1. Quantum Correlations in Nuclear Spin Ensembles
T. S. Mahesh
Indian Institute of Science Education and Research, Pune

2. Contents: Nuclear Spin-Ensembles
Nuclear Magnetic Resonance (NMR) and Quantum Information
Quantumness debate
Leggett-Garg inequality
-- NMR evaluation
Quantum correlations
-- NMR evalutions
Summary

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

4. Nuclear Magnetism p1 p1 p0 p0 Sample 1015 molecules |1 = 1   B0 |0
~ 105 at 300 K, 11.7 T E kT
 =
 = ½ 1 +  z

5. Nuclear Magnetic Resonance (NMR)
|1
Electromagnetic
energy (2pn = gB0)
|1
90 degree pulse B0
|0
|0
 = ½ 1 +  z
 = ½ 1 +  x
Larmor
precession ~

6. Nuclear Magnetic Resonance (NMR)
Spectrometer
Sample
RF coil
Pulse/Detect H0
H1cos(wt) ~
Superconducting
coil

7. NMR Spectrometer, IISER-Pune

8. Addressability: Different Larmor freqencies – i) Heteronuclear
ii) Chemical Shift
Chemical Shift:
O H
C H wH OH wH CH
> w

9. Spin-Spin interactions:H
J-coupling
1-102 Hz C H
Dipolar coupling
(liquid crystals,
solids) Hz

10. NMR spin system 1 3-qubits : Eg. 2,3-dibromopropionic acid 2 time
Signal
 x 3
Fourier
Transform 2 3 1 Hz

11. Benchmarking 12-qubits A 8 A’ 2 11 13 4 5 6 7 9 8 10 11
Benchmarking circuit – Prepares entangled states
AA’ 1 2 3 4 5 6 7 8 9 10 11
Qubits
Time
PRL, 2006

12. NMR and Quantum Information
1. Quantum Logic Gates (1997)
2. Deutsch-Jozsa Algorithm (1998)
3. Grover’s Algorithm (1998)
4. Quantum Error Correction (1998)
5. Quantum Simulation (1999) .

13. Is there a quantum state?

14. List goes on … 5. Quantum Fourier Transform (2001)
6. Order-finding, Quantum Counting (2001)
7. Teleportation (2002)
8. Quantum Games (2002) .

15. Factoring 15 by NMR using Quantum Algorithm

16. How to distinguish quantum and classical behavior?
NMR states - Quantum or Classical ?
How to distinguish quantum and classical behavior?
Leggett-Garg
Inequality
Quantum
correlations

17. Leggett-Garg inequality (LGI)

18. Sir Anthony James Leggett
Leggett-Garg inequality (1985)
Sir Anthony James Leggett
Uni. of Illinois at UC 
Prof. Anupam Garg
Northwestern University, Chicago
A. J. Leggett and A. Garg,
PRL 54, 857 (1985)

19. Leggett-Garg inequality (1985)
Consider a system with a dichotomic quantity 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

20.  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 = 0  No correlation
Cij =1  Perfectly anti-correlated

21. LG string with 3 measurements
time Q1
t = 0 Q2 Q3 t
2t
K3 = C12 + C23  C13
K3 = Q1Q2 + Q2Q3  Q1Q3
Consider: Q1Q2 + (Q2  Q1)Q3
If Q1  Q2 : = 1
Q1  Q2 :  (2) = 1 or 3
 Q1Q2 + Q2Q3  Q1Q3 = 1 or 3
 3 < Q1Q2 + Q2Q3  Q1Q3 < 1 K3
time
Macrorealism
(classical)
3  K3  1
Leggett-Garg Inequality (LGI)

22. TTCC of a spin ½ particle
Consider :
A spin ½ particle precessing about z
Hamiltonian : H = ½ z
Initial State : highly mixed state : 0 = ½ 1 +  x ( ~ 10-5)
Dichotomic observable: x  eigenvalues  1
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)

23. 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)
No violation ! K3
t 2  3
Macrorealism
(classical)
Quantum !! 4
(/3,1.5)
Maxima
cos(t) =1/2

24. LG string with 4 measurements
time Q1
t = 0 Q2 Q3 t
2t
3t Q4
K4 = C12 + C23 + C34  C or,
K4 = Q1Q2 + Q2Q3 + Q3Q4  Q1Q4
Macrorealism
(classical) K4
time
Consider: Q1(Q2  Q4) + Q3(Q2 + Q4)
If Q2  Q4 : (2) = 2
Q2  Q4 : (2) = 2
 Q1Q2 + Q2Q3 + Q3Q4  Q1Q = 2
2  K4  2
Leggett-Garg Inequality (LGI)

25. 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

26. Evaluating K3 K3 = C12 + C23  C13 Hamiltonian : H = ½ z EXPT - 1
time
ENSEMBLE
x(0)x(t) = C12
x(t)x(2t) = C23
x(0)x(2t) = C13 0
EXPT - 2
ENSEMBLE 0
EXPT - 3
ENSEMBLE 0

27. 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 Values
Hamiltonian : H = ½ z
EXPT - 1 0
EXPT - 2 0 0
EXPT - 3
EXPT - 4 0

28. 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)

29. Sample 13CHCl3 (in DMSO) Target: 13C Probe: 1H
Resonance Offset: Hz Hz
T1 (IR) s s
T2 (CPMG) s s
Ensemble of
~1018 molecules

30. Experiment – pulse sequence1H
13C 0
= Ax Aref
Ax(t)+i Ay(t)
Ax(t) = cos(2tij)
Ay(t) = sin(2tij)
Ax(t)  x(t)
 =
V. Athalye, S. S. Roy, and T. S. Mahesh,  PRL 107, (2011). 

31. Experiment – Evaluating K3
time Q1
t = 0 Q2 Q3 t
2t
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 T. S. Mahesh,  PRL 107, (2011). 
( = 2100)

32. Experiment – Evaluating K3
time Q1
t = 0 Q2 Q3 t
2t 50
100
150
200
250
300
t (ms)
LGI violated !!
(Quantum)
LGI satisfied
(Macrorealistic)
t ~ 165 ms
Decay constant of K3 = 288 ms
V. Athalye, S. S. Roy, and T. S. Mahesh,  PRL 107, (2011). 

33. 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 T. S. Mahesh,  PRL 107, (2011). 
( = 2100)

34. Quantum Correlations Pinto et al, Phys. Rev. A 81, 062118 (2010)
Pinto et al, arXiv: v1 [quant-ph]
Hemant et al, (manuscript in preparation)

35. Information content in a two-state systemV
Equal probability of the two states  Maximum Entropy !!
Maximum information !!
P(0) < P(1)
P(0) ~ P(1)
P(0) > P(1)
time V
time V
time

36. Information content and Probability
Shannon Entropy :
eg. H(X) =  p log2p (1  p) log2(1p)
H(X) p

37. Information content in a Quantum State
Von Neumann Entropy :
Eigenvalues
For example: Werner State
H(W)
Eigenvalues: (1)/4 , (1)/4 , (1)/4 , (13)/4
Von Neumann Entropy: H(W) =
2 – ¼ log2(1)3(13) – (3/4) log2(13)/(1) 

38. Conditional Entropy
Conditional entropy

39. Mutual Entropy Mutual entropy Classically Equivalent No measurement
Directly from VN entropy !
Requires measurement (of A)
Depends on measurement basis

40. Captures non-classical correlations !!
Discord
Captures non-classical correlations !!
Classical State or Zero-Discord States D(B|A) = 0:
States of the form are classical w.r.t. A
projection
operators
Measurement of A does not change the correlations of A with B
Ollivier and Zurek, PRL 88, (2002)

41. Distance between the given state and the nearest classical state
Geometric Discord
trace[(  )2]
Distance between the given state and the nearest classical state
Dakic et al, PRL 105, (2010)
S Luo et al, PRA 82, (2010)
Rana et al, PRA 85, (2012)
Hassan et al, arXiv: v3.

42. Discord of Werner state

43. NMR Werner state
Werner State
8  105

44. With Dynamical Decoupling

45. Long-lived Werner-State

46. Rotational Symmetry
F = 0.8
F = 0.91 W
F = 0.93
F = 0.99
F = 0.82

47. Summary
Spin ½ nucleus prepared in a quantum superposition violates LGI, thus confirming
the quantum behavior. However, the gradual decay of coherence amplitudes leads ultimately to the satisfaction of LGI, thus indicating the emergence of macro-realistic behavior.
Two-qubit NMR systems can have non-zero discord, and so exhibit quantum correlations.

48. Acknowledgments Dr. Vikram Athalye, Cummins College, Pune
Soumya Singha Roy,
IISER, Pune
Hemant Katiyar
IISER, Pune
Prof. Anthony J. Leggett,
University of Illinois, UC
Prof. Anil Kumar,
IISc, Bangalore
DST project
Prof. Apoorva Patel
IISc, Bangalore