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Density Matrix Tomography, Contextuality , Future Spin Architectures

Density Matrix Tomography, Contextuality , Future Spin Architectures. T. S. Mahesh Indian Institute of Science Education and Research, Pune. ~. Density Matrix Tomography (1-qubit) . + . = ħ / kT ~ 10 -5.  = . Background Does not lead to signal. Deviation May lead

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Density Matrix Tomography, Contextuality , Future Spin Architectures

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  1. Density Matrix Tomography, Contextuality, Future Spin Architectures T. S. Mahesh Indian Institute of Science Education and Research, Pune

  2. ~ Density Matrix Tomography (1-qubit) +  • = ħ / kT • ~ 10-5  = Background Does not lead to signal Deviation May lead to signal My Mx

  3. ~ Density Matrix Tomography (1-qubit) NMR detection operators: x , y 1. Heterodyne detection x = 2R y = -2S 2. Apply (/2)y + Heterodyne detection x = 2P  = (/2)y 1 = My Mx

  4. Density Matrix Tomography (2-qubit) 15 REAL NUMBERS + + NMR detection operators: x1 , y1 , x2 , y2

  5. Density Matrix Tomography (2-qubit) 15 REAL NUMBERS + + Traditional Method : Requires Spin selective pulses Integration of Transition

  6. Density Matrix Tomography (2-qubit) 15 REAL NUMBERS + + Traditional Method : Requires Spin selective pulses Integration of Transition

  7. Density Matrix Tomography (2-qubit) 15 REAL NUMBERS + + NEW Method Requires No spin selective pulses Integration of spins JMR, 2010

  8. Density Matrix Tomography (2-qubit) SVD tomo

  9. Density Matrix Tomography of singlet state tr(rthrexp) Correlation = = 0.98 [tr(rth2 ) tr(rexp 2)]1/2 Real Imag Theory Expt JMR, 2010

  10. Quantum Contextuality

  11. Non- Contextuality • The result of the measurement of an operator A depends solely on A and on the system being measured. • If operators A and B commute, the result of a measurement of their product AB is the product of the results of separate measurements of A and of B. All classical systems are NON-CONTEXTUAL Physics Letters A (1990), 151, 107-108

  12. Non- Contextuality • Measurement outcomes can be • assigned, in principle, even before • the measurement

  13. Quantum Contextuality • Measurement outcomes can not be • pre-assigned even in principle z1 z2 z1z2 1 Eg. Two spin-1/2 particles x2 x1 x1x2 1 z1x2 x1z2 y1y2 1 1 1 -1 LHVT = 6 QM N. D. Mermin. PRL 65, 3373 (1990). PRL 101,210401(2008)

  14. Laflamme, PRL 2010

  15. NMR demonstration of contextuality Sample: Malonic acid single crystal ~ 5.3 Laflamme PRL 2010

  16. Peres Contextuality Let us consider a system of two spin half particles in singlet state. Singlet state: Physics Letters A (1990), 151, 107-108

  17. Peres Contextuality For a singlet state < σx1 σx2> = -1 < σy1 σy2> = -1 <(σx1 σy2)(σy1 σx2)> = -1 Note: [σx1,σx2] = 0 [σy1,σy2] = 0 [σx1 σy2, σy1 σx2] = 0 Physics Letters A (1990), 151, 107-108

  18. Peres Contextuality For a singlet state Pre-assignment of eigenvalues < σx1 σx2> = -1  x1 x2 = -1 < σy1 σy2> = -1  y1 y2 = -1 <(σx1 σy2)(σy1 σx2)> = -1  x1 y2 y1 x2 = -1 CONTRADICTION !! Note: [σx1,σx2] = 0 [σy1,σy2] = 0 [σx1 σy2, σy1 σx2] = 0 Physics Letters A (1990), 151, 107-108

  19. Experiment • Using three F spins of Iodotrifluoroethylene. Two were used to prepare singlet and one was ancilla.

  20. Pseudo-singlet state Iz1+Iz2+Iz3 • Pure singlet state is hard to prepare in NMR

  21. Pseudo-singlet state Iz1+Iz2+Iz3 • Pure singlet state is hard to prepare in NMR No Signal !! <σx1+σx2>=0

  22. Pseudo-singlet state Imaginary Part Real Part Theory Fidelity=0.97 Experiment

  23. Moussa Protocol A B Target (ρ) <AB> Probe(ancilla)|+ <AB> Target (ρ) Physical Review Letters (2010), 104, 160501 A B

  24. NMR circuit for Moussa Protocol <σx>=<AB> 1 (Ancilla) 2 3 |+ PPS A B Singlet

  25. Manvendra Sharma, 2012 Results

  26. Future Architectures ?

  27. Criteria for Physical Realization of QIP • Scalable physical system with mapping of qubits • A method to initialize the system • Big decoherence time to gate time • Sufficient control of the system via time-dependent Hamiltonians • (availability of a universal set of gates). • 5. Efficient measurement of qubits DiVincenzo, Phys. Rev. A 1998

  28. NMR Circuits - Future • xx - qubits Time Decoherence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . . . Qubits Transverse relaxation Longitudinal relaxation Loss of q. memory Loss of c. memory • |00 + b|11 |0110010 {|00 , |11} |000000 Larger Quantum register • Addressability • Week coupling • Controllability T2 T1 <

  29. Liquid-state NMR systems • Advantages • High resolution • Slow decoherence • Ease of control • Disadvantages • Smaller resonance dispersion • Small indirect (J) couplings • Smaller quantum register Random, isotropic tumbling

  30. Single-crystal NMR systems • Advantages • Large dipole-dipole couplings ( > 100 times J) • Orientation dependent Hamiltonian • Longer longitudinal relaxation time • Larger quantum register (???) • Disadvantages • Shorter transverse relaxation time • Challenging to control the spin dynamics

  31. Single-crystal NMR systems • Active spins in a bath of inactive molecules J. Baugh, PRA 2006 • Large couplings • High resolution • Hopefully – • Larger quantum register

  32. QIP with Single Crystals Cory et al, Phys. Rev. A 73, 022305 (2006) Malonic Acid Two-molecules per unit center: Inversion symmetry – P1 space group So, the two molecules are magnetically equivalent Inter-molecular interactions ?

  33. QIP with Single Crystals Malonic Acid Natural Abundance Cory et al, Phys. Rev. A 73, 022305 (2006)

  34. Malonic Acid Pseudopure States Cory et al, Phys. Rev. A 73, 022305 (2006)

  35. Malonic Acid Pseudopure States Cory et al, Phys. Rev. A 73, 022305 (2006)

  36. Quantum Gates Eg. C2-NOT Cory et al, Phys. Rev. A 73, 022305 (2006)

  37. R. Laflamme, PRL 2010 ~ 5.3

  38. Glycine Single Crystal Mueller, JCP 2003 000 PPS

  39. Floquet Register More qubits More coupled Nuclear Spins More Resolved Transitions Side-bands? S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014

  40. S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014

  41. Solid-State NMR and next generation QIP Pseudo-Pure States 13C spectra of aromatic carbons of Hexamethylbenzene spinning at 3.5 kHz

  42. Grover’s Algorithm Methyl 13C S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014

  43. Electron Spin vs Nuclear Spin Spin e n Magnetic moment 103 1 Sensitivity High Low Coherence Time 1 103 Measurement Processing

  44. e-n Entanglement Entanglement in a solid-state spin ensemble • Stephanie Simmons et al Nature 2011 Mehring, 2004

  45. Electron spin actuators Cory et al

  46. Detection of single Electron Spin by Magnetic Resonance Force Microscopy D. Rugar, R. Budakian, H. J. Mamin & B. W. Chui Nature 329, 430 (2004)

  47. Cooling of nuclear spins eq = ee  IN Up = SWAP (e,n1) Ie 11 I(N-1) Measure e-spin If e invert ee  11  I(N-1) Up = SWAP (e,n2) Cory et al, PRA 07

  48. Anisotropic Hyperfine Interaction Nuclear Local Fields under Anisotropic Hyperfine Interaction e-n system B0

  49. Anisotropic Hyperfine Interaction Coherent oscillations between nuclear coherence on levels 1 & 2 driven by Microwave The nuclear p pulse : 520 ns e-n CNOT gate : 2ms (0.98 Fidelity)

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