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Entropic uncertainty lower bound for a two-qubit system coupled to a spin chain with Dzyaloshinskii–Moriya interaction

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Abstract

The uncertainty principle is one of the key concepts in quantum theory. This principle states that it is not possible to measure two incompatible observables simultaneously and accurately. In quantum information theory, the uncertainty principle is formulated using the concept of entropy. In this work we consider the entropic uncertainty relation in the presence of quantum memory. We study the dynamics of entropic uncertainty bound for a two-qubit quantum system coupled to a spin chain with Dzyaloshinskii–Moriya interaction and we investigate the effect of environmental parameters on the entropic uncertainty bound. Notably, our results reveal that there exist some environmental parameters which can be changed to suppress the entropic uncertainty bound.

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References

  • Adabi, F., Salimi, S., Haseli, S.: Tightening the entropic uncertainty bound in the presence of quantum memory. Phys. Rev. A 93, 062123 (2016a)

    ADS  Google Scholar 

  • Adabi, F., Salimi, S., Haseli, S.: Reducing the entropic uncertainty lower bound in the presence of quantum memory via LOCC. Europhys. Lett. 115, 60004 (2016b)

    ADS  Google Scholar 

  • Ballester, M.A., Wehner, S.: Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases. Phys. Rev. A 75, 022319 (2007)

    ADS  Google Scholar 

  • Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957)

    ADS  MathSciNet  MATH  Google Scholar 

  • Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659–662 (2010)

    Google Scholar 

  • Bialynicki-Birula, I.: Formulation of the uncertainty relations in terms of the Rnyi entropies. Phys. Rev. A 74, 052101 (2006)

    ADS  MathSciNet  Google Scholar 

  • Carvalho, A.R.R., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004)

    ADS  Google Scholar 

  • Chen, P.F., Sun, W.Y., Ming, F., Huang, A.J., Wang, D., Ye, L.: Observation of quantum-memory-assisted entropic uncertainty relation under open systems, and its steering. Laser Phys. Lett. 15, 015206 (2018a)

    ADS  Google Scholar 

  • Chen, M.N., Sun, W.Y., Huang, A.J., Ming, F., Wang, D., Ye, L.: Unveiling the decoherence effect of noise on the entropic uncertainty relation and its control by partially collapsed operations. Laser Phys. Lett. 15, 015207 (2018b)

    ADS  Google Scholar 

  • Chen, P.F., Ye, L., Wang, D.: The effect of non-Markovianity on the measurement-based uncertainty. Eur. Phys. J. D 73, 108 (2019a)

    ADS  Google Scholar 

  • Chen, M.N., Wang, D., Ye, L.: Characterization of dynamical measurement’s uncertainty in a two-qubit system coupled with bosonic reservoirs. Phys. Lett. A 383, 977–984 (2019b)

    ADS  Google Scholar 

  • Coles, P.J., Piani, M.: Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)

    ADS  Google Scholar 

  • De Vicente, J.I., Sanchez-Ruiz, J.: Improved bounds on entropic uncertainty relations. Phys. Rev. A 77, 042110 (2008)

    ADS  Google Scholar 

  • Degiorgi, L.: The electrodynamic response of heavy-electron compounds. Rev. Mod. Phys. 71, 687–734 (1999)

    ADS  Google Scholar 

  • Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)

    ADS  MathSciNet  Google Scholar 

  • Ding, Z.Y., Yang, H., Yuan, H., Wang, D., Yang, J., Ye, L.: Experimental investigation of linear-entropy-based uncertainty relations in all-optical systems. Phys. Rev. A 101, 022116 (2020a)

    ADS  Google Scholar 

  • Ding, Z.Y., Yang, H., Wang, D., Yuan, H., Yang, J., Ye, L.: Experimental investigation of entropic uncertainty relations and coherence uncertainty relations. Phys. Rev. A 101, 032101 (2020b)

    ADS  Google Scholar 

  • Dolatkhah, H., Haseli, S., Salimi, S., Khorashad, A.S.: Tightening the entropic uncertainty relations for multiple measurements and applying it to quantum coherence. Quantum. Inf. Process. 18, 13 (2019)

    ADS  MATH  Google Scholar 

  • Ekert, A.K.: Quantum cryptography based on Bells theorem. Phys. Rev. Lett. 67, 661–663 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  • Fang, B.L., Shi, J., Wu, T.: Quantum-memory-assisted entropic uncertainty relation and quantum coherence in structured reservoir. Int. J. Theor. Phys. 59, 763–771 (2020)

    MathSciNet  MATH  Google Scholar 

  • Feng, L.C., Xu, J.S., Xu, X.Y., Li, K., Guo, G.C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752–756 (2011)

    Google Scholar 

  • Guo, Y.N., Fang, M.F., Zeng, K.: Entropic uncertainty relation in a two-qutrit system with external magnetic field and Dzyaloshinskii-Moriya interaction under intrinsic decoherence. Quantum Inf. Process. 17, 187 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  • Haddadi, S., Pourkarimi, M.R., Akhound, A., Ghominejad, M.: Quantum correlations and quantum-memory-assisted entropic uncertainty relation in two kinds of spin squeezing models. Laser Phys. Lett. 16, 095202 (2019)

    ADS  MATH  Google Scholar 

  • Haddadi, S., Ghominejad, M., Akhound, A., Pourkarimi, M.R.: Exploring entropic uncertainty relation and dense coding capacity in a two-qubit X-state. Laser Phys. Lett. 17, 095205 (2020)

    ADS  Google Scholar 

  • Haldane, F.D.M.: Model for a quantum Hall effect without landau levels: condensed-matter realization of the “Parity Anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988)

    ADS  MathSciNet  Google Scholar 

  • Haseli, S., Dolatkhah, H., Salimi, S., Khorashad, A.S.: Controlling the entropic uncertainty lower bound in two-qubit systems under decoherence. Laser Phys. Lett. 16, 045207 (2019)

    ADS  MATH  Google Scholar 

  • Haseli, S., Ahmadi, F.: Protecting the entropic uncertainty lower bound in Markovian and non-Markovian environment via additional qubits. Eur. Phys. J. D 74, 170 (2020a)

    ADS  Google Scholar 

  • Haseli, S., Dolatkhah, H., Jahromi, H.Rangani, Salimi, S., Khorashad, A.S.: The lower bound of quantum memory-assisted entropic uncertainty and secret rate for two topological qubits under environments. Opt. Commun. 461, 125287 (2020b)

    Google Scholar 

  • Heisenberg, W.: ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)

    ADS  MATH  Google Scholar 

  • Hennessy, T., Busch, Th: Detecting atoms trapped in an optical lattice using a tapered optical nanofiber. Opt. Express 22, 32509 (2014)

    ADS  Google Scholar 

  • Hu, M.L.: Environment-induced decay of teleportation fidelity of the one-qubit state. Phys. Lett. A 375, 2140–2143 (2011)

    ADS  Google Scholar 

  • Hu, M.L., Fan, H.: Quantum-memory-assisted entropic uncertainty principle, teleportation, and entanglement witness in structured reservoirs. Phys. Rev. A 86, 032338 (2012)

    ADS  Google Scholar 

  • Hu, M.L., Fan, H.: Competition between quantum correlations in the quantum-memory-assisted entropic uncertainty relation. Phys. Rev. A 87, 022314 (2013a)

    ADS  Google Scholar 

  • Hu, M.L., Fan, H.: Upper bound and shareability of quantum discord based on entropic uncertainty relations. Phys. Rev. A 88, 014105 (2013b)

    ADS  Google Scholar 

  • Hu, M.L., Zhou, W.: Enhancing two-qubit quantum coherence in a correlated dephasing channel. Laser Phys. Lett. 16, 045201 (2019)

    ADS  Google Scholar 

  • Hu, M.L., Hu, X., Wang, J., Peng, Y., Zhang, Y.R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762–764, 1–100 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  • Huang, Y.: Entanglement criteria via concave-function uncertainty relations. Phys. Rev. A 82, 012335 (2010)

    ADS  Google Scholar 

  • Huang, Z.: Quantum-memory-assisted entropic uncertainty in spin models with Dzyaloshinskii-Moriya interaction. Laser Phys. Lett. 15, 025203 (2018)

    ADS  Google Scholar 

  • Huang, A.J., Shi, J.D., Wang, D., Ye, L.: Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations. Quantum Inf. Process. 16, 46 (2017a)

    ADS  MathSciNet  MATH  Google Scholar 

  • Huang, A.J., Wang, D., Wang, J.M., Shi, J.D., Sun, W.Y., Ye, L.: Exploring entropic uncertainty relation in the Heisenberg XX model with inhomogeneous magnetic field. Quantum Inf. Process. 16, 204 (2017b)

    ADS  MathSciNet  MATH  Google Scholar 

  • Korzekwa, K., Lostaglio, M., Jennings, D., Rudolph, T.: Quantum and classical entropic uncertainty relations. Phys. Rev. A 89, 042122 (2014)

    ADS  Google Scholar 

  • Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)

    ADS  MathSciNet  Google Scholar 

  • Kumar, A., Haddadi, S., Pourkarimi, M.R., Behera, B.K., Panigrahi, P.K.: Experimental realization of controlled quantum teleportation of arbitrary qubit states via cluster states. Sci. Rep. 10, 13608 (2020)

    ADS  Google Scholar 

  • Li, J.Q., Bai, L., Liang, J.Q.: Entropic uncertainty relation under multiple bosonic reservoirs with filtering operator. Quantum Inf. Process. 17, 206 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  • Liu, S., Mu, L.Z., Fan, H.: Entropic uncertainty relations for multiple measurements. Phys. Rev. A 91, 042133 (2015)

    ADS  Google Scholar 

  • Lv, W.M., Zhang, C., Hu, X.M., Huang, Y.F., Cao, H., Wang, J., Hou, Z.B., Liu, B.H., Li, C.F., Guo, G.C.: Experimental test of fine-grained entropic uncertainty relation in the presence of quantum memory. Sci. Rep. 9, 8748 (2019)

    ADS  Google Scholar 

  • Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)

    ADS  MathSciNet  Google Scholar 

  • Maccone, L., Pati, A.K.: Stronger uncertainty relations for all incompatible observables. Phys. Rev. Lett. 113, 260401 (2014)

    ADS  Google Scholar 

  • Ming, F., Wang, D., Huang, A.J., Sun, W.Y., Ye, L.: Decoherence effect on quantum-memory-assisted entropic uncertainty relations. Quantum Inf. Process. 17, 9 (2018a)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ming, F., Wang, D., Shi, W.N., Huang, A.J., Sun, W.Y., Ye, L.: Entropic uncertainty relations in the Heisenberg XXZ model and its controlling via filtering operations. Quantum Inf. Process. 17, 89 (2018b)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ming, F., Wang, D., Shi, W.N., Huang, A.J., Du, M.M., Sun, W.Y., Ye, L.: Exploring uncertainty relation and its connection with coherence under the Heisenberg spin model with the Dzyaloshinskii-Moriya interaction. Quantum Inf. Process. 17, 267 (2018c)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ming, F., Wang, D., Ye, L.: Dynamical measurement’s uncertainty in the curved space-time. Ann. Phys. (Berl.) 531, 1900014 (2019)

    ADS  MathSciNet  Google Scholar 

  • Ming, F., Wang, D., Fan, X.G., Shi, W.N., Ye, L., Chen, J.L.: Improved tripartite uncertainty relation with quantum memory. Phys. Rev. A 102, 012206 (2020)

    ADS  MathSciNet  Google Scholar 

  • Ng, N.H.Y., Berta, M., Wehner, S.: Min-entropy uncertainty relation for finite-size cryptography. Phys. Rev. A 86, 042315 (2012)

    ADS  Google Scholar 

  • Oh, S., Lee, S., Lee, H.W.: Fidelity of quantum teleportation through noisy channels. Phys. Rev. A 66, 022316 (2002)

    ADS  MathSciNet  Google Scholar 

  • Partovi, M.H.: Entanglement detection using majorization uncertainty bounds. Phys. Rev. A 86, 022309 (2012)

    ADS  Google Scholar 

  • Pati, A.K., Wilde, M.M., Devi, A.R.Usha, Rajagopal, A.K., Sudha: Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory. Phys. Rev. A 86, 042105 (2012)

    ADS  Google Scholar 

  • Pourkarimi, M.R.: Quantum correlations and entropic uncertainty relation in a three-qubit spin chain under the effect of magnetic field and DM interaction. Int. J. Quantum Inf. 16, 1850057 (2018)

    MathSciNet  MATH  Google Scholar 

  • Pourkarimi, M.R.: Time evolution of quantum-memory-assisted entropic uncertainty relation and quantum correlations under dissipative environment. Int. J. Quantum Inf. 17, 1950008 (2019)

    MATH  Google Scholar 

  • Pourkarimi, M.R., Rahnama, M.: Quantum teleportation under the effect of dissipative environment and hamiltonian XY model. Int. J. Theor. Phys. 53, 1415–1423 (2014)

    MATH  Google Scholar 

  • Pourkarimi, M.R., Haddadi, S.: Quantum-memory-assisted entropic uncertainty, teleportation, and quantum discord under decohering environments. Laser Phys. Lett. 17, 025206 (2020)

    ADS  Google Scholar 

  • Pourkarimi, M.R., Rahnama, M., Rooholamini, H.: Decoherence effect on quantum correlation and entanglement in a two-qubit spin chain. Int. J. Theor. Phys. 54, 1085–1097 (2015)

    MATH  Google Scholar 

  • Pramanik, T., Chowdhury, P., Majumdar, A.S.: Fine-Grained lower limit of entropic uncertainty in the presence of quantum memory. Phys. Rev. Lett. 110, 020402 (2013)

    ADS  Google Scholar 

  • Pramanik, T., Mal, S., Majumdar, A.S.: Lower bound of quantum uncertainty from extractable classical information. Quantum Inf. Process. 15, 981–999 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  • Prevedel, R., Hamel, D.R., Colbeck, R., Fisher, K., Resch, K.J.: Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement. Nat. Phys. 7, 757–761 (2011)

    Google Scholar 

  • Quan, H.T., Song, Z., Liu, X.F., Zanardi, P., Sun, C.P.: Decay of loschmidt echo enhanced by quantum criticality. Phys. Rev. Lett. 96, 140604 (2006)

    ADS  Google Scholar 

  • Rao, D.D.Bhaktavatsala, Panigrahi, P.K., Mitra, C.: Teleportation in the presence of common bath decoherence at the transmitting station. Phys. Rev. A 78, 022336 (2008)

    ADS  Google Scholar 

  • Renes, J.M., Boileau, J.C.: Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009)

    ADS  Google Scholar 

  • Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)

    ADS  Google Scholar 

  • Rudnicki, L.: Majorization approach to entropic uncertainty relations for coarse-grained observables. Phys. Rev. A 91, 032123 (2015)

    ADS  Google Scholar 

  • Rudnicki, L., Walborn, S.P., Toscano, F.: Optimal uncertainty relations for extremely coarse-grained measurements. Phys. Rev. A 85, 042115 (2012)

    ADS  Google Scholar 

  • Rudnicki, L., Puchala, Z., Zyczkowski, K.: Strong majorization entropic uncertainty relations. Phys. Rev. A 89, 052115 (2014)

    ADS  MATH  Google Scholar 

  • Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  • Schrödinger, E.: Sitzungsberichte der preussischen akademie der wissenschaften. Physikalisch-mathematische klasse 14, 296–303 (1930)

    Google Scholar 

  • Shi, W.N., Ming, F., Wang, D., Ye, L.: Entropic uncertainty relations in the spin-1 Heisenberg model. Quantum Inf. Process. 18, 70 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  • Tomamichel, M., Lim, C.C.W., Gisin, N., Renner, R.: Tight finite-key analysis for quantum cryptography. Nat. Commun. 3, 634 (2012)

    ADS  Google Scholar 

  • Wang, J., Batelaan, H., Podany, J., Starace, A.F.: Entanglement evolution in the presence of decoherence. J. Phys. B 39, 4343 (2006)

    ADS  Google Scholar 

  • Wang, D., Huang, A., Ming, F., Sun, W., Lu, H., Liu, C., Ye, L.: Quantum-memory-assisted entropic uncertainty relation in a Heisenberg XYZ chain with an inhomogeneous magnetic field. Laser Phys. Lett. 14, 065203 (2017a)

    ADS  Google Scholar 

  • Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Ye, L.: Entropic uncertainty for spin-1/2 XXX chains in the presence of inhomogeneous magnetic fields and its steering via weak measurement reversals. Laser Phys. Lett. 14, 095204 (2017b)

    ADS  Google Scholar 

  • Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Shi, J.D., Ye, L.: Exploration of quantum-memory-assisted entropic uncertainty relations in a noninertial frame. Laser Phys. Lett. 14, 055205 (2017c)

    ADS  Google Scholar 

  • Wang, D., Huang, A.J., Hoehn, R.D., Ming, F., Sun, W.Y., Shi, J.D., Ye, L., Kais, S.: Entropic uncertainty relations for Markovian and non-Markovian processes under a structured bosonic reservoir. Sci. Rep. 7, 1066 (2017d)

    ADS  Google Scholar 

  • Wang, D., Shi, W.N., Hoehn, R.D., Ming, F., Sun, W.Y., Kais, S., Ye, L.: Effects of Hawking radiation on the entropic uncertainty in a Schwarzschild spacetime. Ann. Phys. (Berlin) 530, 1800080 (2018a)

    ADS  Google Scholar 

  • Wang, D., Shi, W.N., Hoehn, R.D., Ming, F., Sun, W.Y., Ye, L., Kais, S.: Probing entropic uncertainty relations under a two-atom system coupled with structured bosonic reservoirs. Quantum Inf. Process. 17, 335 (2018b)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wang, D., Ming, F., Song, X.K., Ye, L., Chen, J.L.: Entropic uncertainty relation in neutrino oscillations. Eur. Phys. J. C 80, 800 (2020)

    ADS  Google Scholar 

  • Wehner, S., Winter, A.: Entropic uncertainty relationsa survey. New J. Phys. 12, 025009 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wu, S., Yu, S., Mølmer, K.: Entropic uncertainty relation for mutually unbiased bases. Phys. Rev. A 79, 022104 (2009)

    ADS  MathSciNet  Google Scholar 

  • Yan, Y.Y., Qin, L.G., Tian, L.J.: Decoherence from a spin chain with Dzyaloshinskii-Moriya interaction. Chin. Phys. B 21, 100304 (2012)

    ADS  Google Scholar 

  • Yang, Y.Y., Sun, W.Y., Shi, W.N., Ming, F., Wang, D., Ye, L.: Dynamical characteristic of measurement uncertainty under Heisenberg spin models with Dzyaloshinskii-Moriya interactions. Front. Phys. 14, 31601 (2019)

    ADS  Google Scholar 

  • Yang, H., Ding, Z.Y., Wang, D., Yuan, H., Song, X.K., Yang, J., Zhang, C.J., Ye, L.: Experimental certification of the steering criterion based on a general entropic uncertainty relation. Phys. Rev. A 101, 022324 (2020a)

    ADS  Google Scholar 

  • Yang, Y.Y., Ye, L., Wang, D.: Measurement uncertainty and its connection to quantum coherence in an inertial Unruh-DeWitt detector. Ann. Phys. (Berlin) 532, 2000062 (2020b)

    ADS  MathSciNet  Google Scholar 

  • Yao, Y.B., Wang, D., Ming, F., Ye, L.: Dynamics of the measurement uncertainty in an open system and its controlling. J. Phys. B At. Mol. Opt. Phys. 53, 035501 (2020)

    ADS  Google Scholar 

  • Yuan, Z.G., Zhang, P., Li, S.S.: Disentanglement of two qubits coupled to an XY spin chain: Role of quantum phase transition. Phys. Rev. A 76, 042118 (2007)

    ADS  Google Scholar 

  • Zhang, J., Zhang, Y., Yu, C.S.: Entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory. Sci. Rep. 5, 11701 (2015)

    ADS  Google Scholar 

  • Zhang, Y., Fang, M., Kang, G., Zhou, Q.: Controlling quantum memory-assisted entropic uncertainty in non-Markovian environments. Quantum Inf. Process. 17, 62 (2018a)

    ADS  MathSciNet  MATH  Google Scholar 

  • Zhang, Y., Zhou, Q., Fang, M., Kang, G., Li, X.: Quantum-memory-assisted entropic uncertainty in two-qubit Heisenberg XYZ chain with Dzyaloshinskii-Moriya interactions and effects of intrinsic decoherence. Quantum Inf. Process. 17, 326 (2018b)

    ADS  MathSciNet  MATH  Google Scholar 

  • Zozor, S., Bosyk, G.M., Portesi, M.: General entropy-like uncertainty relations in finite dimensions. J. Phys. A Math. Theor. 47, 495302 (2014)

    MathSciNet  MATH  Google Scholar 

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Haseli, S., Haddadi, S. & Pourkarimi, M.R. Entropic uncertainty lower bound for a two-qubit system coupled to a spin chain with Dzyaloshinskii–Moriya interaction. Opt Quant Electron 52, 465 (2020). https://doi.org/10.1007/s11082-020-02589-x

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