On the continuity of quantum correlation quantifiers

Abstract

We show an equivalence relation between different types of continuity of the generalized discord function (GDF) that leads to the continuity of the generalized quantum discord (GQD) in the finite-dimensional case. We extend the definition of the GQD to the case where the GDF is optimized over the set of all states with zero quantum discord and prove its continuity by showing that this set is topologically compact. However, for an unmeasured subsystem with infinite dimension, we find that this set is no longer compact while the set of locally measured states is shown to maintain this property in the space of Hilbert-Schmidt (HS) operators. This allows us to prove the continuity of the GQD when the GDF is jointly continuous in the infinite case. As an application, we obtain that the geometric discord is continuous (HS topology) and has the zero set given by the zero quantum discord set in the infinite-dimensional case as a consequence of our previous results.

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

References

  1. 1.

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

    ADS  MATH  Article  Google Scholar 

  2. 2.

    Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Camb. Philosophi. Soc. 31(4), 555 (1935)

    ADS  MATH  Article  Google Scholar 

  3. 3.

    Bell, J.S.: On the einstein podolsky rosen paradox. Physics 1(3), 195 (1964)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (1984)

  5. 5.

    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Deutsch, D.: Quantum theory, the church-turing principle and the universal quantum computer. Proc. R. Soc. A 400, 97 (1985)

    ADS  MathSciNet  MATH  Google Scholar 

  7. 7.

    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on einstein-podolsky-rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Ekert, A.K.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

    ADS  MATH  Article  Google Scholar 

  11. 11.

    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A 34, 6899 (2001)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press (1994)

  13. 13.

    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings, 28th Annual ACM Symposium on the Theory of Computing, p. 212 (1996)

  14. 14.

    Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672 (1998)

    ADS  Article  Google Scholar 

  15. 15.

    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)

    ADS  Article  Google Scholar 

  16. 16.

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

    ADS  Article  Google Scholar 

  17. 17.

    Dakić, B., et al.: Quantum discord as resource for remote state preparation. Nature Phys. 8, 666 (2012)

    ADS  Article  Google Scholar 

  18. 18.

    Horodecki, P., Tuziemski, J., Mazurek, P., Horodecki, R.: Can communication power of separable correlations exceed that of entanglement resource? Phys. Rev. Lett. 112, 140507 (2014)

    ADS  Article  Google Scholar 

  19. 19.

    Madhok, V., Datta, A.: Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011)

    ADS  Article  Google Scholar 

  20. 20.

    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: Discord and related measures. Reviews of Modern Physics 84, 1655 (2012)

    ADS  Article  Google Scholar 

  21. 21.

    Donald, M.J., Horodecki, M.: Continuity of relative entropy of entanglement. Phys. Lett. A 264, 257 (1999)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Christandl, M., Winter, A.: Squashed entanglement: an additive entanglement measure. J. Math. Phys. 45, 829 (2004)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Alicki, R., Fannes, M.: Continuity of quantum conditional information. J. Phys. A Math. General 37, L55 (2004)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Guo, Y.: Any entanglement of assistance is polygamous. Quantum Inf. Process. 17, 222 (2018)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Guo, Y., Hou, J., Wang, Y.: Concurrence for infinite-dimensional quantum systems. Quantum Inf. Process. 12, 2641 (2013)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Brodutch, A., Modi, K.: Criteria for measures of quantum correlations. Quantum Inf. Comput. 12, 721 (2012)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Dakić, B., Vedral, V., Brukner, Č.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)

    ADS  MATH  Article  Google Scholar 

  28. 28.

    Rossignoli, R., Canosa, N., Ciliberti, L.: Generalized entropic measures of quantum correlations. Phys. Rev. A 82, 052342 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Datta, A.: Quantum discord between relatively accelerated observers. Phys. Rev. A 80, 052304 (2009)

    ADS  Article  Google Scholar 

  30. 30.

    Brown, E.G., Cormier, K., Martin-Martinez, E., Mann, R.B.: Vanishing geometric discord in noninertial frames. Phys. Rev. A 86, 032108 (2012)

    ADS  Article  Google Scholar 

  31. 31.

    Tian, Z., Jing, J.: Measurement-induced-nonlocality via the unruh effect. Ann. Phys. 333, 76 (2013)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Ferraro, A., Aolita, L., Cavalcanti, D., Cucchietti, F.M., Acín, A.: Almost all quantum states have nonclassical correlations. Phys. Rev. A 81, 052318 (2010)

    ADS  Article  Google Scholar 

  33. 33.

    Guo, Y., Hou, J.: A class of separable quantum states. J. Phys. A Math. Theor. 45, 505303 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Bera, A., Das, T., Sadhukhan, D., Roy, S.S., De Sen, A., Sen, U.: Quantum discord and its allies: a review of recent progress. Rep. Progr. Phys. 81, 024001 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Zhou, T., Cui, J., Long, G.L.: Measure of nonclassical correlation in coherence-vector representation. Phys. Rev. A 84, 062105 (2011)

    ADS  Article  Google Scholar 

  36. 36.

    Carrijo, T.M., Avelar, A.T.: Weak quantum correlation quantifiers with generalized entropies. Quantum Inf. Process. 18, 308 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Guo, Z., Cao, H., Chen, Z.: Distinguishing classical correlations from quantum correlations. J. Phys. A Math. Theor. 45, 145301 (2012)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Luo, S.: Using measurement-induced disturbance to characterize correlations as classical or quantum. Phys. Rev. A 77, 022301 (2008)

    ADS  Article  Google Scholar 

  39. 39.

    Guo, J.-L., Lin-Wang, G.-L. Long: Measurement-induced disturbance and thermal negativity in 1d optical lattice chain. Ann. Phys. 330, 192 (2013)

  40. 40.

    Luo, S., Fu, S.: Measurement-induced nonlocality. Phys. Rev. Lett. 106, 120401 (2011)

    ADS  MATH  Article  Google Scholar 

  41. 41.

    Guo, Y., Li, X., Li, B., Fan, H.: Quantum correlation induced by the average distance between the reduced states. Int. J. Theor. Phys. 54, 2022 (2015)

    MATH  Article  Google Scholar 

  42. 42.

    Maziero, J., Celeri, L.C., Serra, R.M.: Symmetry aspects of quantum discord. arXiv:1004.2082 [quant-ph] (2010)

  43. 43.

    Feng-Jian, J., Hai-Jiang, L., Xin-Hu, Y., Ming-Jun, S.: A symmetric geometric measure and the dynamics of quantum discord. Chin. Phys. B 22, 040303 (2013)

    Article  Google Scholar 

  44. 44.

    Singh, U., Pati, A.K.: Quantum discord with weak measurements. Ann. Phys. 343, 141 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Rulli, C.C., Sarandy, M.S.: Global quantum discord in multipartite systems. Phys. Rev. A 84, 042109 (2011)

    ADS  Article  Google Scholar 

  46. 46.

    Dieguez, P.R., Angelo, R.M.: Weak quantum discord. Quantum Inf. Process. 17, 194 (2018)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Luo, S., Fu, S.: Geometric measure of quantum discord. Phys. Rev. A 82, 034302 (2010)

    ADS  MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

We thank the Brazillian agencies CNPq (GRANT PQ#312723/2018-0, INCT-IQ #465469/2014-0), FAPEG (GRANT PRONEX #201710267000503, PRONEN #201710267000540), CAPES(PROCAD2013) for partial support and CAPES/FAPEG(GRANT DOCFIX #201810267001518) for the fellowship of T. M. Carrijo.

Author information

Affiliations

Authors

Corresponding author

Correspondence to T. M. Carrijo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix

Lemma 1

Let \(C\subset \mathbb {R}^{p}\) be a compact convex set. The function \(F:C\rightarrow \mathbb {R}\) is continuous if, and only if, there exists a continuous function \(g:[0,1]\rightarrow \mathbb {R}\), with \(g(0)=0\), such that \(|F(x)-F((1-r)x+ry)|\le g(r)\) for all \(x,y\in C\) with \(r\in [0,1]\).

Proof

First, we suppose that F is continuous. Let \(g:[0,1]\rightarrow \mathbb {R}\) be defined as \(g(r)\equiv \sup _{(x,y)\in C\times C}|F(x)-F((1-r)x+ry)|\). Set \(H(x,y,r)\equiv x\), \(J(x,y,r)\equiv (1-r)x+ry\) and \(G(x,y,r)=|F\circ H(x,y,r)-F\circ J(x,y,r)|\). Then \(G=|.|\circ (F\circ H-F\circ J)\). Using, as a metric on \(C\times C\times [0,1]\), the function \(d((x,y,r),(x',y',r'))\equiv \Vert x-x'\Vert +\Vert y-y'\Vert +|r-r'|\), it is easy to see that H and J are continuous, implying that G is continuous on \(C\times C\times [0,1]\). The set \(C\times C\) is compact in the product topology induced by the metric \(d((x,y),(x',y'))\equiv \Vert x-x'\Vert +\Vert y-y'\Vert \). Then \(g(r)=\sup _{(x,y)\in C\times C}G(x,y,r)\) is a continuous function. Thus for all \(x,y\in C\) and \(r\in [0,1]\),

$$\begin{aligned} |F(x)-F((1-r)x+ry)|\le g(r). \end{aligned}$$
(23)

Now, suppose that Eq. (23) is valid with a continuous function \(g:[0,1]\rightarrow \mathbb {R}\) such that \(g(0)=0\). First, we prove the continuity of F on the relative interior of C, relint(C), and then prove the continuity on the relative boundary of C: relbd(C). Here, we define a metric on \({{\,\mathrm{Aff}\,}}(C)\subseteq \mathbb {R}^{n}\) by the function \(d(x,y)\equiv \Vert x-y\Vert \), where \(\Vert \cdot \Vert \) is the norm on \(\mathbb {R}^{n}\). As the relative interior of any nonempty convex set is nonempty, \(\exists x\in \) relint(C). Then, \(\exists \delta '>0\) such that \(B_{\delta '}(x)\subseteq C\), where \(B_{\delta '}(x)\equiv \{w\in {{\,\mathrm{Aff}\,}}(C):\Vert w-x\Vert <\delta '\}\). As g is continuous and \(g(0)=0\), for any \(\epsilon >0\), there is \(r\in (0,1)\) such that \(g(r)<\epsilon \). Defining \(\delta \equiv r\delta '<\delta '\), for any \(y\in B_{\delta }(x)\in C\) we have \(z\equiv (1-r^{-1})x+r^{-1}y\in {{\,\mathrm{Aff}\,}}(C)\) and \(\Vert z-x\Vert =r^{-1}\Vert x-y\Vert <\delta '\), implying that \(z\in B_{\delta '}(x)\). This means that for any \(y\in B_{\delta }(x)\), we can find \(z\in C\) such that \(y=(1-r)x+rz\). By the continuity of g and \(g(0)=0\), for any \(\epsilon >0\) there is \(r\in (0,1)\) such that \(g(r)<\epsilon \). If \(x\in \) relint(C), by the previous considerations, there is \(\delta >0\) such that \(B_{\delta }(x)\subset C\) and for any \(y\in B_{\delta }(x)\) we can find \(z\in C\) where \(y=(1-r)x+rz\). Then,

$$\begin{aligned} \forall \epsilon>0,x\in {{\,\mathrm{relint}\,}}(C),\exists \delta >0: y\in B_{\delta }(x)\subset C\Rightarrow |F(x)-F(y)|\le g(r)<\epsilon . \end{aligned}$$
(24)

Thus F is continuous on \({{\,\mathrm{relint}\,}}(C)\). Now, we prove the continuity of F on relbd(C). For any \(\epsilon \in (0,1)\) (we can put an upper bound on \(\epsilon \) w.l.o.g), there exists \(0<r<\epsilon /3\) such that \(g(r)<\epsilon /3\). With such r, for any \(x\in \) relbd(C), we can choose an arbitrary \(z\in \) relint(C) and define \(z_{x}\equiv (1-r)x+rz\in \) intrel(C). As F is continuous on \({{\,\mathrm{relint}\,}}(C)\),

$$\begin{aligned} \exists \delta ''>0:\Vert z_{x}-w\Vert<\delta '',w\in C\Rightarrow |F(z_{x})-F(w)|< \epsilon /3. \end{aligned}$$
(25)

For any \(y\in B_{\delta ''}(x)\cap C\) and defining \(z_{y}\equiv (1-r)y+rz\in C\), we have

$$\begin{aligned} \Vert z_{x}-z_{y}\Vert =(1-r)\Vert x-y\Vert<\delta ''\Rightarrow |F(z_{x})-F(z_{y})|<\epsilon /3. \end{aligned}$$
(26)

Using Proposition (26),

$$\begin{aligned} |F(x)-F(y)|&=|F(x)-F(z_{x})+F(z_{x})-F(z_{y})+F(z_{y})-F(y)| \nonumber \\&\le |F(x)-F((1-r)x+rz)|+|F(y)\nonumber \\&\quad -\,F((1-r)y+rz)|+|F(z_{x})-F(z_{y})|\nonumber \\&< 2g(r)+\epsilon /3<\epsilon . \end{aligned}$$
(27)

Then

$$\begin{aligned} \forall \epsilon>0,x\in {{\,\mathrm{relbd}\,}}(C), \exists \delta ''>0:y\in B_{\delta ''}(x)\cap C\Rightarrow |F(x)-F(y)|<\epsilon , \end{aligned}$$
(28)

which proves that F is continuous on \({{\,\mathrm{relbd}\,}}(C)\cup {{\,\mathrm{relint}\,}}(C)=C\).\(\square \)

Appendix

Lemma 2

The set \(\mathcal {P}^{B}\subset H^{\oplus n}\) is compact.

Proof

Let’s define the set \(\tilde{\mathcal {P}}^{B}\). The operator \(\tilde{X}^{B}\in H_{B}^{\oplus n}\), where \(H_{B}\) is the space of hermitian operators on \(\mathcal {H}_{B}\), is an element of \(\tilde{\mathcal {P}}^{B}\) if, and only if, Eq. (29) is satisfied for any kl:

$$\begin{aligned} F_{k}(\tilde{X}^{B})&\equiv (\tilde{\Pi }_{k}(\tilde{X}^{B}))^2-\tilde{\Pi }_{k}(\tilde{X}^{B})=0,\quad G_{k,l}(\tilde{X}^{B})\equiv {{\,\mathrm{Tr}\,}}(\tilde{\Pi }_{k}(\tilde{X}^{B})\tilde{\Pi }_{l}(\tilde{X}^{B}))-\delta _{k,l}=0,\nonumber \\ J(\tilde{X}^{B})&\equiv \sum _{k}\tilde{\Pi }_{k}(\tilde{X}^{B})-\mathbb {1}^{B}=0, \end{aligned}$$
(29)

where \(\tilde{\Pi _{k}}\) projects \(\tilde{X}^{B}=\bigoplus _{l}\tilde{X}^{B}_{l}\) on its kth component \(\tilde{\Pi _{k}}(\tilde{X}^{B})\equiv \tilde{X}^{B}_{k}\). Eq. (29) implies that the elements of \(\tilde{\mathcal {P}}^{B}\) have the form \(\tilde{P}^{B}=\bigoplus _{l} \tilde{P}^{B}_{l}\) such that \(\{\tilde{P}^{B}_{l}:l\in \{1,\ldots ,n\}\}\) is a set of rank one orthogonal projections with \(\sum _{k}\tilde{P}^{B}_{l}=\mathbb {1}^{B}\). As \(\Vert \tilde{P}^{B}_{l}\Vert _{2}=1\), we have \(\Vert \tilde{P}^{B}\Vert =\sum _{l}\Vert \tilde{P}^{B}_{l}\Vert _{2}=n\), implying that \(\tilde{\mathcal {P}}^{B}\) is a bounded set. Suppose \((\tilde{P}^{B,m})_{m\in \mathbb {N}}\) is a convergent sequence in \(H_{B}^{\oplus n}\) such that \(\tilde{P}^{B,m}\in \tilde{\mathcal {P}}^{B}\) for any m. If \(\lim _{m\rightarrow \infty }\tilde{P}^{B,m}=\tilde{X}^{B}\), by the continuity of the functions \(F_{k}\), \(G_{k,l}\) and J, we have \(F_{k}(\tilde{X}^{B})=0\), \(G_{k,l}(\tilde{X}^{B})=0\) and \(J(\tilde{X}^{B})=0\) by Eq. (29), which implies that \(\tilde{X}^{B}\in \tilde{\mathcal {P}}^{B}\). It means that \(\tilde{\mathcal {P}}^{B}\) is a closed set and, as it is also bounded, we conclude that \(\tilde{\mathcal {P}}^{B}\) is compact. Now, consider the function \(L:H_{B}^{\oplus n}\rightarrow H^{\oplus n}\) given by \(L(\tilde{X}^{B})\equiv X^{B}=\bigoplus _{l}\mathbb {1}^{A}\otimes \tilde{X}^{B}_{l}\). As L is a linear function, it is continuous, which implies that \(L(\tilde{\mathcal {P}}^{B})=\mathcal {P}^{B}\) is a compact set.\(\square \)

Appendix

Lemma 3

The function \(\tilde{\mathcal {K}}_{k}:H\times H^{\oplus n}\rightarrow \mathbb {R}\) defined as \(\tilde{\mathcal {K}}_{k}(X,Y)\equiv \Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\)\(\Pi _{k}(Y))\Vert _{1}\)\(\times \Arrowvert {{\,\mathrm{Tr}\,}}_{B}(X)-{{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))/\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}\Arrowvert ^{2}_{2}\) if \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)\)\(X\Pi _{k}(Y))\Vert _{1}\ne 0\) and \(\tilde{\mathcal {K}}_{k}(X,Y)=0\) if \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}=0\), is jointly continuous in the product topology of \(H\times H^{\oplus n}\).

Proof

Defining \(f_{B}(X,Y)\equiv {{\,\mathrm{Tr}\,}}_{B}(YXY)\) and \(\Pi ^{\beta }_{k}(X,Y)\equiv (X,\Pi _{k}(Y))\), we have \(f_{B}\circ \Pi ^{\beta }_{k}(X,\)Y) \(={{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\). As \(f_{B}\) is jointly continuous and \(\Pi _{k}\) is linear, then \(f_{B}\circ \Pi ^{\beta }_{k}\) is jointly continuous. For any point \((X,Y)\in H\times H^{\oplus n}\) such that \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}\ne 0\), \(1/\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}\) is jointly continuous. As \(\tilde{\mathcal {K}}_{k}\) is composition of jointly continuous functions, \(\tilde{\mathcal {K}}_{k}\) also has this property for any point \((X,Y)\in H\times H^{\oplus n}\) such that \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}\ne 0\).

Now, suppose that (XY) is a point such that \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y)X\Pi _{k}(Y))\Vert _{1}=0\). For arbitrary \(\delta _{x}>0\) and \(\delta _{y}>0\), suppose that \(\Vert X-X'\Vert _{1}<\delta _{x}\) and \(\Vert Y-Y'\Vert _{1}<\delta _{y}\), where \(X,X'\in H\) and \(Y,Y'\in H^{\oplus n}\). Define \(\Delta X\equiv X'-X\) and \(\Delta Y\equiv Y'-Y\). By \(|\tilde{\mathcal {K}}_{k}(X+\Delta X,Y+\Delta Y)-\tilde{\mathcal {K}}_{k}(X,Y)|=|\tilde{\mathcal {K}}_{k}(X+\Delta X,Y+\Delta Y)|\), we have

$$\begin{aligned}&|\tilde{\mathcal {K}}_{k}(X+\Delta X,Y+\Delta Y)|\nonumber \\&\quad ={{\,\mathrm{Tr}\,}}_{A}\left( |{{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y))|\right) \Big \Vert {{\,\mathrm{Tr}\,}}_{B}(X\nonumber \\&\qquad +\,\Delta X)-\frac{{{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y))}{{{\,\mathrm{Tr}\,}}_{A}\left( |{{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y))|\right) }\Big \Vert _{2}^{2} \end{aligned}$$
(30)

By the equivalence of norms, there exists a constant c such that, for any \(Z\in H_{A}\), \(\Vert Z\Vert _{2}\le c\Vert Z\Vert _{1}\). Defining \(Z^{B}\equiv {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y)) \), we have

$$\begin{aligned} \frac{\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y))\Vert _{2}}{{{\,\mathrm{Tr}\,}}_{A}\left( |{{\,\mathrm{Tr}\,}}_{B}(\Pi _{k}(Y+\Delta Y)(X+\Delta X)\Pi _{k}(Y+\Delta Y))|\right) }=\frac{\Vert Z^{B}\Vert _{2}}{\Vert Z^{B}\Vert _{1}}\le c. \end{aligned}$$
(31)

Equations (30) and (31) imply that

$$\begin{aligned} |\tilde{\mathcal {K}}_{k}(X+\Delta X,Y+\Delta Y)|&\le \left( \Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi ^{B}(Y)X\Pi ^{B}(Y))\Vert _{1}+\ldots \right) (\Vert {{\,\mathrm{Tr}\,}}_{B}(X+\Delta X)\Vert _{2}+c)^{2}. \end{aligned}$$
(32)

As \(Tr_{B}\) is a bounded linear operator, there exists \(M>0\) such that \(\Vert {{\,\mathrm{Tr}\,}}_{B}(XYZ)\Vert _{1}\le M\Vert XYZ\Vert _{1}\le M\Vert X\Vert _{1}\Vert Y\Vert _{1}\Vert Z\Vert _{1}\). By the same argument, there exists \(N>0\) such that \(\Vert \Pi _{k}(Y)\Vert _{1}\le N\Vert Y\Vert _{1}\). In Inequality (32), the “\(\ldots \)” represents several terms with the form \(\Vert {{\,\mathrm{Tr}\,}}_{B}(X'Y'Z')\Vert _{1}\), such that at least one variable is \(\Delta X\) or \(\Pi _{k}(\Delta Y)\). By these considerations and knowing that \(\Vert {{\,\mathrm{Tr}\,}}_{B}(\Pi ^{B}(Y)X\Pi ^{B}(Y))\Vert _{1}=0\), there exists a constant R such that

$$\begin{aligned} |\tilde{\mathcal {K}}_{k}(X+\Delta X,Y+\Delta Y)|&\le R\max \{\delta _{x},\delta _{y},\delta _{x}\delta _{y},\delta _{x}\delta _{y}^2\}(\Vert {{\,\mathrm{Tr}\,}}_{B}(X+\Delta X)\Vert _{2}+c)^{2}\nonumber \\&\le R\max \{\delta _{x},\delta _{y},\delta _{x}\delta _{y},\delta _{x}\delta _{y}^2\}(\Vert {{\,\mathrm{Tr}\,}}_{B}(X)\Vert _{2}+S\delta _{x}+c)^2, \end{aligned}$$
(33)

where \(S>0\) is a constant. Clearly, Inequality (33) implies that \(\tilde{\mathcal {K}}_{k}\) is jointly continuous.\(\square \)

Lemma 3 implies that:

Corollary 3

The function \(\tilde{\mathcal {K}}:D(\mathcal {H})\times \mathcal {P}^{B}\rightarrow \mathbb {R}_{\ge 0}\) defined as \(\tilde{\mathcal {K}}(\rho ,P^{B})\equiv \sum _{k}\tilde{\mathcal {K}}_{k}(\rho ,P^{B})\), where \(D(\mathcal {H})\times \mathcal {P}^{B}\subset H\times H^{\oplus n}\), is jointly continuous.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carrijo, T.M., Avelar, A.T. On the continuity of quantum correlation quantifiers. Quantum Inf Process 19, 214 (2020). https://doi.org/10.1007/s11128-020-02709-2

Download citation

Keywords

  • Quantum correlation
  • Generalized quantum discord
  • Continuity