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

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1/2 Density Matrix Tomography (1-qubit) = ~ MxMx MyMy PC = R+iS -P + = ħ / kT ~ 10 -5 Background Does not lead to signal Deviation May lead to signal

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PC = R+iS -P 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 = ~ MxMx MyMy ( /2) y - RP+iS R 1 =

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P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I6 + 15 REAL NUMBERS Density Matrix Tomography (2-qubit) NMR detection operators: x 1, y 1, x 2, y 2

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P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I6 + 15 REAL NUMBERS Traditional Method : Requires 1.Spin selective pulses 2.Integration of Transition Spin 1Spin 2 II 90 x I I 90 y I I 90 x 90 y 90 x 90 y Density Matrix Tomography (2-qubit)

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P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I6 + 15 REAL NUMBERS Traditional Method : Spin 1Spin 2 II 90 x I I 90 y I I 90 x 90 y 90 x 90 y Requires 1.Spin selective pulses 2.Integration of Transition

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P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I6 + 15 REAL NUMBERS NEW Method Requires 1.No spin selective pulses 2.Integration of spins Density Matrix Tomography (2-qubit) JMR, 2010

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Density Matrix Tomography (2-qubit) SVD tomo

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Density Matrix Tomography of singlet state Theory Expt RealImag Correlation = = 0.98 tr( th exp ) [tr( th 2 ) tr( exp 2 )] 1/2 JMR, 2010

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Quantum Contextuality

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Non- Contextuality 1. The result of the measurement of an operator A depends solely on A and on the system being measured. 2. 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

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

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Quantum Contextuality x2x2 x1x1 x1x2x1x2 z1z1 z2z2 z1z2z1z2 z1x2z1x2 x1z2x1z2 y1y2y1y2 1 1 1 11 Measurement outcomes can not be pre-assigned even in principle N. D. Mermin. PRL 65, 3373 (1990). = 6 LHVT QM Eg. Two spin-1/2 particles PRL 101,210401(2008)

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Laflamme, PRL 2010

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

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

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Peres Contextuality For a singlet state = -1 Note: [σ x 1,σ x 2 ] = 0 [σ y 1,σ y 2 ] = 0 [σ x 1 σ y 2, σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151, 107-108

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Peres Contextuality For a singlet state Pre-assignment of eigenvalues = -1 x 1 x 2 = -1 = -1 y 1 y 2 = -1 = -1 x 1 y 2 y 1 x 2 = -1 CONTRADICTION !! Note: [σ x 1,σ x 2 ] = 0 [σ y 1,σ y 2 ] = 0 [σ x 1 σ y 2, σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151, 107-108

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

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Pseudo-singlet state Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3

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Pseudo-singlet state Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3 No Signal !! =0

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

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Moussa Protocol Target (ρ) Probe(ancilla)|+ Target (ρ) Physical Review Letters (2010), 104, 160501 AB AB

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NMR circuit for Moussa Protocol Singlet 1 (Ancilla) 2 3 B |+ A =

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Results Manvendra Sharma, 2012

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Future Architectures ?

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Criteria for Physical Realization of QIP 1.Scalable physical system with mapping of qubits 2.A method to initialize the system 3.Big decoherence time to gate time 4.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

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NMR Circuits - Future 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15. Time Qubits xx - qubits Decoherence Transverse relaxation |00 + |11 Loss of q. memory {|00 , |11 } Longitudinal relaxation |0110010 |000000 Loss of c. memory T2T1 < Addressability Week coupling Controllability Larger Quantum register

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

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

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

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

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

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

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

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Quantum Gates Eg. C 2 -NOT Cory et al, Phys. Rev. A 73, 022305 (2006)

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~ 5.3 R. Laflamme, PRL 2010

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Glycine Single Crystal Mueller, JCP 2003 000 PPS

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

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

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

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

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Electron Spin vs Nuclear Spin Spin e n Magnetic moment 10 3 1 Sensitivity High Low Coherence Time 1 10 3 Measurement Processing

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e-n Entanglement Mehring, 2004 Entanglement in a solid-state spin ensemble Stephanie Simmons et alStephanie Simmons Nature 2011

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Electron spin actuators Cory et al

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

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eq = e e I N U p = SWAP (e,n 1 ) I e 1 1 I (N-1) Measure e-spin If e invert U p = SWAP (e,n 2 ) e e 1 1 I (N-1) Cooling of nuclear spins Cory et al, PRA 07

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Nuclear Local Fields under Anisotropic Hyperfine Interaction B0B0 e-n system

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

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