Experimental quantum estimation using NMR

Diogo de Oliveira Soares Pinto

Instituto de Física de São CarlosUniversidade de São Paulo

(dosp@ifsc.usp.br)

NMR – QIP in RioNovember 2013

Operational significance of discord in quantum metrology:

Theory and Experiment*

*Title inspired in Nat.Phys. 8, 671 (2012)

Outline:

1) (Very brief) Introduction to quantum metrology

2) Results: Theory

3) Result: Experiment

4) Conclusions

(Very brief) Introduction to

quantum metrology

Entangled state?

In the lab...

Quantum state tomography = experimental data

*

1 2 3 4

max 0,

y y y yR

C

Eigenvalues of R ordered from the highest to the lowest

Entangled or not?

Estimation problem!

Data analysis

Simplest version of a typical quantum estimation problem:

→ Recover the phase introduced by the unitary operator

H is a known Hamiltonian that generates the phase .

i HU e

1) Prepare the N-probe system in a state

2) Apply the unitary transformation U to the state

3) Measure the final state = U U

5) Check the estimation accuracy through the Root Mean Square Error*:

Repeat these steps times to improve accuracy

Stepwise process:

2 2 221;

2H H H

H

4) From the data find the estimator

* C.W. Helstrom Quantum Detection and Estimation Theory (1976).

Two important limits for this “interferometric-measurement scheme” for phase estimation* ( 1, g the largest Hamiltonian gap):

* V. Giovannetti, S. Lloyd, L. Maccone, Nature Photonics 5, 222 (2011).

2

2

2

1 cos|

2

in

i

in

a b

a e b

p

g

N

N probes, repetitions.

N-entangled probes, repetitions.

2

2

2

1 cos|

2

N N

in

N NiN

in

a b

a e b

Np

g

N

Standard Quantum Limit (SQL) or “shot” noise limit

Heisenberg limit

In usual estimation problems, obey the Cramér-Rao bound:

2

1,

( )

|1( )

|

F

pF

p

where F() is the Fisher information.

In quantum estimation problems, this bound (quantum Cramér-Rao bound) is given by:

2

1,

( ; )

( ; ) , ,

1

2

j j jj

F H

F H L L l

L L

tr

Symmetric Logarithm Derivative (optimal measurement)

Is entanglement the only resource for enhanced estimation that Quantum Mechanics can give us?

Fortunately no! We also have...

Nature 474, 24-26 (2011).For a review see: K. Modi et al. Rev. Mod. Phys. 84, 1655 (2012).

Results:

Theory

1inf ;4 A

AAB AB A

HF H P

Let’s go back to the interferometric scheme. Suppose that the Hamiltonian HA that generate the phase over the partition A is given by

A AH n

and we don’t know a priori the direction ‘n’. Consequently the Hamiltonian itself is unknown for us (blind quantum metrology).

From the worst case scenario we can define a figure of merit for this interferometric scheme:

Interferometric Power of the input state AB

Guarantees the usefulness of the input state for quantum estimation and is a measure of discord! Discord as a resourse for quantum metrology! Details in ArXiv:1309.1472.

1inf ;4 A

AAB AB A

HF H P

• Invariant under local unitaries and nonincreasing under local operations on B;

• Vanishes iff AB is classically correlated;

• Reduces to an entanglement monotone for pure states;

• It is analytically computable if A is a qubit.

Characteristics of

Examples for two qubits (obs: idAB = 4x4 identity matrix):1) Werner states

1; 0 1 .

4WAB AB

ff f

Bell

ABid 22

1A

AB

f

f

P

2) Bell diagonal states

3

, 1

1

4BDAB ij iA jB

i j

C

ABid

2 2

22

2 2 2 21 2 32

2 2 2 21 2 3

2det;

1

,

max , , .

AAB

C C C

C

C c c c

C c c c

P

Details in ArXiv:1309.1472.

2

2

2

2

1 0 0 2

0 1 0 01

4 0 0 1 0

2 0 0 1

QAB

p p

p

p

p p

Suppose two families of states*:

2

2

2

2

1

11

4 1

1

CAB

p p p

p p p

p p p

p p p

with quantum discord.classically correlated.

2, 2 21(1 ) ;4

0 1.

Q CAB p

p

tr

*K. Modi et al. PRX 1, 021022 (2011).

2

24 4

4

2

,

log 1 log 1

log 1 ,

1 1 .

A QAB

A QAB

A QAB

p

p p p

p p

p

P

D

U

Results:

Experiment

What shall we measure? What shall we test experimentally?

†

,

two families of statesA

AB

i nA

AB A in A

U e

U U

First: interferometric scheme

1inf ;4 A

AAB AB A

HF H P

Second: check discord in the initial states

2; ,

,

1

2

trAB A AB

j j jj

F H L

L l

L L

Third: verify the metrological quantities

Compare and check if discord can be seen as a resourse for quantum metrology!

NMR system:

@ CBPF

2

2

2

2

1

11

4 1

1

CAB

p p p

p p p

p p p

p p p

2

2

2

2

1 0 0 2

0 1 0 01

4 0 0 1 0

2 0 0 1

QAB

p p

p

p

p p

Target:

Prepare

Start preparing:

After preparing state , we implement the circuits below to obtain the desired states. It is important to note that

0 0 0 0AB A B A B cos .p

CAB

QAB

0.5p

Fidelity above 99% for initial states!

How to implement unknown phase shift?

Setting the phase to be estimated as

4

We can choose three directions to rotate

,CAB

,QAB

Ok. But what is the (optimal) measurement?

We must measure in the eigenbasis of the symmetric logarithm derivative to obtain the maximum allowed precision.

,sAB

Since:

2 2

, , , ,

, , ,

,

,

, ,

, ; 1, 2,3.

;

tr

k s k s k ss kAB A j j

j

k s k s k s k sj j j

j

k s k s k ssj AB j j

L l

d

s C Q k

F H L l d

We can map the eigenvectors onto the computational basis of two qubits. Doing so, the ensemble expectation values can be directly observed in the diagonal elements of the density matrix.

But how?

,k sjd

The answer: Global rotation dependent on s and k!

Example for s = C, Q and k = 1:

This can be done also for s = C, Q and k = 2, 3. ArXiv:1309.1472.

From the experiment (ArXiv: 1309.1472):

A zAH 2

xA yAAH A xAH

Conclusions

Operational interpretation of quantum discord in terms of a resourse for quantum estimation problems when is considered the worst case scenario!

In settings like NMR, where disorder is high, quantum correlations even without entanglement can be a promising resourse for quantum technology.

Taking advantage of the name proposed for the protocol (blind quantum metrology), I can finish citing:

“Perhaps only in a world of the blind will things be what they truly are.” Saramago – Blindness.

or better:

“Perhaps only in a [quantum mixed] world of the blind will things be what they truly are.”

Fisher Information as a Measure of Quantum Discord.

People involved:

• Davide Girolami – NUS (Singapore)

• Vittorio Giovannetti – SNS (Italy)

• Tommaso Tufarelli – Imperial College (UK)

• Jefferson G. Filgueiras – TUD (Germany)

• Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira –

CBPF (Brazil)

• Me – IFSC/USP (Brazil)

• Gerardo Adesso – UoN (UK)

These guys are around here!

Thanks for the attention!