density matrix tomography, contextuality , future spin architectures
DESCRIPTION
Density Matrix Tomography, Contextuality , Future Spin Architectures. T. S. Mahesh Indian Institute of Science Education and Research, Pune. ~. Density Matrix Tomography (1-qubit) . + . = ħ / kT ~ 10 -5. = . Background Does not lead to signal. Deviation May lead - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/1.jpg)
Density Matrix Tomography,
Contextuality,
Future Spin Architectures
T. S. Mahesh
Indian Institute of Science Education and Research, Pune
![Page 2: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/2.jpg)
1/2
1/2
Density Matrix Tomography (1-qubit)
=
~
Mx
My
P C = R+iS
-P+ = ħ / kT ~ 10-5
Background
Does not leadto signal
Deviation
May leadto signal
![Page 3: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/3.jpg)
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 =
![Page 4: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/4.jpg)
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
![Page 5: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/5.jpg)
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)
![Page 6: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/6.jpg)
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
![Page 7: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/7.jpg)
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
![Page 8: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/8.jpg)
Density Matrix Tomography (2-qubit)
SVD
tomo
![Page 9: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/9.jpg)
Density Matrix Tomography of singlet state
Theory
Expt
Real Imag
Correlation = = 0.98tr(th exp)
[tr(th 2 ) tr(exp
2)]1/2 JMR, 2010
![Page 10: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/10.jpg)
Quantum Contextuality
![Page 11: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/11.jpg)
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
![Page 12: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/12.jpg)
Measurement outcomes can be
assigned, in principle, even before
the measurement
Non- Contextuality
![Page 13: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/13.jpg)
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)
![Page 14: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/14.jpg)
Laflamme,PRL 2010
![Page 15: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/15.jpg)
~ 5.3 LaflammePRL 2010
NMR demonstration of contextuality
Sample: Malonic acid single crystal
![Page 16: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/16.jpg)
Peres Contextuality Let us consider a system of two spin half particles in singlet
state.
Singlet state:
Physics Letters A (1990), 151, 107-108
21001
![Page 17: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/17.jpg)
Peres ContextualityFor a singlet state < σx
1 σx2 > = -1
< σy1 σy
2 > = -1 < (σx
1 σy2)(σy
1 σx2)> = -1
Note:[σx
1,σx2 ] = 0
[σy1,σy
2] = 0[σx
1 σy2 , σy
1 σx2 ] = 0
Physics Letters A (1990), 151, 107-108
![Page 18: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/18.jpg)
Peres ContextualityFor a singlet state Pre-assignment of eigenvalues < σx
1 σx2 > = -1 x1 x2 = -1
< σy1 σy
2 > = -1 y1 y2 = -1 < (σx
1 σy2)(σy
1 σx2)> = -1 x1 y2 y1 x2 = -1
CONTRADICTION !!Note:[σx
1,σx2 ] = 0
[σy1,σy
2] = 0[σx
1 σy2 , σy
1 σx2 ] = 0
Physics Letters A (1990), 151, 107-108
![Page 19: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/19.jpg)
ExperimentUsing three F spins of Iodotrifluoroethylene. Two were
used to prepare singlet and one was ancilla.
![Page 20: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/20.jpg)
Pseudo-singlet statePure singlet state is hard to prepare in NMR
02
1001
81)-(1
Iz1+Iz
2+Iz3
00000081)-(1
![Page 21: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/21.jpg)
Pseudo-singlet statePure singlet state is hard to prepare in NMR
02
1001
81)-(1
Iz1+Iz
2+Iz3
00000081)-(1
No Signal !!<σx
1+σx2>=
0
![Page 22: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/22.jpg)
Pseudo-singlet state
81)-(1
Theory
Experiment
Real Part Imaginary Part
Fidelity=0.97
![Page 23: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/23.jpg)
Moussa Protocol Target (ρ) <AB>
Probe(ancilla)|+ <AB> Target (ρ) Physical Review Letters (2010), 104, 160501
A B
A B
![Page 24: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/24.jpg)
NMR circuit for Moussa Protocol
PPS Single
t
1 (Ancilla)
2
3B
|+
A
<σx>=<AB>
![Page 25: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/25.jpg)
Results Manvendra Sharma, 2012
![Page 26: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/26.jpg)
Future Architectures ?
![Page 27: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/27.jpg)
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
![Page 28: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/28.jpg)
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
![Page 29: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/29.jpg)
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
![Page 30: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/30.jpg)
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
![Page 31: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/31.jpg)
Single-crystal NMR systems Active spins in a bath of inactive molecules
• Large couplings
• High resolution
• Hopefully –
Larger quantum register
J. Baugh, PRA 2006
![Page 32: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/32.jpg)
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 CrystalsCory et al, Phys. Rev. A 73, 022305 (2006)
![Page 33: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/33.jpg)
Malonic Acid
QIP with Single Crystals
Cory et al, Phys. Rev. A 73, 022305 (2006)Natural Abundance
![Page 34: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/34.jpg)
Pseudopure StatesMalonic Acid
Cory et al, Phys. Rev. A 73, 022305 (2006)
![Page 35: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/35.jpg)
Pseudopure StatesMalonic Acid
Cory et al, Phys. Rev. A 73, 022305 (2006)
![Page 36: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/36.jpg)
Quantum GatesEg. C2-NOT
Cory et al, Phys. Rev. A 73, 022305 (2006)
![Page 37: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/37.jpg)
~ 5.3
R. Laflamme,PRL 2010
![Page 38: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/38.jpg)
Glycine Single Crystal Mueller, JCP 2003
000 PPS
![Page 39: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/39.jpg)
Floquet Register
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
More qubits
More coupled Nuclear Spins
More Resolved Transitions
Side-bands?
![Page 40: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/40.jpg)
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
![Page 41: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/41.jpg)
![Page 42: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/42.jpg)
Solid-State NMR and next generation QIP
Pseudo-Pure States
13C spectra of aromatic carbons ofHexamethylbenzenespinning at 3.5 kHz
![Page 43: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/43.jpg)
Grover’s Algorithm
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
Methyl 13C
![Page 44: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/44.jpg)
Electron Spin vs Nuclear Spin
Spin e n
Magnetic moment 103 1
Sensitivity High Low
Coherence Time 1 103
Measurement
Processing
![Page 45: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/45.jpg)
e-n Entanglement
Mehring, 2004
Entanglement in a solid-state spin ensemble•Stephanie Simmons et alNature 2011
![Page 46: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/46.jpg)
Electron spin actuators
Cory et al
![Page 47: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/47.jpg)
Detection of single Electron Spin
D. Rugar, R. Budakian, H. J. Mamin & B. W. ChuiNature 329, 430 (2004)
by Magnetic Resonance Force Microscopy
![Page 48: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/48.jpg)
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
![Page 49: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/49.jpg)
Nuclear Local Fieldsunder
Anisotropic Hyperfine Interaction
B0
Anisotropic Hyperfine Interaction
e-n system
![Page 50: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/50.jpg)
Coherent oscillations between nuclear coherence on levels 1 & 2 driven by Microwave
The nuclear pulse : 520 ns e-n CNOT gate : 2ms (0.98 Fidelity)
Anisotropic Hyperfine Interaction
![Page 51: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/51.jpg)
![Page 52: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/52.jpg)
![Page 53: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/53.jpg)
![Page 54: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/54.jpg)
![Page 55: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/55.jpg)
![Page 56: Density Matrix Tomography, Contextuality , Future Spin Architectures](https://reader035.vdocuments.net/reader035/viewer/2022062814/56816717550346895ddb8695/html5/thumbnails/56.jpg)