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Density Matrix Tomography,
Contextuality,
Future Spin Architectures
T. S. Mahesh
Indian Institute of Science Education and Research, Pune
1/2
1/2
Density Matrix Tomography (1-qubit)
=
~
Mx
My
P C = R+iS
-P+ e = ħ / kT ~ 10-5
Background
Does not leadto signal
Deviation
May leadto signal
P C = R+iS
-P
Density Matrix Tomography (1-qubit)
NMR detection operators: x , y
1. Heterodyne detection
x = 2R
y = -2S
2. Apply (/2)y
+ Heterodyne detection
x = 2P
=
~
Mx
My
(/2)y
- R P+iS
R1 =
P0
P1
P2
R1 R2 R3
R4 R5
R6+I1 I2 I3
I4 I5
I6+ 15 REAL NUMBERS
Density Matrix Tomography (2-qubit)
NMR detection operators: x1 , y
1 , x2 , y
2
P0
P1
P2
R1 R2 R3
R4 R5
R6+I1 I2 I3
I4 I5
I6+ 15 REAL NUMBERS
Traditional Method : Requires
1. Spin selective pulses
2. Integration of Transition
Spin 1 Spin 2
I I
90x I
I 90x
90y I
I 90y
90x 90x
90x 90y
90y 90x
90y 90y
Density Matrix Tomography (2-qubit)
Density Matrix Tomography (2-qubit)
P0
P1
P2
R1 R2 R3
R4 R5
R6+I1 I2 I3
I4 I5
I6+ 15 REAL NUMBERS
Traditional Method :
Spin 1 Spin 2
I I
90x I
I 90x
90y I
I 90y
90x 90x
90x 90y
90y 90x
90y 90y
Requires
1. Spin selective pulses
2. Integration of Transition
P0
P1
P2
R1 R2 R3
R4 R5
R6+I1 I2 I3
I4 I5
I6+ 15 REAL NUMBERS
NEWMethod Requires
1. No spin
selective pulses
2. Integration of
spins
Density Matrix Tomography (2-qubit)
JMR, 2010
Density Matrix Tomography (2-qubit)
SVD
tomo
Density Matrix Tomography of singlet state
Theory
Expt
Real Imag
Correlation = = 0.98tr(rth rexp)
[tr(rth 2 ) tr(rexp
2)]1/2 JMR, 2010
Quantum Contextuality
Non- Contextuality1. The result of the measurement of an
operator A depends solely on A and on the system being measured.
2. If operators A and B commute, the result of a measurement of their product AB is the product of the results of separate measurements of A and of B.
All classical systems are NON-CONTEXTUAL
Physics Letters A (1990), 151, 107-108
Measurement outcomes can be
assigned, in principle, even before
the measurement
Non- Contextuality
Quantum Contextuality
x2 x
1 x1x
2
z1 z
2 z1z
2
z1x
2 x1z
2 y1y
2
1
1
1
1 1 -1
Measurement outcomes can not be
pre-assigned even in principle
N. D. Mermin. PRL 65, 3373 (1990).
= 6
LHVT
QM
Eg. Two spin-1/2 particles
PRL 101,210401(2008)
Laflamme,PRL 2010
~ 5.3 LaflammePRL 2010
NMR demonstration of contextuality
Sample: Malonic acid single crystal
Peres Contextuality Let us consider a system of two spin half particles in singlet
state.
Singlet state:
Physics Letters A (1990), 151, 107-108
2
1001
Peres ContextualityFor a singlet state < σx
1 σx2 > = -1
< σy1 σy
2 > = -1
< (σx1 σy
2)(σy1 σx
2)> = -1
Note:[σx
1,σx2 ] = 0
[σy1,σy
2] = 0
[σx1 σy
2 , σy1 σx
2 ] = 0
Physics Letters A (1990), 151, 107-108
Peres ContextualityFor a singlet state Pre-assignment of eigenvalues < σx
1 σx2 > = -1 x1 x2 = -1
< σy1 σy
2 > = -1 y1 y2 = -1
< (σx1 σy
2)(σy1 σx
2)> = -1 x1 y2 y1 x2 = -1
CONTRADICTION !!Note:[σx
1,σx2 ] = 0
[σy1,σy
2] = 0
[σx1 σy
2 , σy1 σx
2 ] = 0
Physics Letters A (1990), 151, 107-108
ExperimentUsing three F spins of Iodotrifluoroethylene. Two were
used to prepare singlet and one was ancilla.
Pseudo-singlet statePure singlet state is hard to prepare in NMR
02
1001
8
1)-(1
Iz1+Iz
2+Iz3
0000008
1)-(1
Pseudo-singlet statePure singlet state is hard to prepare in NMR
02
1001
8
1)-(1
Iz1+Iz
2+Iz3
0000008
1)-(1
No Signal !!<σx
1+σx2>=
0
Pseudo-singlet state
8
1)-(1
Theory
Experiment
Real Part Imaginary Part
Fidelity=0.97
Moussa Protocol Target (ρ)
<AB>
Probe(ancilla)|+ <AB> Target (ρ) Physical Review Letters (2010), 104, 160501
A B
A B
NMR circuit for Moussa Protocol
PPS Single
t
1 (Ancilla)
2
3
B
|+
A
<σx>=<AB>
Results Manvendra Sharma, 2012
Future Architectures ?
Criteria for Physical Realization of QIP
1. Scalable physical system with mapping of qubits
2. A method to initialize the system
3. Big decoherence time to gate time
4. Sufficient control of the system via time-dependent Hamiltonians
(availability of a universal set of gates).
5. Efficient measurement of qubits
DiVincenzo, Phys. Rev. A 1998
NMR Circuits - Future
123456789
101112131415
.
.
.
Time
Qubits
xx - qubitsDecoherence
Transverserelaxation
a|00 + b |11
Loss of q. memory
{|00 , |11}
Longitudinalrelaxation
|0110010
|000000
Loss of c. memory
T2 T1<
• Addressability• Week coupling• Controllability
Larger Quantum
register
Liquid-state NMR systemsAdvantages
High resolution
Slow decoherence
Ease of control
Disadvantages
o Smaller resonance dispersion
o Small indirect (J) couplings
o Smaller quantum registerRandom, isotropic
tumbling
Single-crystal NMR systemsAdvantages
Large dipole-dipole couplings ( > 100 times J)
Orientation dependent Hamiltonian
Longer longitudinal relaxation time
Larger quantum register (???)
Disadvantages
o Shorter transverse relaxation time
o Challenging to control the spin dynamics
Single-crystal NMR systems Active spins in a bath of inactive molecules
• Large couplings
• High resolution
• Hopefully –
Larger quantum register
J. Baugh, PRA 2006
Two-molecules per unit center:
Inversion symmetry – P1 space group
So, the two molecules are magnetically equivalent
Inter-molecular interactions ?
Malonic Acid
QIP with Single Crystals
Cory et al, Phys. Rev. A 73, 022305 (2006)
Malonic Acid
QIP with Single Crystals
Cory et al, Phys. Rev. A 73, 022305 (2006)Natural Abundance
Pseudopure StatesMalonic Acid
Cory et al, Phys. Rev. A 73, 022305 (2006)
Pseudopure StatesMalonic Acid
Cory et al, Phys. Rev. A 73, 022305 (2006)
Quantum GatesEg. C2-NOT
Cory et al, Phys. Rev. A 73, 022305 (2006)
~ 5.3
R. Laflamme,PRL 2010
Glycine Single Crystal Mueller, JCP 2003
000 PPS
Floquet Register
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
More qubits
More coupled Nuclear Spins
More Resolved Transitions
Side-bands?
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
Solid-State NMR and next generation QIP
Pseudo-Pure States
13C spectra of aromatic carbons ofHexamethylbenzenespinning at 3.5 kHz
Grover’s Algorithm
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
Methyl 13C
Electron Spin vs Nuclear Spin
Spin e n
Magnetic moment 103 1
Sensitivity High Low
Coherence Time 1 103
Measurement
Processing
e-n Entanglement
Mehring, 2004
Entanglement in a solid-state spin ensemble•Stephanie Simmons et alNature 2011
Electron spin actuators
Cory et al
Detection of single Electron Spin
D. Rugar, R. Budakian, H. J. Mamin & B. W. ChuiNature 329, 430 (2004)
by Magnetic Resonance Force Microscopy
eq = ee IN
Up = SWAP (e,n1)
Ie 11 I(N-1)
Measure e-spin
If e invert
Up = SWAP (e,n2)
ee 11 I(N-1)
Cooling of nuclear spins
Cory et al, PRA 07
Nuclear Local Fieldsunder
Anisotropic Hyperfine Interaction
B0
Anisotropic Hyperfine Interaction
e-n system
Coherent oscillations between nuclear coherence on levels 1 & 2 driven by Microwave
The nuclear p pulse : 520 ns e-n CNOT gate : 2ms (0.98 Fidelity)
Anisotropic Hyperfine Interaction