1. 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

2. 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: [quant-ph]

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

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

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

6. 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

7. 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

8. 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

9. 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 ?

10. 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

11. Correlation LGI
(CLGI)

12. 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.”

13. 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

14. 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

15.  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

16. 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 K3
time
Macrorealism
(classical)
3  K3  1
Leggett-Garg Inequality (LGI)

17. 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

18. 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
cos(t) =1/2
No violation !

19. 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
cos(2t) =0

20. 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

21. 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

22. 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, (2010)

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

24. 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, (2011). 

25. 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, (2011). 
( = 2100)

26. Experiment – Evaluating K350
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, (2011). 

27. 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, (2011). 
( = 2100)

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

29. 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: [quant-ph]

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

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

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

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

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

35. System Two-time joint probability ancilla systemH
system
Hemant, Abhishek, Koteswar, TSM, arXiv: [quant-ph]

36. Two-time joint probabilitiesQ1 Q2 Q3 t2 t3 t1
P(q1,q2)
P(q1,q3)
Hemant, Abhishek, Koteswar, TSM, arXiv: [quant-ph]

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

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

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

40. Legitimate Grand Probability
A. R. Usha Devi et al,
arXiv: [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

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

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

43. 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 !!