bibkey: davinoetal2025energyconstrained
authors: Raffaele D'Avino; Gabriel Senno; Mir Alimuddin; Antonio Acín
year: 2025
title: "Entanglement in the energy-constrained prepare-and-measure scenario: applications to randomness certification and channel discrimination"
doi: null
url: https://arxiv.org/abs/2510.27559v3
claim: "The paper studies energy-constrained prepare-and-measure behaviors with a shared state between the devices, shows that entanglement enlarges the set of behaviors compared with separable shared states, derives that separable devices deterministic on input 0 satisfy E_1 >= 2(1-2 omega)^2 - 1, and conjectures that for every positive energy bound some entangled behavior deterministic on input 0 violates this inequality."
strata_touched:
  - D5/S3/QuantumBounds/EnergyConstrainedDeterministicViolation
license: citation-only
triage: anchor

Entanglement in the energy-constrained prepare-and-measure scenario

R. D’Avino, G. Senno, M. Alimuddin and A. Acín, arXiv:2510.27559v3 (15 January 2026; v1 31 October 2025). Quotations are from the v3 TeX source, with the macros \Tr, \Qsep and \Qent written as $\mathrm{Tr}$, ${\mathcal{Q}}_{\omega }^{\mathrm{sep}}$, ${\mathcal{Q}}_{\omega }^{\mathrm{ent}}$ and citations omitted.

Behaviors, for a shared state ${\sigma }_{PM}$, channels ${\Lambda }_x$ on $P$ and a POVM on $S\otimes M$:

$p(b|x,y)=\mathrm{Tr}\left[{\Pi }_{SM}^{b|y}({\Lambda }_x\otimes 1_M)({\sigma }_{PM})\right].$

The paper uses two inputs $x\in \{0,1\}$ and one binary measurement $\{{\Pi }_{SM}^b{\}}_{b\in \{0,1\}}$.

The energy constraint:

$\exists H\text{s.t.}\mathrm{Tr}\left[H{\Lambda }_x({\sigma }_P)\right]\le {\omega }_x,$ where ${\omega }_x\in [0,1/2)$ and $H$ is some hermitian operator (i.e., the Hamiltonian) with a non-degenerate ground state and a unit gap, i.e. $H={\sum }_{i\ge 0}{E}_i|E_i⟩⟨E_i|$ with $E_0=0$ and $E_i\ge 1$ for all $i\ge 1.$

The paper then sets ${\omega }_0={\omega }_1=\omega$. It denotes by ${\mathcal{Q}}_{\omega }^{\mathrm{sep}}$ the behaviors with separable ${\sigma }_{PM}$ and by ${\mathcal{Q}}_{\omega }^{\mathrm{ent}}$ those “when entanglement is allowed”; the correlators are $E_x=p(0|x)-p(1|x)$.

The inequality for separable devices deterministic on input 0:

$E_1\ge 2(1-2\omega {)}^2-1=:I_{\det }^{\omega ,\mathrm{sep}}$ … for all $\mathbf{p}\in {\mathcal{Q}}_{\omega }^{\mathrm{sep}}$.

The conjecture:

Numerically, we find that for all tested positive values of $\omega$, there exist behaviors $\mathbf{p}\in {\mathcal{Q}}_{\omega }^{\mathrm{ent}}$ that are deterministic for input $x=0$ but violate Eq.~(eq:deterministic-ineq-for-fixed-input-0). We conjecture that this holds for all $\omega >0$.

Verified locator

Source: https://arxiv.org/abs/2510.27559v3

• URL: https://arxiv.org/abs/2510.27559v3 (the e-print is a tar archive; member paper_Quantum.tex, 849 lines, md5 012a72f4d8cfd30e14dc3813eda8f546):
  • behaviors, l. 137–139;
  • two inputs and one binary measurement, l. 148;
  • the energy constraint, l. 151–154;
  • the sets of behaviors, l. 158;
  • the correlators, l. 165–166;
  • the inequality, l. 266–270;
  • the conjecture, l. 284.
• Crossref lists no journal version; the author list is from the TeX \author fields and the arXiv metadata.