bibkey: bibakdelsantodakic2024coherence
authors: Fatemeh Bibak; Flavio Del Santo; Borivoje Dakić
year: 2024
title: "Quantum coherence in networks"
doi: 10.1103/PhysRevLett.133.230201
url: https://arxiv.org/abs/2407.11122v1
claim: "The paper defines a network figure of merit C_n distinguishing coherent n-party distributions from classical mixtures of two product distributions, computes the quantum value 1/2 - 2^{-n}, and from numerical optimization for n = 2, …, 10 conjectures that the classical maximum of C_n is 1/4 for all n."
strata_touched:
  - D5/S3/QuantumBounds/NetworkCoherenceClassicalBound
license: citation-only
triage: anchor

Quantum coherence in networks

F. Bibak, F. Del Santo and B. Dakić, arXiv:2407.11122v1 [quant-ph]; Phys. Rev. Lett. 133 (2024). Quotations are from the v1 TeX source.

The classical model:

$p_{\text{cl}}(a_1,a_2,a_3,\dots ,a_n)=\gamma p_1(a_1)p_2(a_2)\dots p_n(a_n)+(1-\gamma )q_1(a_1)q_2(a_2)\dots q_n(a_n)$, where $\gamma$ is the probability of particles being distributed clockwise, and $p_j(a_i)$ ($q_j(a_i)$) is the probability that $j$th station outcomes $a_i$ given that the particles are sent in clockwise (counterclockwise) direction.

The figure of merit:

$C_n=\frac{1}{2}|{\sum }_{\vec{A},{\vec{A}}^{\prime }\in \{0,1{\}}^n,{\vec{A}}^{\prime }\ne \vec{A},{⨁}_{i=1}^n{a}_i^{(\prime )}=0}p(\vec{A})p({\vec{A}}^{\prime })-{\sum }_{\vec{B},{\vec{B}}^{\prime }\in \{0,1{\}}^n,{\vec{B}}^{\prime }\ne \vec{B},{⨁}_{i=1}^n{b}_i^{(\prime )}=1}p(\vec{B})p({\vec{B}}^{\prime })|$

(the summation subscripts are written on one line here).

The quantum value:

One can observe that the logical maximum of ( C_n ), achieved with quantum resources (Eq.~\eqref{psiinn}), is given by $C_n^Q=\left(\genfrac{}{}{0ex}{}{2^{n-1}}{2}\right){\left(\frac{1}{2^{n-1}}\right)}^2=\frac{1}{2}-2^{-n}.$

The conjecture:

We obtained classical bound ( C_n^{\text{cl}} ) by substituting ( p_{\text{cl}} ) (Eq.~\eqref{pncl}) into the expression for ( C_n ) and performing extensive numerical optimization. For all tested values of ( n \in {2, \ldots, 10} ), we consistently obtained (\text{max}(C_n^{\text{cl}}) = \frac{1}{4} ). Consequently, we conjecture that (\text{max}(C_n^{\text{cl}}) = \frac{1}{4}) for all ( n ).

Verified locator

Source: https://arxiv.org/abs/2407.11122v1

• URL: https://arxiv.org/abs/2407.11122v1 (source main.tex, md5 c47930e35162d18bc08d65fb1201100d):
  • the classical model, l. 248–253;
  • the figure of merit, l. 259–268;
  • the quantum value, l. 272–276;
  • the conjecture, l. 277–278.
• DOI: 10.1103/PhysRevLett.133.230201 (https://doi.org/10.1103/PhysRevLett.133.230201; Crossref lists Physical Review Letters, volume 133, issued 2 December 2024).