bibkey: mukhopadhyay2018coherencemagic
authors: Chiranjib Mukhopadhyay; Sk Sazim; Arun Kumar Pati
year: 2018
title: "Coherence makes quantum systems 'magical'"
doi: 10.1088/1751-8121/aac8e8
url: https://arxiv.org/abs/1801.04807v3
claim: "For a qutrit density matrix rho with rho_12 = lambda_1 + i mu_1, rho_13 = lambda_2 + i mu_2, rho_23 = lambda_3 + i mu_3, the paper gives the nine entries of the discrete Wigner function and the sum negativity M_SN = sum |W_u| - 1 (Eq. (13)). From numerical simulation it states, for pure qutrit states, M_SN >= (C/2) sqrt(1 - C/2) with C the l1-norm coherence (Eq. (18)), and for qutrit-qubit states the trade-off 16 E_AB^2 + 9 M_SN_A^2 < 4 (Eq. (19))."
strata_touched:
  - D5/S3/Quantum/Information/QutritMagicCoherenceBound
license: citation-only
triage: anchor

Coherence makes quantum systems ‘magical’

Chiranjib Mukhopadhyay, Sk Sazim, Arun Kumar Pati, J. Phys. A: Math. Theor. 51, 414006 (2018), arXiv:1801.04807v3 (quant-ph). Quotations are from the arXiv v3 TeX source; cohbound and entbound label the panels (a) and (b) of Fig. 4.

The qutrit Wigner function:

For the qutrit case, after some algebra, we have the following discrete Wigner distribution corresponding to a qutrit density matrix $\rho$. $W_{(1,1)}=\frac{1}{3}(2{\lambda }_3+{\rho }_{11})$, $W_{(1,2)}=\frac{1}{3}(2{\lambda }_2+{\rho }_{22})$, $W_{(1,3)}=\frac{1}{3}(2{\lambda }_1+{\rho }_{33})$, $W_{(2,1)}=\frac{1}{3}(-{\lambda }_3-\sqrt{3}{\mu }_3+{\rho }_{11})$, $W_{(2,2)}=\frac{1}{3}(-{\lambda }_2+\sqrt{3}{\mu }_2+{\rho }_{22})$, $W_{(2,3)}=\frac{1}{3}(-{\lambda }_1-\sqrt{3}{\mu }_1+{\rho }_{33})$, $W_{(3,1)}=\frac{1}{3}(-{\lambda }_3+\sqrt{3}{\mu }_3+{\rho }_{11})$, $W_{(3,2)}=\frac{1}{3}(-{\lambda }_2-\sqrt{3}{\mu }_2+{\rho }_{22})$, $W_{(3,3)}=\frac{1}{3}(-{\lambda }_1+\sqrt{3}{\mu }_1+{\rho }_{33})$. Where density matrix elements ${\rho }_{12}={\lambda }_1+i{\mu }_1,{\rho }_{13}={\lambda }_2+i{\mu }_2,{\rho }_{23}={\lambda }_3+i{\mu }_3$.

Now the sum negativity $M_{SN}[\rho ]$ is simply given by $M_{SN}[\rho ]={\sum }_{\text{u}}|W_{\text{u}}|-1$.

The coherence lower bound (Eq. (18)):

However, for qutrit pure states, numerical simulation in Fig. \ref{cohbound} leads us to conjecture the following inequality, which gives a reverse, i.e., lower bound to the magic in terms of quantum coherence. Proposition - The following condition on quantum coherence, quantified via the $l_1$-norm, and ‘magic’, quantified by the sum negativity, holds for qutrit pure states $M_{SN}[|\psi ⟩]\ge \frac{C_{l_1}[|\psi ⟩]}{2}\sqrt{1-\frac{C_{l_1}[|\psi ⟩]}{2}}$

The entanglement trade-off (Eq. (19)):

Proposition - The negativity of entanglement $E_{AB}$ and the sum negativity $S{N}_A$ satisfies the following trade off relation $16E_{AB}^2+9M_{S{N}_A}^2<4$. Although an analytical proof is lacking, the numerical result furnished in Fig. \ref{entbound} strongly suggests that the proposition above is true and indeed, almost tight for pure $3\times 2$ states.

Verified locator

• DOI: https://doi.org/10.1088/1751-8121/aac8e8 (the published version was not read).
• URL: https://arxiv.org/abs/1801.04807v3 (v3, 2018-05-30, the latest version; source coherence_vs_magic.tex, md5 4da28ae07308172cbefcc4b09ae2b7b6): the Wigner function (l. 272–282), the sum negativity (l. 284), Eq. (18) (l. 368–371) and Eq. (19) (l. 374–382).