those limits, teleported ergotropy denotes the remote creation of a work-extractable battery state at Bob’s site, accomplished by Alice’s local expenditure of energy, a shared topological code, continuous error correction, and classical syndrome communication. The central equations summarizing the proposal are

$\rho$41

$\rho$42

$\rho$43

$\rho$44

$\rho$45

and

$\rho$46

Taken together, these relations define teleported ergotropy as a thermodynamically consistent, topologically protected remote work-enabling scheme whose reliability can be exponentially protected below threshold, yet whose useful range is fundamentally limited by the causal and energetic cost of maintaining the channel (Abd-Rabbou et al., 14 May 2026).