Nonlinear steering criteria for arbitrary two-qubit quantum systems


By employing Pauli measurements, we present some nonlinear steering criteria applicable for arbitrary two-qubit quantum systems and optimized ones for symmetric quantum states. These criteria provide sufficient conditions to witness steering, which can recover the previous elegant results for some well-known states. Compared with the existing linear steering criterion and entropic criterion, ours can certify more steerable states without selecting measurement settings or correlation weights, which can also be used to verify entanglement as all steerable quantum states are entangled.

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  1. 1.

    Einstein, A., Podolsky, B., Rosen, N.: Can it quantum mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777 (1935)

    ADS  Article  Google Scholar 

  2. 2.

    Schrödinger, E.: Probability relations between separated systems. Proc. Cambridge Philos. Soc. 32(3), 446 (1936)

    ADS  Article  Google Scholar 

  3. 3.

    Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox. Phys. Rev. Lett. 98(14), 140402 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Reid, M.D.: Signifying quantum benchmarks for qubit teleportation and secure quantum communication using Einstein-Podolsky-Rosen steering inequalities. Phys. Rev. A 88(6), 062338 (2013)

    ADS  Article  Google Scholar 

  5. 5.

    He, Q., Rosales-Zárate, L., Adesso, G., Reid, M.D.: Secure Continuous Variable Teleportation and Einstein-Podolsky-Rosen Steering. Phys. Rev. Lett. 115(18), 180502 (2015)

    ADS  Article  Google Scholar 

  6. 6.

    Walk, N., Hosseini, S., Geng, J., Thearle, O., Haw, J.Y., Armstrong, S., Assad, S.M., Janousek, J., Ralph, T.C., Symul, T., Wiseman, H.M., Lam, P.K.: Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution. Optica 3(6), 634 (2016)

    ADS  Article  Google Scholar 

  7. 7.

    Kogias, I., Xiang, Y., He, Q., Adesso, G.: Unconditional security of entanglement-based continuous-variable quantum secret sharing. Phys. Rev. A 95(1), 012315 (2017)

    ADS  Article  Google Scholar 

  8. 8.

    Branciard, C., Cavalcanti, E.G., Walborn, S.P., Scarani, V., Wiseman, H.M.: One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering. Phys. Rev. A 85(1), 010301(R) (2012)

    ADS  Article  Google Scholar 

  9. 9.

    Piani, M., Watrous, J.: Necessary and Sufficient Quantum Information Characterization of Einstein-Podolsky-Rosen Steering. Phys. Rev. Lett. 114(6), 060404 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Bell, J.S.: On the Einstein-Podolsky-Rosen Porodox. Physics 1, 195 (1964)

    Article  Google Scholar 

  12. 12.

    Jones, S.J., Wiseman, H.M., Doherty, A.C.: Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering. Phys. Rev. A 76(5), 052116 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Brunner, N., Cavalcanti, D., Pironio, S., Scarant, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86(2), 419 (2014)

    ADS  Article  Google Scholar 

  14. 14.

    Bowles, J., Vértesi, T., Quintino, M.T., Brunner, N.: One-way Einstein-Podolsky-Rosen Steering. Phys. Rev. Lett. 112(20), 200402 (2014)

    ADS  Article  Google Scholar 

  15. 15.

    Händchen, V., Eberle, T., Steinlechner, S., Samblowski, A., Franz, T., Werner, R.F., Schnabel, R.: Observation of one-way Einstein-Podolsky-Rosen steering. Nature Photonics 6, 596 (2012)

    ADS  Article  Google Scholar 

  16. 16.

    Wollmann, S., Walk, N., Bennet, A.J., Wiseman, H.M., Pryde, G.J.: Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering. Phys. Rev. Lett. 116(16), 160403 (2016)

    ADS  Article  Google Scholar 

  17. 17.

    Saunders, D.J., Jones, S.J., Wiseman, H.M., Pryde, G.J.: Experimental EPR-steering using Bell-local states. Nat. Phys. 6, 845 (2010)

    Article  Google Scholar 

  18. 18.

    Schneeloch, J., Broadbent, C.J., Walborn, S.P., Cavalcanti, E.G., Howell, J.C.: Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations. Phys. Rev. A 87(6), 062103 (2013)

    ADS  Article  Google Scholar 

  19. 19.

    Ji, S.W., Lee, J., Park, J., Nha, H.: Steering criteria via covariance matrices of local observables in arbitrary-dimensional quantum systems. Phys. Rev. A 92(6), 062130 (2015)

    ADS  Article  Google Scholar 

  20. 20.

    Zhen, Y.Z., Zheng, Y.L., Cao, W.F., Li, L., Chen, Z.B., Liu, N.L., Chen, K.: Certifying Einstein-Podolsky-Rosen steering via the local uncertainty principle. Phys. Rev. A 93(1), 012108 (2016)

    ADS  Article  Google Scholar 

  21. 21.

    Zheng, Y.L., Zhen, Y.Z., Chen, Z.B., Liu, N.L., Chen, K., Pan, J.W.: Efficient linear criterion for witnessing Einstein-Podolsky-Rosen nonlocality under many-setting local measurements. Phys. Rev. A 95(1), 012142 (2017)

    ADS  Article  Google Scholar 

  22. 22.

    Zheng, Y.L., Zhen, Y.Z., Cao, W.F., Li, L., Chen, Z.B., Liu, N.L., Chen, K.: Optimized detection of steering via linear criteria for arbitrary-dimensional states. Phys. Rev. A 95(3), 032128 (2017)

    ADS  Article  Google Scholar 

  23. 23.

    Yu, S.X., Chen, Q., Zhang, C.J., Lai, C.H., Oh, C.H.: All Entangled Pure States Violate a Single Bell’s Inequality. Phys. Rev. Lett. 109(12), 120402 (2012)

    ADS  Article  Google Scholar 

  24. 24.

    Pearson, K.: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of. Science 2, 559–572 (1901)

    Google Scholar 

  25. 25.

    Hotelling, H.: Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 24(6), 417 (1933)

    Article  Google Scholar 

  26. 26.

    Jolliffe, I.T.: Principal Component Analysis. Series: Springer Series in Statistics, 2nd ed., Springer (2002)

  27. 27.

    Gross, D.J.: Symmetry in Physics: Wigner’s Legacy. Phys. Today 48(12), 46 (1995)

    ADS  Article  Google Scholar 

  28. 28.

    Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64(6), 062307 (2001)

    ADS  Article  Google Scholar 

  29. 29.

    Stockton, J.K., Geremia, J.M., Doherty, A.C., Mabuchi, H.: Characterizing the entanglement of symmetric many-particle spin-1/2 systems. Phys. Rev. A 67(2), 022112 (2003)

    ADS  Article  Google Scholar 

  30. 30.

    Táth, G., Gühne, O.: Entanglement and Permutational Symmetry. Phys. Rev. Lett. 102(7), 170503 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40(8), 4277 (1989)

    ADS  Article  Google Scholar 

  32. 32.

    Gisin, N.: Hidden quantum nonlocality revealed by local filters. Phys. Lett. A 210(3), 151 (1996)

    ADS  MathSciNet  Article  Google Scholar 

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This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11947102, the Natural Science Foundation of Anhui Province under Grant Nos. 2008085MA16 and 2008085QA26, the Key Program of West Anhui University under Grant No. WXZR201819, and the Research Fund for high-level talents of West Anhui University under Grant No. WGKQ202001004.

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Appendix: Proof of the equation \(\sum _{k=1}^{9}\lambda _{k}=\sum _{i=1}^{3}\sum _{j=1}^{3}\delta ^{2}(\sigma _{i}\otimes \sigma _{j})\)

Appendix: Proof of the equation \(\sum _{k=1}^{9}\lambda _{k}=\sum _{i=1}^{3}\sum _{j=1}^{3}\delta ^{2}(\sigma _{i}\otimes \sigma _{j})\)

In order to prove the equation \(\sum _{k=1}^{9}\lambda _{k}=\sum _{i=1}^{3} \sum _{j=1}^{3}\delta ^{2}(\sigma _{i}\otimes \sigma _{j})\), we extend principal components analysis to quantum correlation matrix \(\gamma _{mm^{'}}(\rho _{AB})\) of local observables \(\{O_{m}\}=\{\sigma _{i}\otimes \sigma _{j}\} (i,j=1,2,3, m=3(i-1)+j)\). As in classical correlation analysis, the principal components on a matrix space can be expressed as

$$\begin{aligned} P_{j}=a_{1j}O_{1}+a_{2j}O_{2}+\cdots +a_{9j}O_{9}, \end{aligned}$$

where \(j=1,2,...,9\). \(\sum _{i}a_{ij}^{*}a_{ij}=1\), and \(\sum _{i}a_{ij}^{*}a_{ik}=0\) for \(j\ne k\).

To achieve the first principal component, we use the Lagrange multiplier technique to find the maximum of a function. The Lagrangian function is defined as

$$\begin{aligned} L(a)\!=\! & {} tr[\rho (a_{11}O_{1}\!+\!a_{21}O_{2}\!+\!\cdots \!+\!a_{91}O_{9})^{2}] \!-\!\{tr[\rho (a_{11}O_{1}\!+\!a_{21}O_{2}\!+\!\cdots +a_{91}O_{9})]\}^{2}\nonumber \\&+\lambda _{1}(1-a_{11}^{2}-a_{21}^{2}-\cdots -a_{91}^{2}), \end{aligned}$$

where \(\lambda _{1}\) are the Lagrange multipliers. The necessary conditions for the maximum are

$$\begin{aligned} \frac{\partial L}{\partial a_{11}}=0; \frac{\partial L}{\partial a_{21}}=0;...; \frac{\partial L}{\partial a_{91}}=0. \end{aligned}$$

By using the properties of the trace, we obtain

$$\begin{aligned} \frac{\partial L}{\partial a_{i1}}= & {} 2a_{i1}tr(\rho O_{i}^{2}) -2a_{i1}[tr(\rho O_{i})]^{2}+\mathop {\sum }\limits _{k=1,...,9,k \ne i}a_{k1}[tr(\rho O_{i}O_{k})+tr(\rho O_{k}O_{i})]\nonumber \\&-\mathop {\sum }\limits _{k=1,...,9,k\ne i}a_{k1}[tr(\rho O_{i})tr(\rho O_{k})+tr(\rho O_{k})tr(\rho O_{i})]-2\lambda _{1} a_{i1}=0. \end{aligned}$$

By rearranging the above expression, we get

$$\begin{aligned}&a_{i1}[tr(\rho O_{i}^{2})-(tr(\rho O_{i}))^{2}] +\left\{ \mathop {\sum }\limits _{k=1,...,9,k\ne i}a_{k1} [tr(\rho O_{i}O_{k})+tr(\rho O_{k}O_{i})]\right\} /2\nonumber \\&-\mathop {\sum }\limits _{k=1,...,9,k\ne i}a_{k1}tr(\rho O_{i})tr(\rho O_{k})=\lambda _{1} a_{i1}. \end{aligned}$$

For \(i=1,...,9\), the following eigenvalue problem is obtained in compact form:

$$\begin{aligned} \gamma {{\varvec{a}}}_{1}=\lambda _{1} {{\varvec{a}}}_{1}, \end{aligned}$$

where \({{\varvec{a}}}_{1}=(a_{11},a_{21},...,a_{91})'\), \(\gamma _{ij}=(\langle O_{i}O_{j}\rangle +\langle O_{j}O_{i}\rangle )/2-\langle O_{i}\rangle \langle O_{j}\rangle \), which is exactly the quantum covariance matrix as defined in Eq.(6). It shows that \({{\varvec{a}}}_{1}\) should be chosen to be an eigenvector of the covariance matrix \(\gamma \), with eigenvalue \(\lambda _{1}\). The variance of the first principal component is

$$\begin{aligned} V(P_{1})=tr({{\varvec{a}}}_{1}^{\dagger } \gamma {{\varvec{a}}}_{1})=\lambda _{1}. \end{aligned}$$

Therefore, in order to obtain the maximum of the variance, \({{\varvec{a}}}_{1}\) should be chosen as the eigenvector corresponding to the largest eigenvalue \(\lambda _{1}\) of the covariance matrix. Similarly, for the second principal component, in order to obtain the second maximum of the variance, \({{\varvec{a}}}_{2}\) should be chosen as the eigenvector corresponding to the second largest eigenvalue \(\lambda _{2}\) of the covariance matrix. This is fully consistent with the classical principal components analysis since the variances correspond to the eigenvalues of the covariance matrix.

For a arbitrary covariance matrix \(\gamma _{ij}(\rho _{AB})\) of local observables \(\{O_{m}\}=\{\sigma _{i}\otimes \sigma _{j}\} (i,j=1,2,3, m=3(i-1)+j)\), the variance of the observables \(O_{m}\) can be analytically given as \(\sum _{m=1}^{9}\delta ^{2}(O_{m})=\sum _{i=1}^{N}\delta ^{2}P_{i}\) due to the fact \(\sum _{j}a_{ij}^{*}a_{ij}=1\). As \(\sum _{i=1}^{N}\delta ^{2}P_{i}=\sum _{i=1}^{N}\lambda _{i}\), one achieves \(\sum _{m=1}^{9}\delta ^{2}(O_{m})=\sum _{i=1}^{9}\lambda _{i}\).

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Pan, GZ., Yang, M., Yuan, H. et al. Nonlinear steering criteria for arbitrary two-qubit quantum systems. Quantum Inf Process 20, 48 (2021).

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  • Quantum steering
  • Nonlocality
  • Entanglement
  • Covariance matrices