bibkey: bai2007multipartite authors: Yan-Kui Bai; Dong Yang; Z. D. Wang year: 2007 title: “Multipartite quantum correlation and entanglement in four-qubit pure states” doi: 10.1103/PhysRevA.76.022336 url: https://arxiv.org/abs/quant-ph/0703098v2 claim: “For a four-qubit pure state, tau_k = 2(1 - tr rho_k^2) is the linear entropy of qubit k and C_ij = max(sqrt(lambda_1) - sqrt(lambda_2) - sqrt(lambda_3) - sqrt(lambda_4), 0) is the concurrence of the pair ij, where lambda_1 >= … >= lambda_4 are the eigenvalues of rho_ij (sigma_y x sigma_y) rho_ij^* (sigma_y x sigma_y). The sum of the residual correlations is M = sum_k tau_k - 2 sum_{p>q} C_pq^2 (Eq. (7)), and the authors conjecture that M is an entanglement monotone, that is, does not increase on average under LOCC.” strata_touched:

• D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation license: citation-only triage: anchor

Multipartite quantum correlation and entanglement in four-qubit pure states

Yan-Kui Bai, Dong Yang, Z. D. Wang, Phys. Rev. A 76, 022336 (2007); arXiv:quant-ph/0703098v2. Quotations are from the arXiv v2 source, with its macros kept as printed.

The linear entropy of qubit $k$:

where the linear entropy \tau_{k(R_k)}=2(1-\mbox{tr}\rho_k^2) \cite{san00} characterizes the total quantum correlation between qubit $k$ and the remaining qubits $R_k$.

The concurrence of a pair of qubits:

C_{ij}=\mbox{max}[(\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}- \sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}), 0], where the decreasing positive real numbers ${\lambda }_i$s are the eigenvalues of matrix ${\rho }_{ij}({\sigma }_y\otimes {\sigma }_y){\rho }_{ij}^{\ast }({\sigma }_y\otimes {\sigma }_y)$ \cite{woo01}.

The requirement on a multipartite measure:

(3) it does not increase on average under the LOCC \emph{i.e.}, the measure is entanglement monotone.

The sum of the residual correlations (Eq. (7)):

$M=M_A+M_B+M_C+M_D={\sum }_k{\tau }_{k(R_k)}-2{\sum }_{p>q}{C}_{pq}^2$,

The conjecture:

Nevertheless, we conjecture that the correlation $M$ is an entanglement monotone

The paper supports the conjecture with an argument that the decrease of two components of $M$ under a POVM on qubit $A$ compensates the increase of the third, and with numerical values of the average change of $M$ for nine representative states of the SLOCC classification under diagonal two-outcome POVMs with diagonal entries $\alpha ,\beta \in [0.05,0.95]$ (Fig. 2). It also proves that the single residual correlation $M_A$ does not increase on average under a POVM on qubit $A$ (Eq. (6)), and shows by examples in the Appendix that the single residual correlations $M_k$ are not entanglement monotones.

The encoding takes four-qubit vectors as functions $(\mathrm{Fin}4\to \mathrm{Fin}2)\to \mathbb{C}$, qubits $A,B,C,D$ at the indices $0,1,2,3$; the one- and two-qubit reduced states are the existing partial traces reducedState, the matrix $({\sigma }_y\otimes {\sigma }_y){\rho }^{\ast }({\sigma }_y\otimes {\sigma }_y)$ is the existing timeReversed, the eigenvalues are the roots of the characteristic polynomial, with multiplicity, sorted decreasingly, and a $2\times 2$ matrix $K$ acts on qubit $A$ as the existing localOp 0 K.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.76.022336 (Phys. Rev. A 76, 022336 (2007)).
• URL: https://arxiv.org/abs/quant-ph/0703098v2 (v2, 2007-09-01, the latest version; source Four-qubit-31Jul.tex, md5 aa80185a0631338fccbaf59466008d55): the linear entropy (l. 139–141), the concurrence (l. 199–205), the monotonicity requirement (l. 225–229), Eq. (6) (l. 263–266), Eq. (7) (l. 362–366), the conjecture (l. 374–375), the compensation argument (l. 376–390), the numerical evidence (l. 392–422), the per-qubit measure $E_{ms}=M/4$ (l. 453–465) and the $N$-qubit sum $M_N$ (l. 518–528).