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