Density Matrix Tomography, Contextuality, Future Spin Architectures T. S. Mahesh Indian Institute of...

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