# Measurement Uncertainty, Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect

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## Abstract

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## 1. Introduction

## 2. The Physical Model and Its Dynamics

#### 2.1. Physical Description

#### 2.2. The Solution of the Milburn Equation

## 3. Quantum Information Resources Measures

**Entropic uncertainty**For incompatible observables P and Q, Bob’s uncertainty regarding the two qubits (A and B) measurement outcome is given by [49,50]:$$S(P|B)+S(Q|B)\ge S(A|B)+{log}_{2}\frac{1}{c},$$$$\begin{array}{cc}\hfill UL(t)& =S({\widehat{M}}_{{\sigma}_{x}B})+S({\widehat{M}}_{{\sigma}_{z}B})-2S({\widehat{M}}_{B}),\hfill \end{array}$$$$\begin{array}{cc}\hfill UR(t)& =S({\widehat{M}}_{AB})-S({\widehat{M}}_{B})+1,\hfill \end{array}$$**Two-charge-qubit entropy purity ($EP$)**Here, entropy is used to quantify the amount of two-charge-qubit purity/mixedness [51].The qubit–qubit entropy is defined by:$$\begin{array}{c}\hfill EP(t)=-\sum _{i=1}^{\infty}{\lambda}^{i}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ln({\lambda}^{i}),\end{array}$$**Two-qubit negativity entanglement ($NE$)**:The negativity is a good entanglement monotonic measure. In the current case, $NE(t)$ is used to investigate the two-charge-qubit entanglement [52]. It is equal to the absolute sum of the negative eigenvalues of the density matrix ${({\widehat{M}}_{AB}(t))}^{{T}_{A}}$ that is the partial transpose of the two-charge-qubit density matrix ${\widehat{M}}_{AB}$ with respect to subsystem A. The elements of ${({\widehat{R}}^{AB})}^{{T}_{A}}$ are given by:$$\begin{array}{c}\hfill \langle i,j|{({\widehat{M}}_{AB}(t))}^{{T}_{A}}|m,n\rangle =\langle m,j|{\widehat{M}}_{AB}(t)|i,n\rangle .\end{array}$$When $NE(t)=0$, the state is separable. The function $NE(t)$ is used to estimate the entanglement amount of the quantum state.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wendin, G.; Shumeiko, V.S. Quantum bits with Josephson junctions. Low Temp. Phys.
**2007**, 33, 724–744. [Google Scholar] [CrossRef] [Green Version] - Yurgens, A.A. Intrinsic Josephson junctions: Recent developments. Supercond. Sci. Technol.
**2000**, 13, R85. [Google Scholar] [CrossRef] - Pekola, J.P.; Toppari, J.J. Decoherence in circuits of small Josephson junctions. Phys. Rev. B
**2001**, 64, 172509. [Google Scholar] [CrossRef] [Green Version] - Berkley, A.J.; Xu, H.; Gubrud, M.A.; Ramos, R.C.; Anderson, J.R.; Lobb, C.J.; Wellstood, F.C. Decoherence in a Josephson-junction qubit. Phys. Rev. B
**2003**, 68, 060502. [Google Scholar] [CrossRef] [Green Version] - DiVincenzo, D.P.; Brito, F.; Koch, R.H. Decoherence rates in complex Josephson qubit circuits. Phys. Rev. B
**2006**, 74, 014514. [Google Scholar] [CrossRef] [Green Version] - Sete, E.A.; Eleuch, H. Strong squeezing and robust entanglement in cavity electromechanics. Phys. Rev. A
**2014**, 89, 013841. [Google Scholar] [CrossRef] [Green Version] - Fendley, P.; Schoutens, K. Cooper pairs and exclusion statistics from coupled free-fermion chains. J. Stat. Mech. Theory Exp.
**2007**, 2007, P02017. [Google Scholar] [CrossRef] [Green Version] - Wagner, R., Jr. Position and Temperature Measurements of a Single Atom via Resonant Fluorescence. Ph.D. Thesis, University of Oregon, Eugene, OR, USA, 2019. [Google Scholar]
- You, J.Q.; Nori, F. Atomic physics and quantum optics using superconducting circuits. Nature
**2011**, 474, 589–597. [Google Scholar] [CrossRef] [Green Version] - Blais, A.; Girvin, S.M.; Oliver, W.D. Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat. Phys.
**2020**, 16, 247–256. [Google Scholar] [CrossRef] - Wendin, G. Quantum information processing with superconducting circuits: A review. Rep. Prog. Phys.
**2017**, 80, 106001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Devoret, M.H.; Schoelkopf, R.J. Superconducting circuits for quantum information: An outlook. Science
**2013**, 339, 1169–1174. [Google Scholar] [CrossRef] [Green Version] - Menke, T.; Häse, F.; Gustavsson, S.; Kerman, A.J.; Oliver, W.D.; Aspuru-Guzik, A. Automated design of superconducting circuits and its application to 4-local couplers. NPJ Quantum Inf.
**2021**, 7, 1–8. [Google Scholar] [CrossRef] - You, J.Q.; Nori, F. Superconducting Circuits and Quantum Information. Phys. Today
**2005**, 58, 42. [Google Scholar] [CrossRef] [Green Version] - You, J.Q.; Tsai, J.S.; Nori, F. Quantum information processing with superconducting qubits in a microwave field. Phys. Rev. B
**2003**, 68, 02451. [Google Scholar] [CrossRef] [Green Version] - Obada, A.-S.F.; Hessian, H.A.; Mohamed, A.-B.A.; Homid, A.H. A proposal for the realization of universal quantum gates via superconducting qubits inside a cavity. Ann. Phys.
**2013**, 334, 47. [Google Scholar] [CrossRef] - Wehner, S.; Winter, A. Entropic uncertainty relations—A survey. New J. Phys.
**2010**, 12, 025009. [Google Scholar] [CrossRef] - Coles, P.J.; Berta, M.; Tomamichel, M.; Wehner, S. Entropic uncertainty relations and their applications. Rev. Mod. Phys.
**2017**, 89, 015002. [Google Scholar] [CrossRef] [Green Version] - Son, W. Role of quantum non-Gaussian distance in entropic uncertainty relations. Phys. Rev. A
**2015**, 92, 012114. [Google Scholar] [CrossRef] [Green Version] - Jenkins, J.A. On an inequality considered by Robertson. Proc. Am. Math. Soc.
**1968**, 19, 549–550. [Google Scholar] [CrossRef] - Srinivas, M.D. Entropic formulation of uncertainty relations. Pramana
**1985**, 25, 369–375. [Google Scholar] [CrossRef] - Damgard, I.B.; Fehr, S.; Salvail, L.; Schaffner, C. Cryptography in the bounded-quantum-storage model. SIAM J. Comput.
**2008**, 37, 1865–1890. [Google Scholar] [CrossRef] [Green Version] - Guehne, O.; Lewenstein, M. Entropic uncertainty relations and entanglement. Phys. Rev. A
**2004**, 70, 022316. [Google Scholar] [CrossRef] [Green Version] - Awasthi, N.; Haseli, S.; Johri, U.C.; Salimi, S.; Dolatkhah, H.; Khorashad, A.S. Quantum speed limit time for correlated quantum channel. Quantum Inf. Process.
**2020**, 19, 1–17. [Google Scholar] [CrossRef] [Green Version] - Chen, Z.; Zhang, Y.; Wang, X.; Yu, S.; Guo, H. Improving parameter estimation of entropic uncertainty relation in continuous-variable quantum key distribution. Entropy
**2019**, 21, 652. [Google Scholar] [CrossRef] [Green Version] - Luis, A.; Rodil, A. Alternative measures of uncertainty in quantum metrology: Contradictions and limits. Phys. Rev. A
**2013**, 87, 034101. [Google Scholar] [CrossRef] [Green Version] - Orlikowski, W.J.; Scott, S.V. The Entanglement of Technology and Work in Organizations; LSE: London, UK, 2008. [Google Scholar]
- Berrada, K.; Chafik, A.; Eleuch, H.; Hassouni, Y. Concurrence in the framework of coherent states. Quantum Inf. Process.
**2010**, 9, 13–26. [Google Scholar] [CrossRef] - Mohamed, A.-B.A.; Eleuch, H.; Ooi, C.H.R. Quantum coherence and entanglement partitions for two driven quantum dots inside a coherent micro cavity. Phys. Lett. A
**2019**, 383, 125905. [Google Scholar] [CrossRef] - Hu, X.M.; Guo, Y.; Liu, B.H.; Huang, Y.F.; Li, C.F.; Guo, G.C. Beating the channel capacity limit for superdense coding with entangled ququarts. Sci. Adv.
**2018**, 4, eaat9304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saffman, M.; Walker, T.G. Entangling single-and N-atom qubits for fast quantum state detection and transmission. Phys. Rev. A
**2005**, 72, 042302. [Google Scholar] [CrossRef] [Green Version] - Yin, J.; Ren, J.G.; Lu, H.; Cao, Y.; Yong, H.L.; Wu, Y.P.; Pan, J.W. Quantum teleportation and entanglement distribution over 100-kilometre free-space channels. Nature
**2012**, 488, 185–188. [Google Scholar] [CrossRef] [PubMed] - Asjad, M.; Qasymeh, M.; Eleuch, H. Continuous-Variable Quantum Teleportation Using a Microwave-Enabled Plasmonic Graphene Waveguide. Phys. Rev. Appl.
**2021**, 16, 034046. [Google Scholar] [CrossRef] - Zidan, M. A novel quantum computing model based on entanglement degree. Mod. Phys. Lett. B
**2020**, 34, 2050401. [Google Scholar] [CrossRef] - Fan, P.; Rahman, A.U.; Ji, Z.; Ji, X.; Hao, Z.; Zhang, H. Two-party quantum private comparison based on eight-qubit entangled state. Mod. Phys. Lett. A
**2022**, 37, 2250026. [Google Scholar] [CrossRef] - Thagard, P. Explanatory coherence. Behav. Brain Sci.
**1989**, 12, 435–467. [Google Scholar] [CrossRef] - Streltsov, A.; Adesso, G.; Plenio, M.B. Colloquium: Quantum coherence as a resource. Rev. Mod. Phys.
**2017**, 89, 041003. [Google Scholar] [CrossRef] [Green Version] - Rahman, A.U.; Haddadi, S.; Pourkarimi, M.R.; Ghominejad, M. Fidelity of quantum states in a correlated dephasing channel. Laser Phys. Lett.
**2022**, 19, 035204. [Google Scholar] [CrossRef] - Bluhm, H.; Foletti, S.; Neder, I.; Rudner, M.; Mahalu, D.; Umansky, V.; Yacoby, A. Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs. Nat. Phys.
**2011**, 7, 109–113. [Google Scholar] [CrossRef] - Chiorescu, I.; Bertet, P.; Semba, K.; Nakamura, Y.; Harmans, C.J.P.M.; Mooij, J.E. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature
**2004**, 431, 159–162. [Google Scholar] [CrossRef] [Green Version] - Fonseca-Romero, K.M.; Kohler, S.; Hänggi, P. Coherence stabilization of a two-qubit gate by ac fields. Phys. Rev. Lett.
**2005**, 95, 140502. [Google Scholar] [CrossRef] [Green Version] - Luthi, F.; Stavenga, T.; Enzing, O.W.; Bruno, A.; Dickel, C.; Langford, N.K.; DiCarlo, L. Evolution of nanowire transmon qubits and their coherence in a magnetic field. Phys. Rev. Lett.
**2018**, 120, 100502. [Google Scholar] [CrossRef] [Green Version] - Milburn, G.J. Intrinsic decoherence in quantum mechanics. Phys. Rev. A
**1991**, 44, 5401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anwar, S.J.; Ramzan, M.; Usman, M.; Khan, M.K. Thermal and intrinsic decoherence effects on the dynamics of two three-level moving atomic system. Phys. A
**2020**, 549, 124297. [Google Scholar] [CrossRef] - Khalil, E.M.; Mohamed, A.-B.A.; Obada, A.-S.F.; Eleuch, H. Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity. Mathematics
**2020**, 8, 1830. [Google Scholar] [CrossRef] - Karpat, G.; Piilo, J.; Maniscalco, S. Controlling entropic uncertainty bound through memory effects. EPL (Europhys. Lett.)
**2015**, 111, 50006. [Google Scholar] [CrossRef] - Duty, T.; Gunnarsson, D.; Bladh, K.; Delsing, P. Coherent dynamics of a Josephson charge qubit. Phys. Rev. B
**2004**, 69, 140503. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.-X.; Wei, L.F.; Nori, F. Measuring the quality factor of a microwave cavity using superconducting qubit devices. Phys. Rev. A
**2005**, 72, 033818. [Google Scholar] [CrossRef] [Green Version] - Zidan, N.; Bakry, H.; Rahman, A.U. Entanglement and Entropic Uncertainty of Two Two-Level Atoms. Annalen der Physik
**2022**, 2100555. [Google Scholar] [CrossRef] - Berta, M.; Christandl, M.; Colbeck, R.; Renes, J.M.; Renner, R. The uncertainty principle in the presence of quantum memory. Nat. Phys.
**2010**, 6, 659. [Google Scholar] [CrossRef] - Phoenix, S.J.D.; Knight, P.L. Establishment of an entangled atom-field state in the Jaynes-Cummings model. Phys. Rev. A
**1991**, 44, 6023. [Google Scholar] [CrossRef] - Vidal, G.; Werner, R.F. A computable measure of entanglement. Phys. Rev. A
**2002**, 65, 032314. [Google Scholar] [CrossRef] [Green Version] - Mohamed, A.B.; Metwally, N. Quantifying the non-classical correlation of a two-atom system nonlinearly interacting with a coherent cavity: Local quantum Fisher information and Bures distance entanglement. Nonlinear Dyn.
**2021**, 104, 2573–2582. [Google Scholar] [CrossRef] - Wang, C.Z.; Li, C.X.; Nie, L.Y.; Li, J.F. Classical correlation and quantum discord mediated by cavity in two coupled qubits. J. Phys. B
**2010**, 44, 015503. [Google Scholar] [CrossRef] - Fang, B.L.; Shi, J.; Wu, T. Quantum-memory-assisted entropic uncertainty relation and quantum coherence in structured reservoir. Int. J. Theor. Phys.
**2020**, 59, 763–771. [Google Scholar] [CrossRef] - 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.
**2018**, 17, 1–23. [Google Scholar] [CrossRef] - Khedr, A.N.; Mohamed, A.B.A.; Abdel-Aty, A.H.; Tammam, M.; Abdel-Aty, M.; Eleuch, H. Entropic Uncertainty for Two Coupled Dipole Spins Using Quantum Memory under the Dzyaloshinskii-Moriya Interaction. Entropy
**2021**, 23, 1595. [Google Scholar] [CrossRef] - Rahman, A.U.; Noman, M.; Javed, M.; Ullah, A.; Luo, M.X. Effects of classical fluctuating environments on decoherence and bipartite quantum correlations dynamics. Laser Phys.
**2021**, 31, 115202. [Google Scholar] [CrossRef] - Mishra, U.; Prabhu, R.; Rakshit, D. Quantum correlations in periodically driven spin chains: Revivals and steady-state properties. J. Magn. Magn. Mater.
**2019**, 491, 165546. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Dynamics of the quantum memory-assisted entropic uncertainty ($UR(t)$ and $UL(t)$), entropy purity $EP(t)$, and negativity entanglement $NE(t)$ are shown for the initial maximally correlated state $\frac{1}{\sqrt{2}}(|{1}_{A}{0}_{B}\rangle -|{0}_{A}{1}_{B}\rangle )$ in the absence of the decoherence $\gamma =0$ and detunings ${\delta}_{i}=0$. When the cavities are initially in CEC cavity state in (

**a**) and in EC cavity state in (

**b**) for small coherent strengths, $|{\alpha}_{A}{|}^{2}=0.9$ and $|{\alpha}_{B}{|}^{2}=0.5$.

**Figure 3.**Dynamics of the quantum memory-assisted entropic uncertainty ($UR(t)$ and $UL(t)$), entropy purity $EP(t)$, and negativity entanglement $NE(t)$ are shown in (

**a**) and in (

**b**) with the same parameters as Figure 1a and Figure 2a, respectively, for CEC configuration but in the presence of the intrinsic decoherence $\gamma =0.06\lambda $.

**Figure 4.**Dynamics of the quantum memory-assisted entropic uncertainty ($UR(t)$ and $UL(t)$), entropy purity $EP(t)$, and negativity entanglement $NE(t)$ are shown for the initial maximally correlated state in the absence of the decoherence $\gamma =0$ and detunings ${\delta}_{i}=0$. When the cavities are initially prepared as CEC configuration in (

**a**) and in EC in (

**b**) for large coherent strengths $|{\alpha}_{A}{|}^{2}=8$ and $|{\alpha}_{B}{|}^{2}=10$.

**Figure 5.**Dynamics of the quantum memory-assisted entropic uncertainty ($UR(t)$ and $UL(t)$), entropy purity $EP(t)$, and negativity entanglement $NE(t)$ are shown as Figure 4b for EC configuration, but under the effects of the two-charge-qubit detunings ${\delta}_{A}={\delta}_{B}=2\lambda $ in (

**a**) and of the intrinsic decoherence $\gamma =0.06\lambda $ in (

**b**).

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**MDPI and ACS Style**

Mohamed, A.-B.A.; Rahman, A.U.; Eleuch, H.
Measurement Uncertainty, Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect. *Entropy* **2022**, *24*, 545.
https://doi.org/10.3390/e24040545

**AMA Style**

Mohamed A-BA, Rahman AU, Eleuch H.
Measurement Uncertainty, Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect. *Entropy*. 2022; 24(4):545.
https://doi.org/10.3390/e24040545

**Chicago/Turabian Style**

Mohamed, Abdel-Baset A., Atta Ur Rahman, and Hichem Eleuch.
2022. "Measurement Uncertainty, Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect" *Entropy* 24, no. 4: 545.
https://doi.org/10.3390/e24040545