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Quantification of the energy consumption of entanglement distribution

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A quantum channel that irreversibly degrades entanglement cannot distribute it for free: the paper proves an energy floor per ebit proportional to $2\hbar\omega(E_C/E_D - 1)$.

desk verdict ECRED and the Hamiltonian-model energy cost are genuine contributions, but the advertised 'fundamental' lower bound is really only a lower bound on a restricted standard quantity, so the strongest conclusion needs rewording. read the letter →

arxiv 2507.23108 v1 pith:LZIX3TAR submitted 2025-07-30 quant-ph

Karol Horodecki , Marek Winczewski , Leonard Sikorski , Pawe{l} Mazurek , Miko{l}aj Czechlewski , Raja Yehia This is my paper · ORCID
classification quant-ph MSC 81P4081P4581P68 PACS 03.67.-a03.67.Mn
keywords entanglementdistillationenergyconsumptionrateirreversibilityquantumchannelsChoistatenetworksLandauerprincipleHamiltonianmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a measure of the fundamental energy cost of distributing entanglement through a noisy quantum channel, in joules per ebit, and proves that entanglement irreversibility forces a positive energy floor on standard distribution protocols. Its central result is a lower bound: for a channel whose Choi state has entanglement cost $E_C$ and distillable entanglement $E_D$, the standard energy consumption rate is at least $2\hbar\omega(E_C/E_D - 1)$, so whenever $E_C > E_D$ the energy per ebit cannot be zero. The paper also builds a Hamiltonian-model definition of the energy cost of a quantum operation, and uses it to compute upper bounds for three concrete distillation protocols on depolarizing channels. A sympathetic reader should care because the result turns a structural fact of entanglement theory, irreversibility, into a concrete resource bill for future quantum networks.

What carries the argument

The Energy Consumption Rate of Entanglement Distribution (ECRED), $C(\Phi|\mathrm{Ent})$, is the infimum over physical realizations, input states, and LOCC distillation protocols of the net energy balance (input energy plus distillation energy minus output energy) divided by the number of distilled ebits. The standard ECRED fixes photons, a maximally entangled input, and the 'send half through the channel' protocol, giving an upper bound on the fundamental ECRED. The lower bound uses the Choi state $\rho_\Phi = (1 \otimes \Phi)(\psi^+_{d_{\rm in}})$ and the non-negativity of the energy cost of a quantum operation, which follows from a Hamiltonian-model definition of that cost satisfying subadditivity in time and locality in space. The upper bounds come from decomposing distillation protocols into building blocks such as CNOT gates, measurements, erasure, and classical communication, whose energy costs are bounded using Landauer-type arguments.

What would settle it

Exhibit a standard protocol, meaning a maximally entangled input, half sent through a passive channel with $E_C(\rho_\Phi) > E_D(\rho_\Phi)$, and LOCC distillation, whose net energy per ebit is strictly below $2\hbar\omega(E_C(\rho_\Phi)/E_D(\rho_\Phi) - 1)$ under the paper's Hamiltonian energy model; a concrete starting point is the qubit erasure channel, where the claimed bound reads $C_{\rm std} \ge 2\hbar\omega\, p/(1-p)$.

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Extended reading notes

Core claim

The central claim is a quantitative bridge between entanglement irreversibility and energy consumption. For the standard way of distributing entanglement, where a maximally entangled state of dimension $d_{\rm in}$ is prepared, half is sent through a passive channel $\Phi$, and the parties distill pure entanglement by LOCC, the paper proves $C^{\epsilon}_{\rm std}(\Phi|\mathrm{Ent}) \ge 2E(\lceil \log_2 d_{\rm in}\rceil / \lceil E^\epsilon_D(\rho_\Phi)\rceil - 1)$ in the single-shot case and $C_{\rm std}(\Phi|\mathrm{Ent}) \ge 2\hbar\omega(\log_2 d_{\rm in}/E_D(\rho_\Phi) - 1)$ in the asymptotic limit. Since $E_C(\rho_\Phi) \le \log_2 d_{\rm in}$, this implies $C_{\rm std}(\Phi|\mathrm{Ent}) \ge 2\hbar\omega(E_C(\rho_\Phi)/E_D(\rho_\Phi) - 1)$. If the Choi state is irreversible in the sense $E_C > E_D$, the standard energy consumption rate is strictly positive.

Load-bearing premise

The bound assumes a maximally entangled state of local dimension $d$ is carried by $2\lceil\log_2 d\rceil$ non-interacting particles of fixed energy $E$, with input and output energies exactly those values; platforms whose qubits share energy levels, interact, or have different energy accounting escape the bound.

Editorial extensions

If this is right

  • Any standard photonic entanglement distribution over a channel whose Choi state satisfies $E_C > E_D$ pays at least $2\hbar\omega(E_C/E_D - 1)$ joules per ebit; the cost is a fundamental floor, not an engineering overhead.
  • For channels with zero distillable entanglement, including separable and bound-entangled Choi states, the standard ECRED diverges: no energy investment can produce even one ebit by LOCC.
  • The gap between the lower bound, of order $10^{-34}\,\mathrm{J/ebit}$, and the upper bounds for three concrete distillation protocols, of order $10^{-12}\,\mathrm{J/ebit}$, means protocol choice matters more than channel physics for current energy budgets.
  • The energy-cost measure for quantum operations satisfies subadditivity in time and locality in space, so it can be applied outside entanglement distribution, for example to estimate the fundamental energy cost of parts of quantum computations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the same per-particle energy accounting and irreversibility logic should apply to any pure resource state distributed through a resource-degrading channel, so analogous energy floors are expected for magic-state or coherence distillation on photonic platforms.
  • Editorial extension: because the lower bound is for the standard ECRED and the fully minimized ECRED can only be smaller, a testable prediction is that non-standard protocols, such as encoding multiple logical qubits per particle or recycling output energy, could beat the photonic standard floor.
  • Editorial extension: the upper-bound analysis suggests that reducing the number of probabilistic CNOTs and measurements, rather than maximizing distillation rate, is the most direct route to lower energy per ebit; the paper's numerical results already show that the highest-rate protocol is not the most energy-efficient.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript introduces a framework for quantifying the energy cost of entanglement distribution through quantum channels, defining the Energy Consumption Rate of Entanglement Distribution (ECRED) and a 'standard' version (standard ECRED) that fixes a photonic realization, a maximally entangled input, and the usual half-singlet-sending protocol. It derives lower bounds on the standard ECRED in terms of the distillable entanglement, entanglement cost, two-way capacity, and logarithmic negativity of the channel's Choi state, under the assumption that entanglement is encoded in non-interacting particles of fixed energy. It also provides numerical upper bounds on the fundamental ECRED for depolarizing channels by costing the building blocks of three entanglement distillation protocols (BBPSSW, DEJMPS, P1-or-P2). The paper claims that entanglement irreversibility implies a non-zero energy cost of entanglement distribution and emphasizes the hardware-independent, fundamental nature of the cost.

Significance. The paper's formal apparatus is a useful contribution: the axioms for energy costs, the Hamiltonian protocol model, and the explicit derivation of lower bounds under stated assumptions are coherent and clearly presented. The comparison of three distillation protocols on an energy-per-ebit basis is a novel and potentially influential way to benchmark quantum network protocols. If the lower bounds applied to the fully minimized, hardware-independent ECRED, the result would be a significant bridge between entanglement irreversibility and thermodynamics. However, the lower bounds are proven only for the standard ECRED, which is an upper bound on the fundamental ECRED, so the advertised 'fundamental' and 'hardware-independent' lower bound is not established. The upper-bound estimates are also not fully reproducible as presented. For these reasons the manuscript merits a major revision rather than acceptance in its current form.

major comments (3)
  1. [Abstract, §II.C, §V, Eq. (3)] The lower bounds in Theorem 1, Proposition 1, and Corollary 1 (Eqs. (25), (33), (52)) are all lower bounds on Cstd, the standard ECRED, not on the fundamental ECRED C defined in Definition 1. As Eq. (3) states, C(Φ|Ent) ≤ Cstd(Φ|Ent); therefore a lower bound on Cstd does not imply any lower bound on C. The abstract's statement that the paper focuses on the 'hardware-independent, i.e. fundamental cost' and the Discussion's claim (Section V, first paragraph) that 'the fundamental energy cost ... is lower bounded by the so-called distillable entanglement' are not supported by the proofs. This is a load-bearing overclaim: the central conceptual result would only follow from a lower bound on C itself. The authors should either prove such a bound (e.g., by extending the per-particle energy accounting to all physical realizations) or consistently reframe the main results as bounds on the standard ECRED, adjusting the abstract, title, and discussion accordingly.
  2. [§IV.G, Eq. (9), Fig. 3] The numerical upper bounds on C depend on a chain of estimates (p_cnot=1/8, Eaux=ℏω+ELandauer,ub, Emeasurement=2kBT ln2, and the inclusion of classical communication) but no code or complete computational derivation is provided. In particular, the role of the CNOT success probability p_cnot in the energy accounting is unclear: if the probabilistic CNOT implementation of Ref. [25] is used, failed CNOT attempts must be paid for in the expected energy cost, either by multiplying operation counts by 1/p_cnot or by absorbing p_cnot into the success probabilities pk of Eq. (79); the text does not specify which is done. Moreover, Section II.D states that classical communication is neglected, while Table II includes 2ECC terms. As written, the claim of rigorous upper bounds of order O(10^-12) J/ebit is not verifiable. Please provide the code or a detailed step-by-step account, and reconcile these inconsistencies.
  3. [§VI.A, Lemmas 1 and 2] The proofs of Lemmas 1 and 2 rely on the assertion that in the cited erasure and measurement protocols 'heat flows only in one direction', which is essential because the energy cost E(ΛP) in Definition 14 counts only positive ingoing currents. If the protocols involve bidirectional heat or work exchange, bounding the net heat (or energy balance) does not upper-bound the sum of positive currents. The manuscript cites Reeb-Wolf and Abdelkhalek-Nakata-Reeb but does not demonstrate the unidirectional property. Please provide a derivation or a more precise citation showing that the integrated positive currents are bounded by the stated expressions.
minor comments (6)
  1. [§II.E, Eq. (10)] The inequality direction in Eq. (10) is wrong: it should be Cstd(Φ|Ent) ≥ 2ℏω(EC(ρΦ)/ED(ρΦ) − 1), not ≤.
  2. [§II.B] Axiom 1 is stated twice with different content (subadditivity in time and locality of energy); the numbering should be made consistent throughout the main text and the appendix.
  3. [Definition 4] The infimum over dout ∈ N+ includes dout = 1, for which log2 dout = 0; the definition should exclude dout = 1 or handle the zero denominator explicitly.
  4. [§IV.G, item 1] The statement 'We bound ΔS by 1' should read 'by ln2' when Shannon entropies are measured in nats, otherwise the claimed ELandauer,ub = 2kBT ln2 does not follow from Lemma 1.
  5. [Table II, Fig. 9] The protocol name is spelled 'BPPSSW' in Table II and Fig. 9 but 'BBPSSW' elsewhere; please standardize.
  6. [General] No data or code availability statement is provided for the numerical results shown in Figs. 3 and 9, which is important for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bound follows algebraically from the ECRED definition plus standard facts about ED/EC; upper bounds use independent thermodynamic results.

full rationale

The central derivation (Theorem 1 through Proposition 1 and Corollary 1) starts from the definition of the standard ECRED, substitutes the explicitly assumed per-particle energies E(ψ+_d)=2E⌈log2 d⌉, drops the non-negative distillation cost E(Λ,ρ), and uses only standard facts R(Λ) ≤ E^ϵ_D(ρΦ) and EC(ρΦ) ≤ log2 din to obtain the bounds. No parameter is fitted to the target quantity, and no step defines its conclusion into its premises. The relation C(Φ|Ent) ≤ Cstd(Φ|Ent) is explicitly stated, so the lower bounds constrain the standard, not the fully minimized, ECRED; the Discussion's phrasing that the 'fundamental energy cost' is lower bounded goes beyond what the proof establishes, but that is an interpretive overreach rather than a circular reduction. Self-citations such as [5], [6], and [22] are contextual or technical and are not load-bearing for the main inequalities; the upper bounds rely on independent results by Reeb-Wolf and Abdelkhalek-Nakata-Reeb plus explicit protocol estimates. Hence no circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central lower bound depends on standard properties of entanglement measures and on a domain-specific energy accounting for non-interacting photonic qubits. The upper bounds introduce several hand-picked physical parameters (wavelength, temperature, CNOT success). The mesoscopic interface is a modeling device, not a new physical entity.

free parameters (4)
  • Photon energy ℏω (wavelength 1550 nm) = corresponds to ~1.28e-19 J
    Chosen for numerical estimates of the lower and upper bounds; not derived from data.
  • Ambient temperature T = 293 K
    Used in Landauer erasure upper bounds for the distillation protocols.
  • CNOT success probability p_cnot = 1/8
    Taken from the linear-optics CNOT implementation of Ref. [25]; used to estimate the energy cost of CNOT gates.
  • Epsilon for Landauer upper bound = 1/2
    Set to obtain the factor 2 in the upper bound ELandauer,ub = 2 k_B T ln 2.
assumptions (5)
  • standard math LOCC operations do not increase distillable entanglement.
    Used in Theorem 1 to assert ⌈log2 din⌉ ≥ ⌈R(Λϵ,ρΦ)⌉, citing non-increasing property under LOCC (Ref. [51]).
  • domain assumption The channel Φ is LOCC and passive, i.e., it does not increase distillable entanglement.
    The entire standard scenario assumes the channel only degrades entanglement; stated in Section II.A and used in Theorem 1.
  • ad hoc to paper The energy cost E(Λ,ρ) is non-negative and satisfies subadditivity in time and locality.
    These are axioms of the paper's framework (Axiom 1 and 2 in Section IV.B), later proved for the Hamiltonian model; they are used to drop E(Λ,ρ) in the lower bound.
  • domain assumption The physical carriers are non-interacting particles, each with energy E, and a maximally entangled state of dimension d requires 2⌈log2 d⌉ carriers.
    Introduced in Theorem 1 (Section IV.C) to relate state energies to entanglement rates. This is the load-bearing assumption for the numerical value of the lower bound.
  • domain assumption The distillation operation leaves the local Hamiltonian unchanged for polarization-encoded qubits.
    Remark 3 and the text before Theorem 1; ensures input and output energies are measured with the same Hamiltonian.
invented entities (1)
  • Mesoscopic interface
    purpose: A conceptual device with a few controllable quantum degrees of freedom that mediates energy transfer between a quantum system and its classical controller, used to define the fundamental energy cost E(Λ,ρ) of a quantum operation.
    Introduced in Section VI.A (Definition 9 and surrounding text) as part of the Hamiltonian model. It is a theoretical construct without an independent falsifiable prediction.

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Cite this review

Pith. "Pith review of Quantification of the energy consumption of entanglement distribution." pith.science (2026). https://pith.science/paper/LZIX3TAR

@misc{pith2026250723108,
  author       = {Pith},
  title        = {Pith review of: Quantification of the energy consumption of entanglement distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZIX3TAR}},
  note         = {Machine review of arXiv:2507.23108}
}
read the original abstract

Inspired by environmental sciences, we develop a framework to quantify the energy needed to generate quantum entanglement via noisy quantum channels, focusing on the hardware-independent, i.e. fundamental cost. Within this framework, we define a measure of the minimal fundamental energy consumption rate per distributed entanglement (expressed in Joule per ebit). We then derive a lower bound on the energy cost of distributing a maximally entangled state via a quantum channel, which yields a quantitative estimate of energy investment per entangled bit for future quantum networks. We thereby show that irreversibility in entanglement theory implies a non-zero energy cost in standard entanglement distribution protocols. We further establish an upper bound on the fundamental energy consumption rate of entanglement distribution by determining the minimal energy required to implement quantum operations via classical control. To this end, we formulate the axioms for an energy cost measure and introduce a Hamiltonian model for classically-controlled quantum operations. The fundamental cost is then defined as the infimum energy over all such Hamiltonian protocols, with or without specific hardware constraints. The study of the energy cost of a quantum operation is general enough to be naturally applicable to quantum computing and is of independent interest. Finally, we evaluate the energy demands of three entanglement distillation protocols for photonic polarization qubits, finding that, due to entanglement irreversibility, their required energy exceeds the fundamental lower bound by many orders of magnitude. The introduced paradigm can be applied to other quantum resources, with appropriate changes depending on their nature.

Figures

Figures reproduced from arXiv: 2507.23108 by the authors.

Figure 1
Figure 1. FIG. 1: The quantification of the energy consumption rate of entanglement distribution in the landscape of scien [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Visual represenation of the formal definition of energy consumption rate of entanglement distribution. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Upper bounds on the fundamental energy consumption required to obtain one quantum state with fidelity [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The energy consumption rate of entanglement distribution on a single plot, for a fixed channel Φ = [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The scheme of entanglement distillation. It is assumed that the channel Φ is passive, i.e. it does not increase [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Lower bound on the standard fundamental energy consumption of a distillation protocol using the quantum [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Lower bounds on the standard fundamental energy consumption rate of distillation protocols, for the quantum [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: A protocol realizing quantum measurement proposed in [ [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Upper bounds on the fundamental energy consumption required to obtain one quantum state with fidelity [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

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