Transcript Slide 1
Density Matrix Tomography, Contextuality, Future Spin Architectures T. S. Mahesh Indian Institute of Science Education and Research, Pune
Density Matrix Tomography (1-qubit)
=
1/2 1/2
+
P C = R+iS -P
Background Does not lead to signal Deviation May lead to signal = ħ / kT ~ 10 -5 M x
~
M y
Density Matrix Tomography (1-qubit)
=
P C = R+iS -P
( /2) y NMR detection operators: x , y 1. Heterodyne detection x = 2R y = -2S 1 =
- R P+iS R
2. Apply ( /2) y + Heterodyne detection x = 2P M x
~
M y
Density Matrix Tomography (2-qubit) P0 P1 P2
+
R1 R2 R3 R4 R5 R6
+
I1 I2 I4 I3 I5 I6
15 REAL NUMBERS NMR detection operators: x 1 , y 1 , x 2 , y 2
Density Matrix Tomography (2-qubit) P0 P1 P2
+
R1 R2 R4 R3 R5 R6
+
I1 I2 I4 I3 I5 I6
15 REAL NUMBERS Traditional Method :
Spin 1
I 90 x I 90 y I 90 x 90 x 90 y 90 y
Spin 2
I I 90 x I 90 y 90 x 90 y 90 x 90 y Requires 1. Spin selective pulses 2. Integration of Transition
Density Matrix Tomography (2-qubit) P0 P1 P2
+
R1 R2 R4 R3 R5 R6
+
I1 I2 I4 I3 I5 I6
15 REAL NUMBERS Traditional Method :
Spin 1
I 90 x I 90 y I 90 x 90 x 90 y 90 y
Spin 2
I I 90 x I 90 y 90 x 90 y 90 x 90 y Requires 1. Spin selective pulses 2. Integration of Transition
Density Matrix Tomography (2-qubit) P0 P1 P2
+
R1 R2 R4 R3 R5 R6
+
I1 I2 I4 I3 I5 I6
15 REAL NUMBERS
NEW Method
Requires 1. No spin selective pulses 2. Integration of spins JMR, 2010
Density Matrix Tomography (2-qubit)
tomo SVD
Density Matrix Tomography of singlet state Real Imag Theory Expt [tr( tr(
th
exp )
th 2 ) tr(
exp 2 )] 1/2
JMR, 2010
Quantum Contextuality
1.
Non- Contextuality
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
Non- Contextuality
Measurement outcomes can be assigned, in principle, even before the measurement
Quantum Contextuality
Measurement outcomes can not be pre-assigned even in principle Eg. Two spin-1/2 particles z 1 x 2 z 1 x 2 x 1 z x 1 2 z 2 z 1 z 2 x 1 x 2 y 1 y 2
1 1 1 1 1 -1
= 6 LHVT QM N. D. Mermin. PRL 65, 3373 (1990).
PRL 101,210401(2008)
Laflamme, PRL 2010
NMR demonstration of contextuality
Sample: Malonic acid single crystal ~ 5.3
Laflamme PRL 2010
Peres Contextuality
Let us consider a system of two spin half particles in singlet state.
Singlet state : 01 10 2 Physics Letters A (1990), 151, 107-108
Peres Contextuality
For a singlet state < σ x 1 σ x 2 > = -1 < (σ x 1 σ y 2 < σ y 1 σ y 2 )(σ y 1 > = -1 σ x 2 )> = -1 Note: [σ x 1 ,σ x 2 ] = 0 [σ y 1 ,σ y 2 ] = 0 [σ x 1 σ y 2 , σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151, 107-108
Peres Contextuality
For a singlet state Pre-assignment of eigenvalues < (σ x 1 < σ < σ x y 1 1 σ y 2 )(σ y 1 σ σ σ x y x 2 2 2 > = -1 > = -1 )> = -1 x x y 1 1 1 x y y 2 2 2 = -1 y = -1 1 x 2 = -1 CONTRADICTION !!
Note: [σ x 1 ,σ x 2 ] = 0 [σ y 1 ,σ y 2 ] = 0 [σ x 1 σ y 2 , σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151, 107-108
Experiment
Using three F spins of Iodotrifluoroethylene. Two were used to prepare singlet and one was ancilla.
Pseudo-singlet state
Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3 01 10 0 2 (1 ) 1 8 000 000 (1 ) 1 8
Pseudo-singlet state
Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3 01 10 0 2 (1 ) 1 8 000 000 No Signal !!
<σ x 1 +σ x 2 >=0 (1 ) 1 8
Pseudo-singlet state
(1 ) 1 8 Real Part Imaginary Part Theory Fidelity=0.97
Experiment
Moussa Protocol
Target (ρ) A B
NMR circuit for Moussa Protocol
<σ x >=< AB> 1 (Ancilla)
|+
2
P P S Singlet A B
3
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
xx - qubits
NMR Circuits - Future
Time Decoherence Qubits Larger Quantum register
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .
.
.
• • • Addressability Week coupling Controllability Transverse relaxation Loss of q. memory
|00
+
b
|11
{|00
, |11
}
T2
<
Longitudinal relaxation Loss of c. memory
|0110010
|000000
T1
Liquid-state NMR systems
Advantages High resolution Slow decoherence Ease of control Disadvantages o Smaller resonance dispersion o Small indirect (J) couplings o Smaller quantum register Random, isotropic tumbling
Single-crystal NMR systems
Advantages 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 J. Baugh, PRA 2006 • Large couplings • High resolution • Hopefully – Larger quantum register
Malonic Acid QIP with Single Crystals Cory et al, Phys. Rev. A 73, 022305 (2006) Two-molecules per unit center: Inversion symmetry – P1 space group So, the two molecules are magnetically equivalent Inter-molecular interactions ?
QIP with Single Crystals Malonic Acid Natural Abundance Cory et al, Phys. Rev. A 73, 022305 (2006)
Pseudopure States Malonic Acid Cory et al, Phys. Rev. A 73, 022305 (2006)
Pseudopure States Malonic Acid Cory et al, Phys. Rev. A 73, 022305 (2006)
Quantum Gates Eg. C 2 -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 More coupled Nuclear Spins More qubits More Resolved Transitions Side-bands?
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
Solid-State NMR and next generation QIP Pseudo-Pure States 13 C spectra of aromatic carbons of Hexamethylbenzene spinning at 3.5 kHz
Grover’s Algorithm Methyl 13 C S. Ding, C. A. McDowell, … M. Liu, quant-ph/0110014
Electron Spin vs Nuclear Spin
Spin e n Magnetic moment 10 3 1 Sensitivity High Low Coherence Time 1 10 3 Measurement Processing
e-n Entanglement Entanglement in a solid-state spin ensemble •
Stephanie Simmons et al
Nature 2011
Mehring, 2004
Electron spin actuators
Cory et al
Detection of single Electron Spin
by Magnetic Resonance Force Microscopy D. Rugar, R. Budakian, H. J. Mamin & B. W. Chui Nature 329, 430 (2004)
Cooling of nuclear spins
eq =
e
e
I
N U p = SWAP (e,n 1 ) I e
1
1
I
(N-1) Measure e-spin If
e
invert
e
e
1
1
I
(N-1) U p = SWAP (e,n 2 ) Cory et al, PRA 07
Anisotropic Hyperfine Interaction
Nuclear Local Fields under Anisotropic Hyperfine Interaction e-n system B 0
Anisotropic Hyperfine Interaction
Coherent oscillations between nuclear coherence on levels 1 & 2 driven by Microwave The nuclear pulse : 520 ns e-n CNOT gate : 2 m s (0.98 Fidelity)