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Measurements of General Quantum Correlations in
Nuclear Magnetic Resonance Systems
Eduardo Ribeiro deAzevedo
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São Paulo Brazil
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• 75 years• 240 courses• 57.000 undergrad students• ~200 Msc. and PHD programs
UNIVERSITY OF SÃO PAULO - USP
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UNIVERSITY OF SÃO PAULO AT SÃO CARLOS
São Carlos City:
250.000 people.5 universities: 1 Federal University (UFSCAR). 1 State Univesity (USP). 3 Private Univesities.
USP at São Carlos:
2 Campi, ~8.000 undergrad students
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São Carlos Institute of Physics, USP, Brazilwww.ifsc.usp.br
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Al
OC1OC6 - PPV
ETL(ionomer)
ITO
DC V
glassemitted
light
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NMR QIP in Brazil
CBPF NMR group: Ivan Oliveira, Alberto Passos, Roberto Sarthour, Jair C. C Freitas
(magnetism and magnetic materials)
IFSC NMR group: Tito Bonagamba, Eduardo R. deAzevedo(Solid-State NMR, MRI)
2002
First experiments done in São Carlos using quadrupolar nuclei 2003
First thesis defence in NMR QIP (Fabio A. Bonk at IFSC) and (Juan Bulnes at CBPF)
2005
Publication of the Book Quantum Information Processing by Elsevier
2007
2009 Gather with the quantum information theory group at UFABC – Lucas Celeri and Roberto
Serra.
2010
CBPF NMR spectrometer start to operate.
Hiring of new researchers (Alexandre Souza-CBPF, Diogo Pinto IFSC, João Teles-UFSCAR, Ruben Auccaise - UEPG ) tend to strenght this researche area. 2012
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PEOPLE INVOLVED
Isabela AlmeidaRuben AuccaiseAlexandre SouzaIvan S. Oliveira Roberto SarthourTito Bonagamba
Experiments Theory
Diogo S. PintoLucas CéleriRoberto SerraJonas Maziero Felipe FanchiniDavid GirolamiGerardo AdessoF. M. PaulaJ. D. MontealegreA SaguiaMarcelo Sarandy
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NMR and the QIP
• NMR is also an excellent test bench for studies on open quantum systems:– efficient implementation and manipulation of the quantum
states (excellent control of the unitary transformations coming from the radiofrequency pulses);
– presence of real environments, which can be described by phase damping and generalized amplitude damping channels;
• Experimental demonstration of QIP procedures, including quantum protocols, algorithms, quantum simulations etc.;• Development of many useful tools for QIP, including quantum protocols, algorithms, dynamic decoupling schemes, among others;
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?
Quantum Computation
Entanglement
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• In certain schemes of quantum computation where the quantum bits are affected by noise, there seems to be a speed-up over classical scenarios even in the presence of negligibly small or vanishing entanglement.
Knill, E.; Laflamme, R. Power of one bit of quantum information. Physical Review Letters, v. 81, n. 25, p. 5672, 1998.
Datta, A.; Shaji, A. and Caves. C. M. Physical Review Letters 100, p.050502, 2008.
Modi, K., Paterek, T., Son, W., Vedral, V. and Williamson M. Unified View of Quantum and Classical Correlations Physical Review Letters, v. 104, p.080501, 2010.
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• A possible explanation for the speed up would be quantum correlations different for entanglement.
How to detect them?
General Quantum Correlations
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Other types of correlations
?
Quantum Computation
Entanglement
Merali, Z. Nature, v. 474, p. 24, 2011.
Ollivier, H. & Zurek, W. H. Quantum discord: a measure of the quantumness of correlations.Phys. Rev. Lett. 88, 017901 (2001).
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Classification of Quantum and Classical States
All CorrelatedStates
Entangled States
Separable StatesC CQ
Separable
Entangled
ClassicallyCorrelated
A Bij i j
ij
p
A Bij i j
ij
p
, ortonormal basis.
ijij
i j
p i i j j
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Classification of Quantum and Classical Two-Qubit States
All CorrelatedStates
Entangled States
Separable StatesC CQ
Separable
Entangled
ClassicallyCorrelated
A Bij i j
ij
p
A Bij i j
ij
p
, ortonormal basis.
ijij
i j
p i i j j
• Bell diagonal states:3
1
1
4 j j jj
c
1
3 1 2
3 1 2
1 2 3
1 2 3
1 0 0
0 1 0
0 1 0
0 0 1
AB
c c c
c c c
c c c
c c c
1
2
3
0 0
0 0
0 0
c
C c
c
Correlation Matrix:
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Classification of Quantum and Classical States
All CorrelatedStates
Entangled States
Separable StatesC CQ
Separable
Entangled
ClassicallyCorrelated
A Bij i j
ij
p
A Bij i j
ij
p
, ortonormal basis.
ijij
i j
p i i j j
• Bell diagonal states:3
1
1
4 j j jj
c
1
44
3
1
11j
ji
jc
NMR sensitive part of the density matrix
In this sense NMR seems to be the perfect tool for probing quantum correlations of separable states and their interaction with the environment;
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– Entropic Discord*: disturbance made in a system when a measurement is applied.
*Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002.
Quantum Discord
S(ρA)
S(ρB)S(ρAB)
2( ) ( log )S Tr Von Neumann
Entropy
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– Entropic Discord*: disturbance made in a system when a measurement is applied.
• Mutual information: : :A B A B A BI S S S
*Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002.
Quantum Discord
S(ρA)
S(ρB)S(ρAB)
2( ) ( log )S Tr Von Neumann
Entropy
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– Entropic Discord*: disturbance made in a system when a measurement is applied.
• Mutual information:
• Classical Correlation:
: :A B A B A BI S S S
: Bj
Q A B A A BJ S S
*Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002.
Quantum Discord
S(ρA)
S(ρB)S(ρAB)
2( ) ( log )S Tr Von Neumann
Entropy
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– Entropic Discord*: disturbance made in a system when a measurement is applied.
• Mutual information:
• Classical Correlation:
• Quantum Discord:
: :A B A B A BI S S S
: Bj
Q A B A A BJ S S
: :maxBj
AB A B Q A BD I J
*Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002.
Quantum Discord
S(ρA)
S(ρB)S(ρAB)
2( ) ( log )S Tr Von Neumann
Entropy
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– For two-qubits Bell diagonal states*:
*Luo, S. Quantum discord for two-qubit systems. Physical Review A, v. 7, n. 4, p. 042303, 2008.
1 2 3max , ,c c c c
1 2 3 2 1 2 3
1 2 3 2 1 2 3
1 2 3 2 1 2 3
1 2 3 2 1 2 3
2 2
11 log 1
41 log 1
1 log 1
1 log 1
1 1log 1 log 1
2 2
AB AB ABD I C
c c c c c c
c c c c c c
c c c c c c
c c c c c c
c cc c
Quantum Discord
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Probing Quantum Correlations
What is required for probing discord and their degradation upon interaction with the environment?
• To prepare states with different amounts of QCs .
• To perform a reliable read-out of the final states.
• To have a good description and characterization of the system relaxation.
NMR has all that!!!!
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Diogo sets a partnership do study quantum discord by NMR
with Roberto Serra and Lucas Céleri ;
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• NMR system.
23 16
1QL Z zH I I I I
Sodium dodecyl sulfate in water forming a lyotropic liquid crystal – 23Na NMR
Anatoly K. Khitrin and B. M. Fung. The Journal of Chemical Physics, 112(16):6963–6965, 2000.
Neeraj Sinha, T. S. Mahesh, K. V. Ramanathan, and Anil Kumar. The Journal of Chemical Physics, 114(10):4415–4420, 2001.
3/2 spins systemSample: Lyotropic Liquid Crystals -Sodium Dodecyl Sulfate (SDS) - Heavy Water (D2O) - Decanol (C10H21OH)
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• Strong Modulated Pulase (SMP)*:
1
, ,k
SMP n n nn
U U t
*Fortunato, E.; Pravia, M.; Boulant, N.; Teklemariam, G.; Havel, T.; Cory, D. Design of modulating pulses to implement precise effective hamiltonians for quantum information processing. Journal of Chemical Physics, v. 116, n. 17, p. 7599, 2002. Nelder, J.A.; Mead, R. A simplex-method for function minimization. Computer Journal, v. 7, n. 4, p. 308, 1965.
22
arg
argarg
,,
SMPett
SMPettSMPett
TrTr
TrF
Tools for NMR QIP using quadrupolar Nuclei
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Single hard pulse
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11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
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Pure quadrupolar relaxation
Redfield Equations
+
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• Generalized Amplitude Damping Channel (GAD):
– Longitudinal relaxation (T1)
1 2
3 4
1 0 0,
0 1 0 0
0 01 01 ,
00 1
E p E p
E p E p
1
2 2L
B
pk T
Two Qubit System
AT
t
tCJA ee 12 11 2
BT
t
tCJB ee 11 11 2
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– Global phase damping channel (GPD);
0 1
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 01 ,
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
E E
01
12
CJ te
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
2 1 2 1
2 1 2 1
2 1 2 1
2 1 2 1
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
† †0 0 1 1E E E E
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|
randomX
time (ms)
MonotonicalDecay
Different amount of classical and correations in each state
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HOWEVER....
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• Decoherence Process in Bell-diagonal States:→ Local Phase Damping Channel:
*Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009.
3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c
Mutual InformationClassical CorrelationEntropic Discord
Time (s)
Sudden-Change Phenomena:
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• Decoherence Process in Bell-diagonal States:→ Phase Damping Channel:
*Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009.
3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c
Mutual InformationClassical CorrelationEntropic Discord
Time (s) Time (s)
Sudden-Change Phenomena:
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• Decoherence Process in Bell-diagonal States:→ Phase Damping Channel:
*Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009.
3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c
Mutual InformationClassical CorrelationEntropic Discord
Time (s) Time (s)Time (s)
Sudden-Change Phenomena:
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• Decoherence Process in Bell-diagonal States:→ Phase Damping Channel:
*Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009.
3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c
Mutual InformationClassical CorrelationEntropic Discord
Time (s) Time (s)Time (s)
Sudden-Change Phenomena:
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− 2 spins 1/2:
2A BL Z L Z Z zH I S JI S
• Two physical Qubits - NMR representation:
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• Generalized Amplitude Damping Channel:– Longitudinal relaxation (T1)
1 2
3 4
1 0 0,
0 1 0 0
0 01 01 ,
00 1
E p E p
E p E p
11
12
t Te 1
2 2L
B
pk T
Energy exchange between
system and environment
• Phase Damping Channel:- Transversal relaxation (T2):Loss of coherence
without loss of energy
1 2
1 0 1 0, 1
0 1 0 1E E
2211
2
tTe
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0 0 0
0 0 01
0 0 02
0 0 0
0 0
0 01
0 02
0 0
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3 1 20 0 e 0c c c
Mutual information
Classical correlation
Quantum correlation
3 10 0 ec c 2ou 0c
Mutual information
Classical correlation
Quantum correlation
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Hilbert-Schmidt distance between the state and the nearest classical state;
C
Sρ
E
D 2
22minG C
D
*Dakic, B.; Vedral, V.; Brukner, C. Necessary and sufficient condition for nonzero quantum discord. Physical Review Letters, v. 105, n. 19, p. 190502, 2010. Girolami, D.; Adesso, G. Observable measure of bipartite quantum correlations. Physical Review Letters, v. 108, n. 15, p. 150403, 2012. Modi, K. and et al. Unified view of quantum and classical correlations. Physical Review Letters, v. 104, n. 8, p. 080501, 2010.
Geometric Discord
Diogo sets a partneship with Gerardo Adesso
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3 3 3
1 1 , 1
1
4 i i i i ij i ji i i j
x y C
1 1 1 1
12 TrGD S k
• 2 q-bits:
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3 3 3
1 1 , 1
1
4 i i i i ij i ji i i j
x y C
1 1 1 1
12 TrGD S k
1
4t tS xx CC
• 2 q-bits:
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3 3 3
1 1 , 1
1
4 i i i i ij i ji i i j
x y C
1 1 1 1
12 TrGD S k 22
1
6Tr 2TrTrcos
3 3 3
S SSk
1
4t tS xx CC
• Para um sistema de 2 q-bits:
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3 3 3
1 1 , 1
1
4 i i i i ij i ji i i j
x y C
1 1 1 1
12 TrGD S k 22
1
6Tr 2TrTrcos
3 3 3
S SSk
1
4t tS xx CC
• 2 q-bits:
3 2 3322
2arccos 2Tr 9Tr Tr 9Tr
3Tr TrS S S S
S S
Tr 1 1
Tr 1 1
i i i
i i i
x
y
Tr 4ij i j i j i jc I I
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• Direct Measurement Method:3 3 3
1 1 1
11 1 1 1
4 i i i i ij i ji i i
x y c
Tr 1 1
Tr 1 1
i i i
i i i
x
y
NMR
Observables
Tr 4ij i j i j i jc I I Zero and Double
Quantum Coherences
and anti-phase magnetizations
Convert into a local measurement:
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• Direct Measurement Method:3 3 3
1 1 1
11 1 1 1
4 i i i i ij i ji i i
x y c
Tr 1 1
Tr 1 1
i i i
i i i
x
y
NMR
Observables
Tr 4ij i j i j i jc I I
Convert into a local measurement: Tr Tr 1i j i ij
†,, onde i jij ij ij ij A B ijU U U CNOT R
Zero and Double
Quantum Coherences
and anti-phase magnetizations
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Θ
j/i 1 2 3
1 0 3π/2 π/2
2 3π/2 π/2 - π/2
3 π/2 - π/2 π/2
1 2 3 x z y
†,, onde i jij ij ij ij A B ijU U U CNOT R
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– Negativity of Quantumness (QNA)*:
Minimum amount of entanglement created between the system
and its measurement apparatus in a local measurement;
Geometric measurement (trace norm);
J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013).
T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013).
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– Negativity of Quantumness (QNA)*:
Minimum amount of entanglement created between the system
and its measurement apparatus in a local measurement;
Geometric measurement (trace norm);
1,
1min 1
2A
N AB ijB Ai j
Q
• Bell diagonal states:
J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013).
T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013).
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– Negativity of Quantumness (QNA)*:
Minimum amount of entanglement created between the system
and its measurement apparatus in a local measurement;
Geometric measurement (trace norm);
1,
1min 1
2A
N AB ijB Ai j
Q
int
2A
N AB
cQ
3
1
1
4 j j jj
c
1
• Bell diagonal states:
J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013).
T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013).
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• Freezing phenomenon:– Initial state condition*:
– Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4)
*You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012.
0 3 1 2 0 3 1 2
0 1 2 3 0 1 3 2
e 0
ou e 0
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• Freezing phenomenon:– Initial state condition*:
– Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4)
*You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012.
0 3 1 2 0 3 1 2
0 1 2 3 0 1 3 2
e 0
ou e 0
1 1
2 2
3 3
0
0
0
t
t
c t e c
c t e c
c t c
2 2
2 2
A B
A B
T T
T T
Time (s)
2 D
G
DG
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1 1
2 2
3 3
1 1 0
1 1 0
1 0
a b
a b
a b a b
c t c
c t c
c t c
1 te 1T
• Generalized Amplitude Damping Channel:
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• 2 qubits system represented by 2 coupled spins ½:
– Sample: 100 mg of 13C-labeled CHCl3 dissolved
in 0.7 mL CDCl3
– Spectrometer:
– Initial State:
H – 500 MHz, T1 = 9 s, T2 = 1.2 s
C – 125 MHz, T1 = 25 s, T2 =
0.18 s
Acoplamento J – 215.1 Hz
Varian Premium Shielded – 11 T
1 2 30.5, 0.06, 0.24c c c 3 10 0 ec c 2ou 0c
Fidelity = 0.993
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• 2 qubits system represented by 2 coupled spins ½:
– Sample: 100 mg of 13C-labeled CHCl3 dissolved
in 0.7 mL CDCl3
– Spectrometer:
– Initial State:
H – 500 MHz, T1 = 9 s, T2 = 1.2 s
C – 125 MHz, T1 = 25 s, T2 =
0.18 s
Acoplamento J – 215.1 Hz
Varian Premium Shielded – 11 T
1 2 30.5, 0.06, 0.24c c c 3 10 0 ec c 2ou 0c
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– 1º State: 1 2 3 0.2c c c
Fidelity = 0.994
3 1 20 0 e 0c c c
– 2º State: 1 2 30.5, 0.06, 0.24c c c
Fidelity = 0.993
3 10 0 ec c 2ou 0c
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• Geometric Discord:
Time (s)
Time (s)
Time (s)
Time (s)
Direct Measurement Tomography Theoretical
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• Negativity of Quantumness:
Time (s) Time (s)
Time (s) Time (s)
(Theoretical)
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(Theoretical)
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(a) Discord (b) Geometric Discord (c) Trace Distance (d) Bures Distance
Preliminary Results
Aaronson, B.; Lo Franco, R.; Adesso, G. Physical Review A, v. 88, p. 012120, 2013.
Freezing Universality
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Relaxation Process
Phase Damping (PD)
Generalized Amplitude Damping (GAD)
Decoherence Channels:
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• Phase Damping Channel:
- Transversal relaxation (T2):Loss of coherence
without loss of energy
1 2
1 0 1 0, 1
0 1 0 1E E
2211
2
tTe
Two Qubit System
*Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010.
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• Phase Damping Channel:
- Transversal relaxation (T2):
- Global Phase Damping (spin 3/2 system)*:
Loss of coherence without loss of
energy
1 2
1 0 1 0, 1
0 1 0 1E E
2211
2
tTe
0 1
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 01 ,
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
E E
0
11
2CJ te
*Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010.
Two Qubit System
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• Phase Damping Channel:
- Transversal relaxation (T2):
- Global Phase Damping (spin 3/2 system)*:
Loss of coherence without loss of
energy
1 2
1 0 1 0, 1
0 1 0 1E E
2211
2
tTe
0 1
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 01 ,
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
E E
0
11
2CJ te
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
*Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010.
Two Qubit System
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• Phase Damping Channel:
- Transversal relaxation (T2):
- Global Phase Damping (spin 3/2 system)*:
Loss of coherence without loss of
energy
1 2
1 0 1 0, 1
0 1 0 1E E
2211
2
tTe
0 1
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 01 ,
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
E E
0
11
2CJ te
11 12 13 14
12 22 23 24
13 23 33 34
14 24 34 44
*Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010.
Two Qubit System
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Emergence of the Pointer Basis:
SE
A
Measurement
Decoherence
Collapse of A in some classical state which is not altered by decoherence!
Pointer Basis
The pointer basis emerges when
classical correlation between S and A
becomes constant!*
J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013).
Time (s)
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• Phase Damping Channel:
– 2 spins ½ system
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• Generalized Amplitude Damping Channel:
– 3/2 spins system– Sample: Lyotropic Liquid Crystals
• Sodium Dodecyl Sulfate (SDS)• Heavy Water (D2O)
• Decanol (C10H21OH)
– Spectrometer: Varian Inova – 8 T
Na – 92 MHzνQ = 10.4 kHz
A B1
1 2
1T 11.3 ms
2CJ
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• Differences between representing two qubit systems with two spins 1/2 coupled and one spin 3/2.
• Effects of phase damping and generalized amplitude damping channels.
• Experimental observation of Sudden-change, Freezing, Double Sudden-Change phenomena and the emergence of Pointer Basis.
Conclusion
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Acknowledgments