NMR investigations of Leggett-Garg Inequality V. Athalye2, H. Katiyar1, Soumya S. Roy1, Abhishek Shukla1, R. Koteswara Rao3 T. S. Mahesh1 1IISER-Pune, 2Cummins College, Pune, 3IISc, Bangalore Acknowledgements: Usha Devi1, K. Rajagopal2, Anil Kumar3, and G. C. Knee4 1 Bangalore University, 2 HRI & Inspire Inst., Virginia, USA, 3 IISc, Bangalore 4 University of Oxford

Plan NMR as a quantum testbed Correlation Leggett-Garg Inequality Entropic Leggett-Garg Inequality Summary Athalye, Roy, TSM, PRL 2011.  Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Nuclear Spins Many nuclei have ‘spin angular momentum’ and ‘magnetic moment’ ħgB0 B0 |0 |1 |0 + b |1 Coherent Superposition

Nuclear Magnetic Resonance (NMR) Spectrometer Sample: 1015 spins RF coil Pulse/Detect H0 H1cos(wt) ~ Superconducting coil

Pseudopure State p1 p0 p1 p0 |0 |1  B0 = 1   ~ 105 at 300 K, 12 T E kT  = 1015 spins

Pseudopure State p1 p0 p1 p0 |0 |1  B0  =(1-  )1/2+|00|  =(1-  )1/2+|00| pseudopure p1 p0 = 1   ~ 105 at 300 K, 12 T E kT  = 1015 spins

Pseudopure State p1 p0 p1 p0 |0 |1  RF B0  =(1-  )1/2+|00|  =(1-  )1/2+|00|  =(1-  )1/2+|++| pseudopure p1 p0 = 1   ~ 105 at 300 K, 12 T E kT  = 1015 spins

Resources parahydrogens (Jones &Anwar, PRA 2004) q-transducer (Cory et al, PRA 2007) Nonseparable State ~ pure states Resource: Entanglement  > 1/3 2-qubit register  =(1-  )1/2+|00| UW UW   1/3 Cory 1997 Chuang 1997 pseudopure states Separable State Resource: Discord (in units of 2) Hemant, Roy, TSM, A. Patel, PRA2012

NMR systems useful? Pseudopure |0000000 Preparation (scalability?) 7-qubit NMR register Shor’s algorithm No entanglement finite discord Chuang, Nature 2002 15 = 3 x 5 Open question: Is discord sufficient resource for quantum computation ?

NMR system as a quantum testbed Geometric Phases (Suter, 1988) Electromagnetically Induced Transparency (Murali, 2004) Contextuality (Laflamme, 2010) Delayed choice (Roy, 2012) Born’s rule (Laflamme, 2012) Why NMR? Long life-times of quantum coherence Unmatched control on spin dynamics

Correlation LGI (CLGI)

Sir Anthony James Leggett Leggett-Garg (1985) Sir Anthony James Leggett Uni. of Illinois at UC  Prof. Anupam Garg Northwestern University, Chicago How to distinguish Quantum behavior From Classical ? A. J. Leggett and A. Garg, PRL 54, 857 (1985) Macrorealism “A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states.” Non-invasive measurability “It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics.”

LGI studies in various systems N. Lambert et al, PRB 2001 J.-S. Xu et al., Sci. Rep 2011 Palacios-Laloy et al., Nature Phys. 2010 M. E. Goggin et al., PNAS USA 2011 J. Dressel et al., PRL 2011 M. Souza et al, NJP 2011 Roy et al, PRL 2011 G. C. Knee et al., Nat. Commun. 2012 C. Emary et al, PRB 2012 Y. Suzuki et al, NJP 2012 Hemant et al, arXiv 2012

Leggett-Garg (1985) Consider a system with a dynamic dichotomic observable Q(t) Dichotomic : Q(t) =  1 at any given time time Q1 Q2 Q3 t2 t3 . . . t1 A. J. Leggett and A. Garg, PRL 54, 857 (1985) PhD Thesis, Johannes Kofler, 2004

 Two-Time Correlation Coefficient (TTCC) time Q1 t = 0 Q2 Q3 t . . . Temporal correlation: Cij =  Qi Qj  = Qi(r) Qj(r)  N 1 r = 1 = pij+(+1) + pij(1) r  over an ensemble Ensemble Time ensemble (sequential) Spatial ensemble (parallel) 1  Cij  1 Cij = 1  Perfectly correlated Cij =1  Perfectly anti-correlated Cij = 0  No correlation

LG string with 3 measurements time Q1 t = 0 Q2 Q3 t 2t K3 = C12 + C23  C13 K3 = Q1Q2 + Q2Q3  Q1Q3 Q1 Q2 Q3 Q1Q2+Q2Q3-Q1Q3 1 1 1 1 1 1 -1 1 1 -1 1 -3 1 -1 -1 1 -1 1 1 1 -1 1 -1 -3 -1 -1 1 1 -1 -1 -1 1 K3 time Macrorealism (classical) 3  K3  1 Leggett-Garg Inequality (LGI)

TTCC of a spin ½ particle Consider : A spin ½ particle Hamiltonian : H = ½ z Maximally mixed initial State : 0 = ½ 1 Dynamic observable: x  eigenvalues  1 (Dichotomic ) Time Q1 t = 0 Q2 Q3 t 2t C12 = x(0)x(t) =  x e-iHt x eiHt  = x [xcos(t) + ysin(t)]   C12 = cos(t) Similarly, C23 = cos(t) and C13 = cos(2t) PhD Thesis, Johannes Kofler, 2004

Quantum States Violate LGI: K3 with Spin ½ time Q1 t = 0 Q2 Q3 t 2t K3 = C12 + C23  C13 = 2cos(t)  cos(2t) K3 t 2  3 Macrorealism (classical) Quantum !! 4 (/3,1.5) Maxima (1.5) @ cos(t) =1/2 No violation !

Quantum States Violate LGI: K4 with Spin ½ time Q1 t = 0 Q2 Q3 t 2t 3t Q4 K4 = C12 + C23 + C34  C14 = 3cos(t)  cos(3t) K4 Macrorealism (classical) Quantum !! t 2  3 4 (/4,22) (3/4,22) Extrema (22) @ cos(2t) =0

Evaluating K3 K3 = C12 + C23  C13 Hamiltonian : H = ½ z t = 0 t x↗ time ENSEMBLE x(0)x(t) = C12 x(t)x(2t) = C23 x(0)x(2t) = C13 0 0 0

Evaluating K4 K4 = C12 + C23 + C34 C14 t = 0 t 2t time 3t ENSEMBLE x↗ time 3t ENSEMBLE x(0)x(t) = C12 x(t)x(2t) = C23 x(0)x(3t) = C14 x(2t)x(3t) = C34 Joint Expectation Value Hamiltonian : H = ½ z 0 0 0 0

Moussa Protocol Joint Expectation Value AB Target qubit (T) Dichotomic observables  Target qubit (T) Probe qubit (P) A B x↗ |+  AB O. Moussa et al, PRL,104, 160501 (2010)

Sample 13CHCl3 (in DMSO) Target: 13C Probe: 1H Resonance Offset: 100 Hz 0 Hz T1 (IR) 5.5 s 4.1 s T2 (CPMG) 0.8 s 4.0 s V. Athalye, S. S. Roy, and TSM,  Phys. Rev. Lett. 107, 130402 (2011). 

Experiment – pulse sequence 1/2 0 = Ax Aref 1H 90x PFG Ax(t)+i Ay(t) Ax(t) = cos(2tij) Ay(t) = sin(2tij) Ax(t)  x(t) 13C  = V. Athalye, S. S. Roy, and TSM,  Phys. Rev. Lett. 107, 130402 (2011). 

Experiment – Evaluating K3 time Q1 t = 0 Q2 Q3 t 2t t K3 = C12 + C23  C13 = 2cos(t)  cos(2t) Error estimate:  0.05 V. Athalye, S. S. Roy, and TSM,  Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100)

Experiment – Evaluating K3 50 100 150 200 250 300 t (ms) LGI violated !! (Quantum) LGI satisfied 165 ms Decay constant of K3 = 288 ms V. Athalye, S. S. Roy, and TSM, Phys. Rev. Lett. 107, 130402 (2011). 

Experiment – Evaluating K4 time Q1 t = 0 Q2 Q3 t 2t 3t Q4 K4 = C12 + C23 + C34  C14 = 3cos(t)  cos(3t) Error estimate:  0.05 Decay constant of K4 = 324 ms V. Athalye, S. S. Roy, and TSM,  Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100)

Entropic LGI (ELGI) A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, arXiv: 1208.4491 [quant-ph]

System time Q1 Q2 Q3 t2 t3 . . . t1 System state: 1/2 Dynamical observable : Sz(t) = Ut Sz Ut† Time Evolution: Ut = exp(iSxt) A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph]

ELGI bound time Q1 Q2 Q3 t2 t3 . . . Information Deficit: t1   A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph]

Extracting Probabilities time Qk . . . Single-event: tk k For S = 1/2 P(0) = ½ P(1) = ½ Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Extracting Probabilities time Qj . . . Qi Two-time joint: ti tj Invasive j

Extracting Probabilities time Qj . . . Qi Two-time joint: ti tj

Extracting Probabilities time Qj . . . Qi Two-time joint: ti tj Non-Invasive Measurement (NIM) P(0,qj) P(1,qj)

System Two-time joint probability ancilla system H system Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Two-time joint probabilities Q1 Q2 Q3 t2 t3 t1 P(q1,q2) P(q1,q3) Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Information Deficit CNOT Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT

Information Deficit CNOT Anti CNOT Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT Anti CNOT

Information Deficit CNOT Anti CNOT NIM Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT Anti CNOT NIM

Legitimate Grand Probability A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph] time Q1 Q2 Q3 t2 t3 t1 Classical Probability Theory: Marginals Grand P(q1,q2)  P’(q1,q2) = P(q1,q2,q3)  q3 P(q1,q3)  P’(q1,q3) = P(q1,q2,q3)  q2 P(q2,q3)  P’(q2,q3) = P(q1,q2,q3)  q1

Extracting Grand Probability Three-time joint: Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Illegitimate Joint Probability P(q1,q2,q3) is illegitimate !!  Violation of Entropic LGI Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

Summary NMR spin-system violated correlation LGI for short time scales indicating the quantumness of the system. The gradual decoherence lead to the ultimate satisfaction of correlation LGI. NMR spins systems also violated entropic LGI in the expected time interval The experimental grand probability P(q1,q2,q3) could not generate the experimental marginal probability P(q1,q3) supporting the theoretical prediction. Thank You !!