S - CBPF
Measurements of General Quantum Correlations in Nuclear Magnetic Resonance Systems Eduardo Ribeiro deAzevedo São Paulo Brazil UNIVERSITY OF SÃO PAULO - USP • 75 years • 240 courses • 57.000 undergrad students • ~200 Msc. and PHD programs UNIVERSITY OF SÃO PAULO AT SÃO CARLOS São Carlos City: 250.000 people. 5 universities: 1 Federal University (UFSCAR). 1 State Univesity (USP). 3 Private Univesities. USP at São Carlos: 2 Campi, ~8.000 undergrad students São Carlos Institute of Physics, USP, Brazil www.ifsc.usp.br Al DC V ETL (ionomer) OC1OC6 - PPV ITO glass emitted light IFSC NMR group: Tito Bonagamba, Eduardo R. deAzevedo (Solid-State NMR, MRI) First experiments done in São Carlos using quadrupolar nuclei First thesis defence in NMR QIP (Fabio A. Bonk at IFSC) and (Juan Bulnes at CBPF) 2012 2010 2009 2007 2003 CBPF NMR group: Ivan Oliveira, Alberto Passos, Roberto Sarthour, Jair C. C Freitas (magnetism and magnetic materials) 2005 2002 NMR QIP in Brazil Publication of the Book Quantum Information Processing by Elsevier Gather with the quantum information theory group at UFABC – Lucas Celeri and Roberto Serra. CBPF NMR spectrometer start to operate. Hiring of new researchers (Alexandre Souza-CBPF, Diogo Pinto IFSC, João Teles-UFSCAR, Ruben Auccaise - UEPG ) tend to strenght this researche area. PEOPLE INVOLVED Experiments Isabela Almeida Ruben Auccaise Alexandre Souza Ivan S. Oliveira Roberto Sarthour Tito Bonagamba Theory Diogo S. Pinto Lucas Céleri Roberto Serra Jonas Maziero Felipe Fanchini David Girolami Gerardo Adesso F. M. Paula J. D. Montealegre A Saguia Marcelo Sarandy NMR and the QIP • Experimental demonstration of QIP procedures, including quantum protocols, algorithms, quantum simulations etc.; • Development of many useful tools for QIP, including quantum protocols, algorithms, dynamic decoupling schemes, among others; • NMR is also an excellent test bench for studies on open quantum systems: – efficient implementation and manipulation of the quantum states (excellent control of the unitary transformations coming from the radiofrequency pulses); – presence of real environments, which can be described by phase damping and generalized amplitude damping channels; Quantum Computation Entanglement ? • In certain schemes of quantum computation where the quantum bits are affected by noise, there seems to be a speed-up over classical scenarios even in the presence of negligibly small or vanishing entanglement. Knill, E.; Laflamme, R. Power of one bit of quantum information. Physical Review Letters, v. 81, n. 25, p. 5672, 1998. Datta, A.; Shaji, A. and Caves. C. M. Physical Review Letters 100, p.050502, 2008. Modi, K., Paterek, T., Son, W., Vedral, V. and Williamson M. Unified View of Quantum and Classical Correlations Physical Review Letters, v. 104, p.080501, 2010. • A possible explanation for the speed up would be quantum correlations different for entanglement. General Quantum Correlations How to detect them? Other types of correlations Quantum Computation Ollivier, H. & Zurek, W. H. Quantum discord: a measure of the quantumness of correlations.Phys. Rev. Lett. 88, 017901 (2001). Entanglement ? Merali, Z. Nature, v. 474, p. 24, 2011. Classification of Quantum and Classical States All Correlated States Separable States C CQ Entangled States Separable pij iA Bj Entangled pij iA Bj Classically Correlated ij ij pij i i j j ij i , j ortonormal basis. Classification of Quantum and Classical Two-Qubit States All Correlated States Separable States C CQ Entangled States Separable pij iA Bj Entangled pij iA Bj Classically Correlated • Bell diagonal states: 3 1 1 c j j j 4 j 1 c1 0 Correlation Matrix: C 0 c2 0 0 0 0 c3 AB ij ij pij i i j j ij i , j ortonormal basis. 1 c3 0 0 c1 c2 0 0 1 c3 c1 c2 0 c1 c2 1 c3 0 c1 c2 0 0 1 c3 Classification of Quantum and Classical States All Correlated States Separable States C CQ Entangled States Separable pij iA Bj Entangled pij iA Bj Classically Correlated • Bell diagonal states: 3 1 1 c j j j 4 j 1 ij ij pij i i j j ij i , j ortonormal basis. 3 c 1 1 j i j j 1 4 4 In this sense NMR seems to be the perfect tool for probing quantum correlations of separable states and their interaction with the environment; NMR sensitive part of the density matrix Quantum Discord – Entropic Discord*: disturbance made in a system when a measurement is applied. Von Neumann Entropy S(ρA) S ( ) Tr ( log 2 ) *Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002. S(ρAB) S(ρB) Quantum Discord – Entropic Discord*: disturbance made in a system when a measurement is applied. Von Neumann Entropy S(ρA) S ( ) Tr ( log 2 ) • Mutual information: S(ρAB) S(ρB) I A:B S A S B S A:B *Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002. Quantum Discord – Entropic Discord*: disturbance made in a system when a measurement is applied. Von Neumann Entropy S(ρA) S ( ) Tr ( log 2 ) • Mutual information: S(ρAB) S(ρB) I A:B S A S B S A:B • Classical Correlation: J Q A:B S A S *Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002. Bj AB Quantum Discord – Entropic Discord*: disturbance made in a system when a measurement is applied. Von Neumann Entropy S(ρA) S ( ) Tr ( log 2 ) • Mutual information: S(ρAB) S(ρB) I A:B S A S B S A:B • Classical Correlation: J Q A:B S A S Bj AB • Quantum Discord: D AB I A:B max J Q A:B B j *Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002. Quantum Discord – For two-qubits Bell diagonal states*: D AB I AB C AB c max c1 , c2 , c3 1 1 c1 c2 c3 log 2 1 c1 c2 c3 4 1 c1 c2 c3 log 2 1 c1 c2 c3 1 c1 c2 c3 log 2 1 c1 c2 c3 1 c1 c2 c3 log 2 1 c1 c2 c3 1 c 1 c log 2 1 c log 2 1 c 2 2 *Luo, S. Quantum discord for two-qubit systems. Physical Review A, v. 7, n. 4, p. 042303, 2008. Probing Quantum Correlations What is required for probing discord and their degradation upon interaction with the environment? • To prepare states with different amounts of QCs . • To perform a reliable read-out of the final states. • To have a good description and characterization of the system relaxation. NMR has all that!!!! Diogo sets a partnership do study quantum discord by NMR with Roberto Serra and Lucas Céleri ; 3/2 spins system • NMR system. Sample: Lyotropic Liquid Crystals -Sodium Dodecyl Sulfat Sodium dodecyl sulfate in (SDS) - Heavy Water (D2O) - Decanol (C10H21OH) water forming a lyotropic Q 2 23 H I 3 I liquid crystal – Na NMR L Z z I I 1 1 6 Anatoly K. Khitrin and B. M. Fung. The Journal of Chemical Physics, 112(16):6963–6965, 2000. Neeraj Sinha, T. S. Mahesh, K. V. Ramanathan, and Anil Kumar. The Journal of Chemical Physics, 114(10):4415–4420, 2001. Tools for NMR QIP using quadrupolar Nuclei • Strong Modulated Pulase (SMP)*: k U SMP U n , n , tn n 1 F t arg et , SMP Tr t arg et , SMP 2 Tr t2arg et Tr SMP *Fortunato, E.; Pravia, M.; Boulant, N.; Teklemariam, G.; Havel, T.; Cory, D. Design of modulating pulses to implement precise effective hamiltonians for quantum information processing. Journal of Chemical Physics, v. 116, n. 17, p. 7599, 2002. Nelder, J.A.; Mead, R. A simplex-method for function minimization. Computer Journal, v. 7, n. 4, p. 308, 1965. Single hard pulse 11 12 13 14 11 12 13 14 11 12 13 14 11 12 13 14 12 22 23 24 13 14 23 24 33 34 34 44 12 22 23 24 13 14 23 24 33 34 34 44 12 22 23 24 13 14 23 24 33 34 34 44 12 22 23 24 13 14 23 24 33 34 34 44 Pure quadrupolar relaxation + Redfield Equations Two Qubit System • Generalized Amplitude Damping Channel (GAD): – Longitudinal relaxation (T1) 1 E1 p 0 0 0 , E2 p 1 0 1 E3 1 p 0 L 1 p 2 2 k BT 0 0 , E4 p 1 B 1 e 2CJ t 1 e 1 0 0 0 t T1 B A 1 e 2CJ t 1 e 2 t T1 A – Global phase damping channel (GPD); 1 0 0 0 1 0 E0 1 0 0 1 0 0 0 11 12 13 14 12 22 23 24 13 14 23 24 33 34 34 44 1 1 e CJ 0t 2 0 1 0 0 , E1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 E0 E0† E1 E1† 11 2 1 12 2 1 13 14 2 1 12 2 1 13 22 23 2 1 24 23 33 2 1 34 14 2 1 24 2 1 34 44 X random Monotonical Decay | Different amount of classical and correations in each state time (ms) HOWEVER.... Sudden-Change Phenomena: • Decoherence Process in Bell-diagonal States: → Local Phase Damping Channel: Mutual Information Classical Correlation Entropic Discord c3 0 c1 0 e c2 0 Time c3 0 0 (s) *Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009. c3 0 c1 0 e ou c2 0 Sudden-Change Phenomena: • Decoherence Process in Bell-diagonal States: → Phase Damping Channel: Mutual Information Classical Correlation Entropic Discord c3 0 c1 0 e c2 0 Time (s) c3 0 0 Time (s) *Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009. c3 0 c1 0 e ou c2 0 Sudden-Change Phenomena: • Decoherence Process in Bell-diagonal States: → Phase Damping Channel: Mutual Information Classical Correlation Entropic Discord c3 0 c1 0 e c2 0 Time (s) c3 0 0 Time (s) *Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009. c3 0 c1 0 e ou c2 0 Time (s) Sudden-Change Phenomena: • Decoherence Process in Bell-diagonal States: → Phase Damping Channel: Mutual Information Classical Correlation Entropic Discord c3 0 c1 0 e c2 0 Time (s) c3 0 0 Time (s) *Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009. c3 0 c1 0 e ou c2 0 Time (s) • Two physical Qubits - NMR representation: − 2 spins 1/2: H LA I Z LB S Z 2 JI Z S z • Generalized Amplitude Damping Channel: – Longitudinal relaxation (T1) 1 E1 p 0 0 0 , E2 p 1 0 1 E3 1 p 0 L 1 p 2 2 k BT 0 0 , E4 p 1 0 1 1 et T1 2 Energy exchange between system and environment 0 0 • Phase Damping Channel: Loss of coherence without loss of energy - Transversal relaxation (T2): 1 0 1 0 E1 , E2 1 0 1 0 1 t 1 1 e 2T2 2 1 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 c3 0 c1 0 e c2 0 c3 0 c1 0 e ou c2 0 Mutual information Mutual information Classical correlation Classical correlation Quantum correlation Quantum correlation Geometric Discord Hilbert-Schmidt distance between the state and the nearest classical state; E DG 2 min C 2 S D 2 Diogo sets a partneship with Gerardo Adesso ρ C *Dakic, B.; Vedral, V.; Brukner, C. Necessary and sufficient condition for nonzero quantum discord. Physical Review Letters, v. 105, n. 19, p. 190502, 2010. Girolami, D.; Adesso, G. Observable measure of bipartite quantum correlations. Physical Review Letters, v. 108, n. 15, p. 150403, 2012. Modi, K. and et al. Unified view of quantum and classical correlations. Physical Review Letters, v. 104, n. 8, p. 080501, 2010. • 2 q-bits: 3 3 3 1 1 1 xi i 1 yi 1 i Cij i j 4 i 1 i 1 i , j 1 DG 2 Tr S k1 • 2 q-bits: 3 3 3 1 1 1 xi i 1 yi 1 i Cij i j 4 i 1 i 1 i , j 1 DG 2 Tr S k1 S 1 t xx CC t 4 • Para um sistema de 2 q-bits: 3 3 3 1 1 1 xi i 1 yi 1 i Cij i j 4 i 1 i 1 i , j 1 DG 2 Tr S k1 S 1 t xx CC t 4 k1 Tr S 3 6Tr S 2 2Tr S 3 2 cos 3 • 2 q-bits: 3 3 3 1 1 1 xi i 1 yi 1 i Cij i j 4 i 1 i 1 i , j 1 DG 2 Tr S k1 S k1 1 t xx CC t 4 Tr S 3 6Tr S 2 2Tr S 3 3 arccos 2Tr S 9Tr S Tr S 2 9Tr S 3 xi Tr i 1 i 1 yi Tr 1 i 1 i 2 cos 3 2 3Tr S 2 Tr S 2 3 cij Tr i j i j 4 Ii I j • Direct Measurement Method: 3 3 3 1 1 1 xi i 1 yi 1 i cij i j 4 i 1 i 1 i 1 xi Tr i 1 i 1 yi Tr 1 i 1 i NMR Observables cij Tr i j i j 4 Ii I j Convert into a local measurement: Zero and Double Quantum Coherences and anti-phase magnetizations • Direct Measurement Method: 3 3 3 1 1 1 xi i 1 yi 1 i cij i j 4 i 1 i 1 i 1 xi Tr i 1 i 1 yi Tr 1 i 1 i NMR Observables cij Tr i j i j 4 Ii I j Convert into a local measurement: Zero and Double Quantum Coherences and anti-phase magnetizations Tr i j Tr i 1 ij ij U ij U ij† , onde U ij CNOTA B R , ij i j ij U ij U ij† , onde U ij CNOTA B R , ij i j 1 x 2 z 3 y Θ j/i 1 2 3 1 0 3π/2 π/2 2 3π/2 π/2 - π/2 3 π/2 - π/2 π/2 – Negativity of Quantumness (QNA)*: Minimum amount of entanglement created between the system and its measurement apparatus in a local measurement; Geometric measurement (trace norm); J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013). T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013). – Negativity of Quantumness (QNA)*: Minimum amount of entanglement created between the system and its measurement apparatus in a local measurement; Geometric measurement (trace norm); QN A 1 AB min ij 1 1 B A 2 i, j • Bell diagonal states: J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013). T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013). – Negativity of Quantumness (QNA)*: Minimum amount of entanglement created between the system and its measurement apparatus in a local measurement; Geometric measurement (trace norm); QN A 1 AB min ij 1 1 B A 2 i, j • Bell diagonal states: QN A AB cint 3 1 1 c j j j 4 j 1 2 J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013). T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013). • Freezing phenomenon: – Initial state condition*: 0 3 12 e 0 3 1 2 0 ou 0 1 2 3 e 0 1 3 2 0 – Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4) *You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012. • Freezing phenomenon: c1 t e t c1 0 – Initial state condition*: c2 t e t c2 0 0 3 12 e 0 3 1 2 0 c3 t c3 0 ou 0 1 2 3 e 0 1 3 2 0 T2A T2B A B T2 T2 – Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4) 2 DG DG *You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012. Time (s) • Generalized Amplitude Damping Channel: c1 t 1 a 1 b c1 0 c2 t 1 a 1 b c2 0 c3 t 1 a b a b c3 0 1 et T1 • 2 qubits system represented by 2 coupled spins ½: – Sample: 100 mg of 13C-labeled CHCl3 dissolved in 0.7 mL CDCl3 – Spectrometer: Varian Premium Shielded – 11 T H – 500 MHz, T1 = 9 s, T2 = 1.2 s C – 125 MHz, T1 = 25 s, T2 = 0.18 s Acoplamento J – 215.1 Hz – Initial State: c1 0.5, c2 0.06, c3 0.24 c3 0 c1 0 e ou c2 0 Fidelity = 0.993 • 2 qubits system represented by 2 coupled spins ½: – Sample: 100 mg of 13C-labeled CHCl3 dissolved in 0.7 mL CDCl3 – Spectrometer: Varian Premium Shielded – 11 T H – 500 MHz, T1 = 9 s, T2 = 1.2 s C – 125 MHz, T1 = 25 s, T2 = 0.18 s Acoplamento J – 215.1 Hz – Initial State: c1 0.5, c2 0.06, c3 0.24 c3 0 c1 0 e ou c2 0 – 1º State: c1 c2 c3 0.2 c3 0 c1 0 e c2 0 Fidelity = 0.994 – 2º State: c1 0.5, c2 0.06, c3 0.24 c 0 c 0 e ou c 0 3 1 2 Fidelity = 0.993 • Geometric Discord: Direct Measurement Time (s) Time (s) Tomography Theoretical Time (s) Time (s) • Negativity of Quantumness: (Theoretical) Time (s) Time (s) Time (s) Time (s) (Theoretical) Freezing Universality (a) Discord (b) Geometric Discord (c) Trace Distance (d) Bures Distance Aaronson, B.; Lo Franco, R.; Adesso, G. Physical Review A, v. 88, p. 012120, 2013. Preliminary Results Relaxation Process Decoherence Channels: Phase Damping (PD) Generalized Amplitude Damping (GAD) • Phase Damping Channel: Loss of coherence without loss of energy Two Qubit System - Transversal relaxation (T2): 1 0 1 0 E1 , E 1 2 0 1 0 1 t 1 1 e 2T2 2 *Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010. • Phase Damping Channel: Loss of coherence without loss of energy Two Qubit System - Transversal relaxation (T2): 1 0 1 0 E1 , E 1 2 0 1 0 1 - Global Phase Damping (spin 3/2 system)*: 1 0 0 0 1 0 E0 1 0 0 1 0 0 0 1 CJ t 1 e 2 0 0 1 0 0 , E1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 *Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010. t 1 1 e 2T2 2 • Phase Damping Channel: Loss of coherence without loss of energy Two Qubit System - Transversal relaxation (T2): 1 0 1 0 E1 , E 1 2 0 1 0 1 t 1 1 e 2T2 2 - Global Phase Damping (spin 3/2 system)*: 1 0 0 0 1 0 E0 1 0 0 1 0 0 0 1 CJ t 1 e 2 0 0 1 0 0 , E1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 11 12 13 14 12 22 23 24 13 23 33 34 24 34 44 14 *Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010. • Phase Damping Channel: Loss of coherence without loss of energy Two Qubit System - Transversal relaxation (T2): 1 0 1 0 E1 , E 1 2 0 1 0 1 t 1 1 e 2T2 2 - Global Phase Damping (spin 3/2 system)*: 1 0 0 0 1 0 E0 1 0 0 1 0 0 0 1 CJ t 1 e 2 0 0 1 0 0 , E1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 11 12 13 14 12 22 23 24 13 23 33 34 24 34 44 14 *Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010. Emergence of the Pointer Basis: The pointer basis emerges when classical correlation between S and A becomes constant!* Decoherence S A Measurement Collapse of A in some classical state which is not altered by decoherence! Time (s) J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013). E • Phase Damping Channel: – 2 spins ½ system • Generalized Amplitude Damping Channel: – 3/2 spins system – Sample: Lyotropic Liquid Crystals • Sodium Dodecyl Sulfate (SDS) • Heavy Water (D2O) • Decanol (C10H21OH) – Spectrometer: Varian Inova – 8 T Na – 92 MHz νQ = 10.4 kHz T1A B 1 11.3 ms 2CJ1 2 Conclusion • Differences between representing two qubit systems with two spins 1/2 coupled and one spin 3/2. • Effects of phase damping and generalized amplitude damping channels. • Experimental observation of Sudden-change, Freezing, Double Sudden-Change phenomena and the emergence of Pointer Basis. Acknowledgments
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