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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, Bangalore4 University of Oxford
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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]
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ħgB0
Many nuclei have ‘spin angular momentum’ and ‘magnetic moment’
Coherent Superposition
a|0 + b |1 |0 |1
B0
Nuclear Spins
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Spectrometer Sample:1015 spins
RF coilPulse/Detect
Superconductingcoil
H0
H1cos(wt)
~
Nuclear Magnetic Resonance (NMR)
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1015 spins
Pseudopure State
p1
p0
= 1
~ 105 at 300 K, 12 T
E
kT =
B0 |0
|1 p1
p0
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1015 spins
Pseudopure State
p1
p0
= 1
~ 105 at 300 K, 12 T
E
kT =
B0 |0
|1 p1
p0
pseudopure
=(1- )1/2+|00|
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1015 spins
Pseudopure State
p1
p0
= 1
~ 105 at 300 K, 12 T
E
kT =
B0 |0
|1 p1
p0
RF
pseudopure
=(1- )1/2+|++| =(1- )1/2+|00|
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2-qubit register
=(1- )1/2+|00|
> 1/3UW
NonseparableState
Resources
SeparableState
1/3UW
• parahydrogens (Jones &Anwar, PRA 2004)
• q-transducer (Cory et al, PRA 2007)
Resource:Entanglement
Resource:Discord(in units of 2)
Hemant, Roy, TSM, A. Patel, PRA2012
• pseudopure states
• Cory 1997• Chuang 1997
• ~ pure states
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7-qubit NMR register
NMR systems useful?Pseudopure|0000000
Preparation
(scalability?)
Shor’salgorithm
15 = 3 x 5 Chuang, Nature 2002
No entanglementfinite discord
Open question:
Is discord sufficient resource for quantum computation ?
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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
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Correlation LGI(CLGI)
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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.”
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
Leggett-Garg (1985) Sir Anthony James LeggettUni. of Illinois at UC
Prof. Anupam GargNorthwestern University, Chicago
How to distinguishQuantum behaviorFrom Classical ?
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• 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
LGI studies in various systems
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Consider a system with a dynamic dichotomic observable Q(t)
Dichotomic : Q(t) = 1 at any given time
timeQ1 Q2 Q3
t2 t3 . . .
. . .
Leggett-Garg (1985)
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
PhD Thesis, Johannes Kofler, 2004
t1
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timeQ1
t = 0
Q2 Q3
t . . .
. . .
2t
Two-Time Correlation Coefficient (TTCC)
EnsembleTime ensemble (sequential)
Spatial ensemble (parallel)
Temporal correlation: Cij = Qi Qj = Qi(r)
Qj(r)N
1
r = 1
N
1 Cij 1 Cij = 1 Perfectly correlated
Cij =1 Perfectly anti-correlated
Cij = 0 No correlation
= pij+(+1) + pij
(1)
r over an ensemble
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LG string with 3 measurements
K3 = C12 + C23 C13
K3 = Q1Q2 + Q2Q3 Q1Q3
3 K3 1
Leggett-Garg Inequality (LGI)
K3
time
Macrorealism(classical)
timeQ1
t = 0
Q2 Q3
t 2t
Q1 Q2 Q3 Q1Q2+Q2Q3-Q1Q3
1 1 1 11 1 -1 11 -1 1 -31 -1 -1 1-1 1 1 1-1 1 -1 -3-1 -1 1 1-1 -1 -1 1
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TTCC of a spin ½ particle
TimeQ1
t = 0
Q2 Q3
t 2t
Consider :A spin ½ particle
Hamiltonian : H = ½ wz
Maximally mixed initial State : 0 = ½ 1 Dynamic observable: x eigenvalues 1 (Dichotomic )
C12 = x(0)x(t) = x e-iHt x eiHt
= x [xcos(wt) + ysin(wt)]
C12 = cos(wt)
Similarly, C23 = cos(wt)
and C13 = cos(2wt)PhD Thesis, Johannes Kofler, 2004
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Quantum States Violate LGI: K3 with Spin ½
timeQ1
t = 0
Q2 Q3
t 2t
K3 = C12 + C23 C13 = 2cos(wt) cos(2wt)
K3
wt2 3
Macrorealism(classical)
Quantum !!
40
No violation !
(/3,1.5)
Maxima (1.5) @cos(wt) =1/2
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K4 = C12 + C23 + C34 C14 = 3cos(wt) cos(3wt)
Quantum States Violate LGI: K4 with Spin ½
Extrema (22) @cos(2wt) =0
K4 Macrorealism(classical)
Quantum !!
wt2 3 40
(/4,22)
(3/4,22)
time
Q1
t = 0
Q2 Q3
t 2t 3t
Q4
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Evaluating K3
K3 = C12 + C23 C13
t = 0 t 2t
x
↗
x
↗
x
↗
x
↗
x
↗
x
↗
time
ENSEMBLE x(0)x(t) = C12
x(t)x(2t) = C23
x(0)x(2t) = C13
ENSEMBLE
ENSEMBLE
0
Hamiltonian : H = ½ wz
0
0
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Evaluating K4
K4 = C12 + C23 + C34 C14
t = 0 t 2t
x
↗
x
↗
x
↗
x
↗
x
↗
time
x
↗x
↗
x
↗
3t
ENSEMBLE x(0)x(t) = C12
x(t)x(2t) = C23
x(0)x(3t) = C14
x(2t)x(3t) = C34
Joint Expectation Value
ENSEMBLE
ENSEMBLE
ENSEMBLE
Hamiltonian : H = ½ wz
0
0
0
0
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Moussa Protocol
O. Moussa et al, PRL,104, 160501 (2010)
Target qubit (T)
Probe qubit (P)
A B
x
↗|+
AB
Joint Expectation Value
A↗
B↗
ABTarget qubit (T)
Dichotomicobservables
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Sample13CHCl3
(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).
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Experiment – pulse sequence
1H
13C
= Ax Aref
Ax(t)+i Ay(t)
Ax(t) = cos(2tij) Ay(t) = sin(2tij)
Ax(t) x(t)
=
0
V. Athalye, S. S. Roy, and TSM, Phys. Rev. Lett. 107, 130402 (2011).
1/2
90x
PFG
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wt
Experiment – Evaluating K3
timeQ1
t = 0
Q2 Q3
t 2t
K3 = C12 + C23 C13
= 2cos(wt) cos(2wt)
(w = 2100)
Error estimate: 0.05
V. Athalye, S. S. Roy, and TSM, Phys. Rev. Lett. 107, 130402 (2011).
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Experiment – Evaluating K3
50 100 150 200 250 300 t (ms)
LGI violated !!(Quantum)
LGI satisfied
Decay constant of K3 = 288 ms
165 ms
V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011).
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wt
Experiment – Evaluating K4
(w = 2100)
Error estimate: 0.05
K4 = C12 + C23 + C34 C14
= 3cos(wt) cos(3wt)
time
Q1
t = 0
Q2 Q3
t 2t 3t
Q4
Decay constant of K4 = 324 ms
V. Athalye, S. S. Roy, and TSM, Phys. Rev. Lett. 107, 130402 (2011).
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Entropic LGI(ELGI)
A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal,arXiv: 1208.4491 [quant-ph]
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timeQ1 Q2 Q3
t2 t3 . . .
. . .
t1
System
A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]
System state: 1/2
Dynamical observable : Sz(t) = Ut Sz Ut†
Time Evolution: Ut = exp(iwSxt)
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Information Deficit:
timeQ1 Q2 Q3
t2 t3 . . .
. . .
t1
ELGI bound
A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]
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Extracting ProbabilitiesSingle-event:
timeQk
. . .
. . .
tk
For S = 1/2
P(0) = ½P(1) = ½
Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]
k
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Extracting Probabilitiestime
Qj
. . .
. . .
tjti
QiTwo-time joint:
Invasivej
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Extracting Probabilitiestime
Qj
. . .
. . .
tjti
QiTwo-time joint:
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Extracting Probabilitiestime
Qj
. . .
. . .
tjti
QiTwo-time joint:
P(0,qj) P(1,qj)
Non-Invasive Measurement (NIM)
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System
Two-time joint probability
CH
system
ancilla
Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]
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Two-time joint probabilities
P(q1,q2) P(q1,q3)
time
Q1 Q2 Q3
t2 t3t1
Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]
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Information Deficit
CNOT
Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]
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Information Deficit
CNOT
AntiCNOT
Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]
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Information Deficit
CNOT
AntiCNOT
NIM
Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]
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Legitimate Grand Probability A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]
Classical Probability Theory:
P’(q1,q2) = P(q1,q2,q3)q3
P’(q1,q3) = P(q1,q2,q3)q2
P’(q2,q3) = P(q1,q2,q3)q1
P(q1,q2)
P(q1,q3)
P(q2,q3)
Marginals Grand
time
Q1 Q2 Q3
t2 t3t1
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Extracting Grand Probability
Three-time joint:
Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]
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Illegitimate Joint Probability
P(q1,q2,q3)is illegitimate !!
Violation ofEntropic LGI
Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]
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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 !!