One-way deficit and quantum phase transitions in XY model and extended Ising model


Originating in questions regarding work extraction from quantum systems coupled to a heat bath, the quantum deficit, a kind of quantum correlations in addition to entanglement and quantum discord, links quantum thermodynamics with quantum information theory. In this paper, we investigate the one-way deficit of two adjacent spins in the bulk of the XY model and the extended Ising model. We find that the one-way deficit susceptibility is able to characterize quantum phase transitions in the XY model and even topological phase transitions in the extend Ising model. This study will enlighten extensive studies of quantum phase transitions from the perspective of quantum information processing and quantum computation, including finite-temperature phase transitions, topological phase transitions, and dynamical phase transitions in a variety of quantum many-body systems.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Oppenheim, J., Horodecki, M., Horodecki, P., Horodecki, R.: Thermodynamical approach to quantifying quantum correlations. Phys. Rev. Lett. 89, 180402 (2002)

    MATH  ADS  Google Scholar 

  2. 2.

    Horodecki, M., Horodecki, K., Horodecki, P., Horodecki, R., Oppenheim, J., Sen(De), A., Sen, U.: Local information as a resource in distributed quantum systems. Phys. Rev. Lett. 90, 100402 (2003)

    MathSciNet  MATH  ADS  Google Scholar 

  3. 3.

    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)

    ADS  Google Scholar 

  4. 4.

    Horodecki, M., Horodecki, P., Horodecki, R., Oppenheim, J., Sen(De), A., Sen, U., Synak-Radtke, B.: Local versus nonlocal information in quantum-information theory: formalism and phenomena. Phys. Rev. A 71, 062307 (2005)

    MATH  ADS  Google Scholar 

  5. 5.

    Horodecki, M., Horodecki, P., Oppenheim, J.: Reversible transformations from pure to mixed states and the unique measure of information. Phys. Rev. A 67, 062104 (2003)

    MathSciNet  ADS  Google Scholar 

  6. 6.

    Streltsov, A., Kampermann, H., Bruß, D.: Quantum cost for sending entanglement. Phys. Rev. Lett. 108, 250501 (2012)

    ADS  Google Scholar 

  7. 7.

    Chuan, T.K., Maillard, J., Modi, K., Paterek, T., Paternostro, M., Piani, M.: Quantum discord bounds the amount of distributed entanglement. Phys. Rev. Lett. 109, 070501 (2012)

    ADS  Google Scholar 

  8. 8.

    Ladd, T.D., Jelezko, F., Laflamme, R., Nakamura, Y., Monroe, C., O’Brien, J.L.: Quantum computers. Nature 464, 45–53 (2010)

    ADS  Google Scholar 

  9. 9.

    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)

    MathSciNet  MATH  ADS  Google Scholar 

  10. 10.

    Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002)

    ADS  Google Scholar 

  11. 11.

    Wu, L.-A., Sarandy, M.S., Lidar, D.A.: Quantum phase transitions and bipartite entanglement. Phys. Rev. Lett. 93, 250404 (2004)

    MathSciNet  ADS  Google Scholar 

  12. 12.

    Orús, R., Wei, T.C.: Visualizing elusive phase transitions with geometric entanglement. Phys. Rev. B 82, 155120 (2010)

    ADS  Google Scholar 

  13. 13.

    Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)

    ADS  Google Scholar 

  14. 14.

    Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002)

    MathSciNet  ADS  Google Scholar 

  15. 15.

    Dillenschneider, R.: Quantum discord and quantum phase transition in spin chains. Phys. Rev. B 78, 224413 (2008)

    ADS  Google Scholar 

  16. 16.

    Zhang, X.Z., Guo, J.L.: Quantum correlation and quantum phase transition in the one-dimensional extended Ising model. Quantum Inf. Process. 16, 223 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  17. 17.

    Cui, J., Cao, J.P., Fan, H.: Entanglement-assisted local operations and classical communications conversion in quantum critical systems. Phys. Rev. A 85, 022338 (2012)

    ADS  Google Scholar 

  18. 18.

    Cui, J., Gu, M., Kwek, L.C., Santos, M.F., Fan, H., Vedral, V.: Quantum phases with differing computational power. Nat. Commun. 3, 812 (2012)

    ADS  Google Scholar 

  19. 19.

    Cui, J., Amico, L., Fan, H., Gu, M., Hamma, A., Vedral, V.: Local characterization of one-dimensional topologically ordered states. Phys. Rev. B 88, 125117 (2013)

    ADS  Google Scholar 

  20. 20.

    Franchini, F., Cui, J., Amico, L., Fan, H., Gu, M., Korepin, V., Kwek, L.C., Vedral, V.: Local convertibility and the quantum simulation of edge states in many-body systems. Phys. Rev. X 4, 041028 (2014)

    Google Scholar 

  21. 21.

    Liu, S.Y., Quan, Q., Chen, J.J., Zhang, Y.R., Yang, W.L., Fan, H.: Phase diagram of quantum critical system via local convertibility of ground state. Sci. Rep. 6, 29175 (2016)

    ADS  Google Scholar 

  22. 22.

    Chen, J.J., Cui, J., Zhang, Y.R., Fan, H.: Coherence susceptibility as a probe of quantum phase transitions. Phys. Rev. A 94, 022112 (2016)

    ADS  Google Scholar 

  23. 23.

    Wen, X.G., Niu, Q.: Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces. Phys. Rev. B 41, 9377–9396 (1990)

    ADS  Google Scholar 

  24. 24.

    Wen, X.G.: Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons. Oxford University Press on Demand, Oxford (2004)

    Google Scholar 

  25. 25.

    Kitaev, A.Y.: Unpaired Majorana fermions in quantum wires. Phys. Uspekhi 44, 131 (2001)

    ADS  Google Scholar 

  26. 26.

    Kitaev, A., Preskill, J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006)

    MathSciNet  ADS  Google Scholar 

  27. 27.

    Levin, M., Wen, X.G.: Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96, 110405 (2006)

    ADS  Google Scholar 

  28. 28.

    Li, H., Haldane, F.D.M.: Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. Phys. Rev. Lett. 101, 010504 (2008)

    ADS  Google Scholar 

  29. 29.

    Zhang, Y.R., Zeng, Y., Fan, H., You, J.Q., Nori, F.: Characterization of topological states via dual multipartite entanglement. Phys. Rev. Lett. 120, 250501 (2018)

    MathSciNet  ADS  Google Scholar 

  30. 30.

    Zhang, G., Song, Z.: Topological characterization of extended quantum Ising models. Phys. Rev. Lett. 115, 177204 (2015)

    ADS  Google Scholar 

  31. 31.

    Zhang, G., Li, C., Song, Z.: Majorana charges, winding numbers and Chern numbers in quantum Ising models. Sci. Rep. 7, 8176 (2017)

    ADS  Google Scholar 

  32. 32.

    Streltsov, A., Kampermann, H., Bruß, D.: Linking quantum discord to entanglement in a measurement. Phys. Rev. Lett. 106, 160401 (2011)

    ADS  Google Scholar 

  33. 33.

    Mukherjee, V., Divakaran, U., Dutta, A., Sen, D.: Quenching dynamics of a quantum XY spin-\(\frac{1}{2}\) chain in a transverse field. Phys. Rev. B 76, 174303 (2007)

    ADS  Google Scholar 

  34. 34.

    Garnerone, S., Jacobson, N.T., Haas, S., Zanardi, P.: Fidelity approach to the disordered quantum XY model. Phys. Rev. Lett. 102, 057205 (2009)

    ADS  Google Scholar 

  35. 35.

    Franchini, F., Its, A.R., Korepin, V.E.: Renyi entropy of the XY spin chain. J. Phys. A: Math. Theor. 41, 025302 (2008)

    MathSciNet  MATH  ADS  Google Scholar 

  36. 36.

    Liu, S.Y., Zhang, Y.R., Yang, W.L., Fan, H.: Global quantum discord and quantum phase transition in XY model. Ann. Phys. 362, 805–813 (2015)

    MathSciNet  MATH  ADS  Google Scholar 

  37. 37.

    Qin, W., Wang, C., Long, G.L.: High-dimensional quantum state transfer through a quantum spin chain. Phys. Rev. A 87, 012339 (2013)

    ADS  Google Scholar 

  38. 38.

    Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970)

    ADS  Google Scholar 

  39. 39.

    Son, W., Amico, L., Plastina, F., Vedral, V.: Quantum instability and edge entanglement in the quasi-long-range order. Phys. Rev. A 79, 022302 (2009)

    ADS  Google Scholar 

  40. 40.

    Wang, Y.K., Zhang, Y.R.: One-way deficit and quantum phase transitions in XX model. Int. J. Theor. Phys. 57, 363–370 (2018)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Niu, Y.Z., Chung, S.B., Hsu, C.H., Mandal, I., Raghu, S., Chakravarty, S.: Majorana zero modes in a quantum Ising chain with longer-ranged interactions. Phys. Rev. B 85, 035110 (2012)

    ADS  Google Scholar 

  42. 42.

    Vodola, D., Lepori, L., Ercolessi, E., Gorshkov, A.V., Pupillo, G.: Kitaev chains with long-range pairing. Phys. Rev. Lett. 113, 156402 (2014)

    ADS  Google Scholar 

  43. 43.

    Feng, X.Y., Zhang, G.M., Xiang, T.: Topological characterization of quantum phase transitions in a spin-1/2 model. Phys. Rev. Lett. 98, 087204 (2007)

    ADS  Google Scholar 

  44. 44.

    Qin, Y.Q., He, Y.Y., You, Y.Z., Lu, Z.Y., Sen, A., Sandvik, A.W., Xu, C.K., Meng, Z.Y.: Duality between the deconfined quantum-critical point and the bosonic topological transition. Phys. Rev. X 7, 031052 (2017)

    Google Scholar 

  45. 45.

    Ye, B.L., Fei, S.M.: A note on one-way quantum deficit and quantum discord. Quantum Inf. Process. 15, 279–289 (2016)

    MathSciNet  MATH  ADS  Google Scholar 

  46. 46.

    Luo, S.L.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)

    ADS  Google Scholar 

  47. 47.

    Chiu, C.K., Teo, J.C.Y., Schnyder, A.P., Ryu, S.: Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)

    ADS  Google Scholar 

Download references


We would like to thank Yu Zeng, Jin-Jun Chen, and Wei Qin for useful discussions. This work was supported by Ministry of Science and Technology of China (Grants Nos. 2016YFA0302104 and 2016YFA0300600), National Natural Science Foundation of China (Grants Nos. 91536108, 11774406, and U1530401), and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000).

Author information



Corresponding author

Correspondence to Yu-Ran Zhang.


Appendix A: One-way deficit of X states

In this section, we evaluate the one-way deficit of X states in Eq. (3). Let \(\{\Pi _{k}=|k\rangle \langle k|, k=0, 1\}\) be the local measurement on the party b along the computational base \({|k\rangle }\); then any von Neumann measurement for the party b can be written as

$$\begin{aligned} \{B_{k}=V\Pi _{k}V^{\dag }: k=0, 1\} \end{aligned}$$

given some unitary operator \(V\in U(2)\). For any V,

$$\begin{aligned} V=tI+i\vec {y}\cdot \vec {\sigma }=\left( \begin{array}{cc} t+y_{3}i &{} y_{2}+y_{1}i \\ -y_{2}+y_{1}i &{} t-y_{3}i \\ \end{array} \right) , \end{aligned}$$

with \(t\in \mathbb {R}\), \(\vec {y}=(y_{1}, y_{2}, y_{3})\in \mathbb {R}^{3}\), and

$$\begin{aligned} t^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1, \end{aligned}$$

after the measurement \({B_{k}}\), the state \(\rho ^{ab}\) will be changed into the ensemble \(\{{\rho _{k}, p_{k}}\}\) with

$$\begin{aligned} \rho _{k}=\frac{1}{p_{k}}(I\otimes B_{k})\rho (I\otimes B_{k}), \end{aligned}$$
$$\begin{aligned} p_{k}=\text {Tr}(I\otimes B_{k})\rho (I\otimes B_{k}). \end{aligned}$$

To evaluate \(\rho _{k}\) and \(p_{k}\), we write

$$\begin{aligned} p_{k}\rho _{k}= & {} (I\otimes B_{k})\rho (I\otimes B_{k})\nonumber \\= & {} \frac{1}{4}(I\otimes V)(I\otimes \Pi _{k})[I+r\sigma _{3}\otimes I+sI\otimes V^{\dag }\sigma _{3}V^{\dag }\nonumber \\&+\sum _{j=1}^3 c_{j}\sigma _{j}\otimes (V^{\dag } \sigma _{j} V)](I\otimes \Pi _{k})(I\otimes V^{\dag }). \end{aligned}$$

Using the relations [46]

$$\begin{aligned} V^{\dag }\sigma _{1}V= & {} (t^{2}+y_{1}^{2}-y_{2}^{2}-y_{3}^{2})\sigma _{1}+2(ty_{3}+y_{1}y_{2})\sigma _{2}\nonumber \\&+2(-ty_{2}+y_{1}y_{3})\sigma _{3}, \end{aligned}$$
$$\begin{aligned} V^{\dag }\sigma _{2}V= & {} 2(-ty_{3}+y_{1}y_{2})\sigma _{1}+(t^{2}+y_{2}^{2}-y_{1}^{2}-y_{3}^{2})\sigma _{2}\nonumber \\&+2(ty_{1}+y_{2}y_{3})\sigma _{3}, \end{aligned}$$
$$\begin{aligned} V^{\dag }\sigma _{3}V= & {} 2(ty_{2}+y_{1}y_{3})\sigma _{1}+2(-ty_{1}+y_{2}y_{3})\sigma _{2}\nonumber \\&+(t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2})\sigma _{3}, \end{aligned}$$


$$\begin{aligned} \Pi _{0}\sigma _{3}\Pi _{0}=\Pi _{0}, \Pi _{1}\sigma _{3}\Pi _{1}=-\Pi _{1},\Pi _{j}\sigma _{k}\Pi _{j}=0, \end{aligned}$$

for \(j=0, 1, k=1, 2\), we obtain

$$\begin{aligned} p_{0}\rho _{0}= & {} \frac{1}{4}[I+sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+(r+c_{3}z_{3})\sigma _{3}]\nonumber \\&\otimes (V\Pi _{0}V^{\dag }), \end{aligned}$$
$$\begin{aligned} p_{1}\rho _{1}= & {} \frac{1}{4}[I-sz_{3}I-c_{1}z_{1}\sigma _{1}-c_{2}z_{2}\sigma _{2}+(r-c_{3}z_{3})\sigma _{3}]\nonumber \\&\otimes (V\Pi _{1}V^{\dag }), \end{aligned}$$


$$\begin{aligned} z_{1}= & {} 2(-ty_{2}+y_{1}y_{3}),\ z_{2}=2(ty_{1}+y_{2}y_{3}), \end{aligned}$$
$$\begin{aligned} z_{3}= & {} t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2}.\ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

Then, we will evaluate the eigenvalues of \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\) by \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}=p_{0}\rho _{0}+p_{1}\rho _{1}\), and

$$\begin{aligned} p_{0}\rho _{0}+p_{1}\rho _{1}&=\frac{1}{4}(I+r\sigma _{3})\otimes I\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad +\frac{1}{4}(sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+c_{3}z_{3}\sigma _{3}) \otimes V\sigma _{3}V^{\dag }. \end{aligned}$$

The eigenvalues of \(p_{0}\rho _{0}+p_{1}\rho _{1}\) are the same with the eigenvalues of the states \((I\otimes V^{\dag })(p_{0}\rho _{0}+p_{1}\rho _{1})(I\otimes V)\), and

$$\begin{aligned}&(I\otimes V^{\dag })(p_{0}\rho _{0}+p_{1}\rho _{1})(I\otimes V)\nonumber \\&\quad =\frac{1}{4}(sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+c_{3}z_{3}\sigma _{3})\otimes \sigma _{3}\nonumber \\&\qquad +\frac{1}{4}(I+r\sigma _{3})\otimes I. \end{aligned}$$

The eigenvalues of the states in Eq. (A16) are given in Eqs. (8,9). Thus, the entropy of \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\) is

$$\begin{aligned} S\left( \sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) =-\sum _{i=1}^{4}w_{i}\log w_{i}. \end{aligned}$$

When \(\gamma \) and h are fixed, \(r, s, c_{1}, c_{2}\), and \(c_{3}\) are constant. By using \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\), it converts the problem about \(\min _{\{\Pi _{k}\}} S(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k})\) to the problem about the function of three variables \(z_{1},z_{2},z_{3}\) for minimum, that is,

$$\begin{aligned} \min _{\{\Pi _{k}\}} S\left( \sum \limits _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) =\!\!\!\!\min \limits _{\{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\}}\!\!\!\!S\left( \sum \limits _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) . \end{aligned}$$

Therefore, by Eqs. (1), (6), and (A18), the one-way deficit of X states in Eq. (3) is obtained as shown in Eq. (7).

Appendix B: Diagonalization, winding numbers and characteristic functions of extended Ising model

The Hamiltonian of the extended Ising model (10) can be mapped to a spinless fermion Hamiltonian by the Jordan-Wigner transformation \(-c_1=\sigma _1^+=(\sigma ^x_1+i\sigma ^y_1)/2\), \(-c_j=\sigma _j^+\prod _{i=1}^{j-1}\sigma ^z_i\) [30]. In the thermodynamic limit \(L\gg 1\), we can use the Bogoliubov–Fourier transformation to obtain a Bogoliubov–de Gennes (BdG) Hamiltonian as [30]

$$\begin{aligned} H=\sum _{\phi }(c_\phi ^\dag \ c_{-\phi })\mathcal {H}_\phi \left( \begin{array}{c}{c_\phi }\\ {c_{-\phi }^\dag }\end{array}\right) , \end{aligned}$$

where the complete set of wavevectors is \(\phi =2\pi m/L\) with \(m=-\frac{L-1}{2},-\frac{L-3}{2},\cdots ,\frac{L-3}{2},\frac{L-1}{2}\). Here, we can write [30]

$$\begin{aligned} \mathcal {H}_\phi =\varvec{r}(\phi )\cdot \varvec{\sigma }, \end{aligned}$$

with the vector \(\varvec{r}(\phi )=(0~Y(\phi )~Z(\phi ))\) in the auxiliary two-dimensional \(y-z\) space,

$$\begin{aligned} Y(\phi )=\lambda \delta \sin (2\phi )+\gamma \sin \phi , \end{aligned}$$
$$\begin{aligned} Z(\phi )=\lambda \cos (2\phi )+\cos \phi -h, \end{aligned}$$

and \(\varvec{\sigma }=(\sigma _1~\sigma _2~\sigma _3)\). The winding number of the closed loop in auxiliary \(y-z\) plane around the origin point can be written as

$$\begin{aligned} \nu =\frac{1}{2\pi }\oint (YdZ-ZdY)/|\varvec{r}|^2, \end{aligned}$$

which is used to identify different topological phases in the BDI class one-dimensional fermion systems [47].

Using the Bogoliubov transformation \(c_\phi =\cos \frac{\Theta }{2}\eta _\phi +i\sin \frac{\Theta }{2}\eta _{-\phi }^\dag \) with \(\tan \Theta \equiv Y(\phi )/Z(\phi )\), we can diagonalize the Hamiltonian as

$$\begin{aligned} {H}=\sum _\phi \omega _\phi (\eta _\phi ^\dag \eta _\phi -{1}/{2}) \end{aligned}$$

and obtain the ground state as

$$\begin{aligned} |\mathcal {G}\rangle =\prod _{\phi }\left[ \cos \frac{\Theta }{2}+i\sin \frac{\Theta }{2}\eta _\phi ^{\dag }\eta _{-\phi }^\dag \right] |0\rangle , \end{aligned}$$

where \(|0\rangle \) is the vacuum state and the energy spectra are

$$\begin{aligned} \omega _\phi =\sqrt{Y(\phi )^2+Z(\phi )^2}. \end{aligned}$$

Via a substitute \(\zeta (\phi )\equiv \exp (i\phi )\), the characteristic function is defined as [29]

$$\begin{aligned}&g(\zeta )\equiv Z(\phi )+iY(\phi ), \end{aligned}$$
$$\begin{aligned}&=\lambda [\zeta ^2+(1-\delta )\zeta ^{-2}/2]+\zeta +(1-\gamma )\zeta ^{-1}/2-h, \end{aligned}$$

with which we can calculate the critical points for the quantum topological phase transitions by the characteristic equation \(g(\zeta )=0\) with \(|\zeta |=1\) required.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Zhang, Y. & Fan, H. One-way deficit and quantum phase transitions in XY model and extended Ising model. Quantum Inf Process 18, 19 (2019).

Download citation


  • One-way deficit
  • Quantum phase transition
  • Symmetry-protected topological order