[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Quantumness-generating capability of quantum dynamics

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We study quantumness-generating capability of quantum dynamics, where quantumness refers to the noncommutativity between the initial state and the evolving state. In terms of the commutator of the square roots of the initial state and the evolving state, we define a measure to quantify the quantumness-generating capability of quantum dynamics with respect to initial states. Quantumness-generating capability is absent in classical dynamics and hence is a fundamental characteristic of quantum dynamics. For qubit systems, we present an analytical form for this measure, by virtue of which we analyze several prototypical dynamics such as unitary dynamics, phase damping dynamics, amplitude damping dynamics, and random unitary dynamics (Pauli channels). Necessary and sufficient conditions for the monotonicity of quantumness-generating capability are also identified. Finally, we compare these conditions for the monotonicity of quantumness-generating capability with those for various Markovianities and illustrate that quantumness-generating capability and quantum Markovianity are closely related, although they capture different aspects of quantum dynamics.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Dirac, P.A.M.: The physical interpretation of the quantum dynamics. Proc. R. Soc. Lond. 113, 621 (1927)

    Article  ADS  MATH  Google Scholar 

  2. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)

    MATH  Google Scholar 

  3. Zanardi, P., Zalka, C., Faoro, L.: Entangling power of quantum evolutions. Phys. Rev. A 62, 030301 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  4. Cao, X., Li, N., Luo, S.: Decoherent information of quantum operations. Recent Dev. Stoch. Dyn. Stoch. Anal. 8, 23 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Luo, S., Li, N.: Decoherence and measurement-induced correlations. Phys. Rev. A 84, 052309 (2011)

    Article  ADS  Google Scholar 

  6. Luo, S., Fu, S., Li, N.: Decorrelating capabilities of operations with application to decoherence. Phys. Rev. A 82, 052122 (2010)

    Article  ADS  Google Scholar 

  7. Wang, L., Yu, C.S.: The roles of a quantum channel on a quantum state. Int. J. Theor. Phys. 53, 715 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mani, A., Karimipour, V.: Cohering and decohering power of quantum channels. Phys. Rev. A 92, 032331 (2015)

    Article  ADS  Google Scholar 

  9. García-Díaz, M., Egloff, D., Plenio, M.B.: A note on coherence power of N-dimensional unitary operators. Quantum Inf. Comput. 16, 1282 (2016)

    MathSciNet  Google Scholar 

  10. Bu, K., Kumar, A., Zhang, L., Wu, J.: Cohering power of quantum operations. Phys. Lett. A 381, 1670 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Zanardi, P., Styliaris, G., Venuti, C.L.: Coherence-generating power of quantum unitary maps and beyond. Phys. Rev. A 95, 052306 (2017)

    Article  ADS  Google Scholar 

  12. Zanardi, P., Styliaris, G., Venuti, C.L.: Measures of coherence-generating power for quantum unital operations. Phys. Rev. A 95, 052307 (2017)

    Article  ADS  Google Scholar 

  13. Wolf, M.M., Eisert, J., Cubitt, T.S., Cirac, J.I.: Assessing non-Markovian quantum dynamics. Phys. Rev. Lett. 101, 150402 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Rivas, A., Huelga, S.F., Plenio, M.B.: Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett. 105, 050403 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  15. Hou, S.C., Yi, X.X., Yu, S.X., Oh, C.H.: Alternative non-Markovianity measure by divisibility of dynamical maps. Phys. Rev. A 83, 062115 (2011)

    Article  ADS  Google Scholar 

  16. Breuer, H.-P., Laine, E.-M., Piilo, J.: Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  17. Rajagopal, A.K., Devi, A.R.U., Rendell, R.W.: Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms. Phys. Rev. A 82, 042107 (2010)

    Article  ADS  Google Scholar 

  18. Lu, X.-M., Wang, X., Sun, C.P.: Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A 82, 042103 (2010)

    Article  ADS  Google Scholar 

  19. Song, H., Luo, S., Hong, Y.: Quantum non-Markovianity based on the Fisher-information matrix. Phys. Rev. A 91, 042110 (2015)

    Article  ADS  Google Scholar 

  20. Luo, S., Fu, S., Song, H.: Quantifying non-Markovianity via correlations. Phys. Rev. A 86, 044101 (2012)

    Article  ADS  Google Scholar 

  21. Jiang, M., Luo, S.: Comparing quantum Markovianities: distinguishability versus correlations. Phys. Rev. A 88, 034101 (2013)

    Article  ADS  Google Scholar 

  22. Li, N., Luo, S., Mao, Y.: Quantifying the quantumness of quantum ensembles. Phys. Rev. A 96, 022132 (2017)

    Article  ADS  Google Scholar 

  23. Fuchs, C.A.: Just two nonorthogonal quantum states. arXiv:quant-ph/9810032 (1998)

  24. Fuchs, C.A., Sasaki, M.: The quantumness of a set of quantum states. arXiv:quant-ph/0302108 (2003)

  25. Fuchs, C.A., Sasaki, M.: Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states. Quantum Inf. Comput. 3, 377 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Horodecki, M., Horodecki, P., Horodecki, R., Piani, M.: Quantumness of ensemble from no-broadcasting principle. Int. J. Quantum Inf. 4, 105 (2006)

    Article  MATH  Google Scholar 

  27. Oreshkov, O., Calsamiglia, J.: Distinguishability measures between ensembles of quantum states. Phys. Rev. A 79, 032336 (2009)

    Article  ADS  Google Scholar 

  28. Luo, S., Li, N., Cao, X.: Relative entropy between quantum ensembles. Period. Math. Hung. 59, 223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Luo, S., Li, N., Sun, W.: How quantum is a quantum ensemble. Quantum Inf. Process. 9, 711 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Luo, S., Li, N., Fu, S.: Quantumness of quantum ensembles. Theor. Math. Phys. 169, 1724 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhu, X., Pang, S., Wu, S., Liu, Q.: The classicality and quantumness of a quantum ensemble. Phys. Lett. A 375, 1855 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  32. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  34. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  35. Pearle, P.: Simple derivation of the Lindblad equation. Eur. J. Phys. 33, 805–822 (2012)

    Article  MATH  Google Scholar 

  36. de Vega, I., Alonso, D.: Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 89, 015001 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  37. Diósi, L.: Comment on “Uniqueness of the equation for quantum state vector collapse”. Phys. Rev. Lett. 112, 108901 (2014)

    Article  ADS  Google Scholar 

  38. Caiaffa, M., Smirne, A., Bassi, A.: Stochastic unraveling of positive quantum dynamics. Phys. Rev. A 95, 062101 (2017)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China, Grant Nos. 11375259, 11405262, the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Grant No. Y029152K51, the Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences, Grant No. 2008DP173182.

Author information

Authors and Affiliations

  1. RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Nan Li, Shunlong Luo & Yuanyuan Mao

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Nan Li, Shunlong Luo & Yuanyuan Mao

Authors
  1. Nan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shunlong Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuanyuan Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nan Li.

Appendix

Appendix

1.1 A. Proof of Eq. (3)

Consider the quantum ensemble \(\mathcal {E}=\{(p_1,\rho _1),(p_2,\rho _2)\}\) consisting of two-qubit states \(\rho _i=\frac{1}{2}(\mathbf{1}+\mathbf{r}_i\cdot {\varvec{\sigma }})\) with fixed a prior probabilities \(p_i,~i=1,2\), where \(\mathbf{1}\) is the identity operator, \(\mathbf{r}_i\) the Bloch vector of the state \(\rho _i\). By the definition of the quantumness of ensembles Eq. (2), we get

$$\begin{aligned} Q(\mathcal {E})=-2\sqrt{p_1p_2} \mathrm{tr}\left[ \sqrt{\rho _1},\sqrt{\rho _2}\right] ^2=4\sqrt{p_1p_2}\left( \mathrm{tr}\rho _1\rho _2-\mathrm{tr}(\sqrt{\rho _1}\sqrt{\rho _2})^2\right) . \end{aligned}$$

By direct calculations, we obtain that \(\mathrm{tr}\rho _1\rho _2=\frac{1}{2}(1+\mathbf{r}_1\cdot \mathbf{r}_2),\) and

$$\begin{aligned} \mathrm{tr}\left( \sqrt{\rho _1}\sqrt{\rho _2}\right) ^2=\frac{1}{2}(1+\mathbf{r}_1\cdot \mathbf{r}_2)-\left( 1-\sqrt{1-r_1^2}\right) \left( 1-\sqrt{1-r_2^2}\right) \frac{|\mathbf{r}_1\times \mathbf{r}_2|^2}{4 r_1^2 r_2^2}. \end{aligned}$$

Here \({r}_i=|\mathbf{r}_i|\). Combining these two expressions, we have

$$\begin{aligned} Q(\mathcal {E})= & {} \frac{\sqrt{p_1p_2}|\mathbf{r}_1 \times \mathbf{r}_2|^2}{\left( 1+\sqrt{1-r_1^2}\right) \left( 1+\sqrt{1-r_2^2}\right) }. \end{aligned}$$

Then Eq. (3) is derived from the above equation directly.

1.2 B. Proof of \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}|G_t|\le 0\) for amplitude damping dynamics

Let \(\lambda _t=1-|G_t|^2\), then the statement of \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}|G_t|\le 0\) is equivalent to \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}\lambda _t\ge 0\). By numerical analysis, we know that \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}\lambda _t\ge 0\). Analytically, we will prove that

  1. (i)

    \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}\lambda _t\ge 0\) for any initial state when \(1/2<\lambda _t<15/16\).

  2. (ii)

    \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}\lambda _t\ge 0\) holds for any pure initial state.

Let us begin with rewriting \(Q(\varLambda _t|{\rho })\) as

$$\begin{aligned} Q(\varLambda _t|{\rho })=\frac{ r^2 \sin ^2 \theta \left( \lambda _t+(1-\lambda _t)r\cos \theta -\sqrt{1-\lambda _t}r\cos \theta \right) ^2}{2\left( 1+\sqrt{1-r^2}\right) \left( 1+\sqrt{1-r_t^2}\right) }, \end{aligned}$$

where \(1-r_t^2=(1-\lambda _t)(1-r^2)+\lambda _t(1-\lambda _t)(1-\cos \theta )^2\). Its derivative with respect to \(\lambda _t\) is

$$\begin{aligned} \frac{\mathrm{d}Q(\varLambda _t|{\rho })}{\mathrm{d}\lambda _t} =\frac{r^2\sin ^2\theta }{2(1+\sqrt{1-r^2})}\frac{\mathrm{d}f(\lambda _t)}{\mathrm{d} \lambda _t} \end{aligned}$$

where

$$\begin{aligned} f(\lambda _t)=\frac{\left( \lambda _t+(1-\lambda _t)r\cos \theta -\sqrt{1-\lambda _t}r\cos \theta \right) ^2}{1+\sqrt{1-r_t^2}}, \end{aligned}$$

from which we have

$$\begin{aligned} \frac{d f(\lambda _t)}{d \lambda _t}=\frac{ \lambda _t+(1-\lambda _t)r\cos \theta -\sqrt{1-\lambda _t} r\cos \theta }{\left( 1+\sqrt{1-r_t^2}\right) ^2}g(\lambda _t). \end{aligned}$$

Here

$$\begin{aligned} g(\lambda _t)= & {} 2\left( 1+\left( \frac{1}{2\sqrt{1-\lambda _t}}-1\right) r\cos \theta \right) \left( \sqrt{1-r_t^2}+1\right) \\&+\,\frac{(1-r^2)-(1-2\lambda _t)(1-\cos \theta )^2}{2\sqrt{1-r_t^2}}\left( \lambda _t+(1-\lambda _t)r\cos \theta -r\cos \theta \sqrt{1-\lambda _t}\right) . \end{aligned}$$

From the above equation, it is easy to see that when

$$\begin{aligned} -1<\frac{1}{2\sqrt{1-\lambda _t}}-1<1 \text { and } 1-2\lambda _t<0, \end{aligned}$$

which is equivalent to \(1/2<\lambda _t<15/16\), \(\mathrm{d} Q(\varLambda _t|\rho )/\mathrm{d}\lambda _t \ge 0\). This completes the proof of (i).

Next, if the initial state is pure, let \(x=1-\cos \theta \), then \(x \in [0,2]\) and

$$\begin{aligned} Q(\varLambda _t|{\rho })=\frac{ x(2-x)\left( 1-\sqrt{1-\lambda _t}+\left( \sqrt{1-\lambda _t}-1+\lambda _t\right) x\right) ^2}{2\left( 1+\sqrt{\lambda _t(1-\lambda _t)}x\right) } \end{aligned}$$

The derivative can be evaluated as

$$\begin{aligned} \frac{\mathrm{d}Q(\varLambda _t|{\rho })}{\mathrm{d}\lambda _t}= & {} \frac{(2-x)\left( 1-\sqrt{1-\lambda _t}+(\sqrt{1-\lambda _t}-1+\lambda _t)x\right) }{4\sqrt{\lambda _t}(1-\lambda _t)\left( 1+\sqrt{\lambda _t(1-\lambda _t)}x\right) ^2} h(x) \end{aligned}$$

where \(h(x)=a x^3 + b x^2 + c x,\) with

$$\begin{aligned} a&=(1-\lambda _t)\left( \sqrt{1-\lambda _t}+2\lambda _t\sqrt{1-\lambda _t}-1\right) ,\\ b&=\sqrt{1-\lambda _t}\left( \sqrt{1-\lambda _t}-1+2\lambda _t-2\sqrt{\lambda _t}+4\sqrt{\lambda _t(1-\lambda _t)}\right) ,\\ c&=2\sqrt{\lambda _t(1-\lambda _t)}. \end{aligned}$$

After tedious calculations, we have \(h(x)\ge h(0)=0\) when \(x \in [0,2]\), which leads directly to \(\mathrm{d} Q(\varLambda _t|{\rho })/\mathrm{d}\lambda _t\ge 0\).

1.3 C. Proof of \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}p_{0t}\le 0\) for Case 2 of random unitary dynamics

Let \(\lambda _t=1-p_{0t}\), then the quantumness-generating capability \(Q(\varLambda _t|\rho )\) can be rewritten as

$$\begin{aligned} Q(\varLambda _t|\rho )=\frac{2\left( 3\alpha -1\right) ^2\left( r_x^2+r_y^2\right) r_z^2\lambda _t^2}{\left( 1+\sqrt{1-r^2}\right) \left( 1+\sqrt{1-r_t^2}\right) }, \end{aligned}$$

in which \(r_t^2=(1-2(1-\alpha )\lambda _t)^2(r_x^2+r_y^2)+(1-4\alpha \lambda _t)^2 r_z^2\). The derivative of \(Q(\varLambda _t|\rho )\) with respect to \(\lambda _t\) is

$$\begin{aligned} \frac{\mathrm{d}Q(\varLambda _t|{\rho })}{\mathrm{d}\lambda _t}=\frac{4\lambda _t (3\alpha -1)^2(r_x^2+r_y^2)r_z^2}{\left( 1+\sqrt{1-r^2}\right) \left( 1+\sqrt{1-r_t^2}\right) ^2\sqrt{1-r_t^2}}f(\lambda _t), \end{aligned}$$

where

$$\begin{aligned} f(\lambda _t)= & {} 1-r_t^2+\sqrt{1-r_t^2}-\lambda _t\left( 1-\alpha \right) \left( 1-2(1-\alpha )\lambda _t\right) \left( r_x^2+r_y^2\right) \\&-2\alpha \lambda _t(1-4\alpha \lambda _t)r_z^2\\\ge & {} 2-2r^2\left( 1-4\alpha \lambda _t\right) \left( 1-3\alpha \lambda _t\right) +\lambda _t \left( r_x^2+r_y^2\right) \left( 1-3\alpha \right) \left( 7-6 \lambda _t(\alpha +1)\right) . \end{aligned}$$

When \(\alpha \in [0,1/2]\), we have \(f(\lambda _t)\ge 0\), which is equivalent to \(\mathrm{d}Q(\varLambda _t|{\rho })/ \mathrm{d}\lambda _t>0\), and \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}p_{0t}\le 0\) follows directly by the chain rule.

1.4 D. Proof of \(\mathrm{d}Q(\varLambda _t|\rho )/\mathrm{d}t\ge 0\) for Case 3 of random unitary dynamics

For the initial state \(\rho \) with Bloch vector \(\mathbf{r}=(r_x,r_y,r_z)\) and the evolving state \(\rho _t\) with the Bloch vector \(\mathbf{r}_t=(r_x,r_y,e^{-4t}r_z)\), we know

$$\begin{aligned} Q(\varLambda _t|\rho )= & {} \frac{ \left( r_x^2+r_y^2\right) r_z^2\left( 1-e^{-4t}\right) ^2}{2\left( 1+\sqrt{1-r^2}\right) \left( 1+\sqrt{1-r_t^2}\right) }. \end{aligned}$$

Now we prove that

$$\begin{aligned} \frac{\mathrm{d}Q(\varLambda _t|{\rho })}{\mathrm{d}t}\ge 0 \ \mathrm{for\ any}\ t\ge 0. \end{aligned}$$

Let \(f_t=(1-e^{-4t})^2/(1+\sqrt{1-r_t^2})\), then

$$\begin{aligned} \frac{\mathrm{d}Q(\varLambda _t|{\rho })}{\mathrm{d}t}=\frac{(r_x^2+r_y^2)r_z^2}{2(1+\sqrt{1-r^2})}\frac{\mathrm{d}f_t}{\mathrm{d}t} \end{aligned}$$

where

$$\begin{aligned} \frac{\mathrm{d}f_t}{\mathrm{d}t}= & {} \frac{4(1-e^{-4t})}{\sqrt{1-r_t^2} \left( 1+\sqrt{1-r_t^2}\right) ^2}\left( 2e^{-4t}\left( 1-r_t^2+\sqrt{1-r_t^2}\right) -e^{-8t}(1-e^{-4t})r_z^2\right) \\\ge & {} \frac{4(1-e^{-4t})}{\sqrt{1-r_t^2}\left( 1+\sqrt{1-r_t^2}\right) ^2}\left( 4e^{-4t}(1-r_t^2)-e^{-8t}(1-e^{-4t})r_z^2\right) \\= & {} \frac{4(1-e^{-4t})}{\sqrt{1-r_t^2}\left( 1+\sqrt{1-r_t^2}\right) ^2}\left( 4e^{-4t}(1-r^2)+e^{-4t}(3e^{-4t}+4)(1-e^{-4t})r_z^2\right) \end{aligned}$$

which is positive when \(t\ge 0\). This completes the proof.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, N., Luo, S. & Mao, Y. Quantumness-generating capability of quantum dynamics. Quantum Inf Process 17, 74 (2018). https://doi.org/10.1007/s11128-018-1829-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-1829-6

Keywords

Search

Navigation