Experimental quantum estimation using NMR

Report
Operational significance of discord in
Experimental
quantum
estimation
quantum metrology:
using NMR
Theory and Experiment*
Diogo de Oliveira Soares Pinto
Instituto de Física de São Carlos
Universidade de São Paulo
([email protected])
NMR – QIP in Rio
November 2013
*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
In the lab...

Entangled state?
Quantum state tomography = experimental data
R    y  
Data analysis
 
1 

y 
2 
*

y
3 

4
C  m ax  0,  
Entangled or not?
Estimation problem!
y

Eigenvalues of R
ordered from the
highest to the lowest
Simplest version of a typical quantum estimation problem:
→ Recover the phase
introduced by the unitary operator U   e  i  H
H is a known Hamiltonian that generates the phase
.
Stepwise process:
1) Prepare the N-probe system in a state
Repeat these
steps  times
to improve
accuracy
2) Apply the unitary transformation U to the state
3) Measure the final state
=U
U
4) From the data find the estimator 
5) Check the estimation accuracy through the Root Mean Square
Error*:
 

 
2

1
2  H
;  H

2
 H
* C.W. Helstrom Quantum Detection and Estimation Theory (1976).
2
 H
2
Two important limits for this “interferometric-measurement scheme”
for phase estimation* ( 1, g the largest Hamiltonian gap):
a  b
 in 
2
a e
 
i
2
p   in |  
N probes,  repetitions.
 in 
 
a
 
2
N
a
1  cos  

Standard Quantum
Limit (SQL) or “shot”
noise limit
N
 b
N
2
N
e
iN 
b
N
2
p   in |  
N-entangled probes, 
repetitions.
2
g
 
b
g
N
* V. Giovannetti, S. Lloyd, L. Maccone, Nature Photonics 5, 222 (2011).
2

1  cos N  
2
Heisenberg limit
In usual estimation problems,
 
obey the Cramér-Rao bound:
1
F ( )

F ( ) 

,
  p   |   
1


p   |   


2
where F( ) is the Fisher information.
In quantum estimation problems, this bound (quantum Cramér-Rao
bound) is given by:
1
 
 F ( ; H )

Symmetric Logarithm Derivative
(optimal measurement)
,

F (  ; H )  tr   L , L 
2
l
j



1
2


L  L  

j
j
j ,
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
Let’s go back to the interferometric scheme. Suppose that the
Hamiltonian HA that generate the phase
over the partition A is
given by
H A  n 
A
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:
P
A
  AB  
1
4
inf F   A B ; H A 
HA
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.
• Invariant under local unitaries and
nonincreasing under local operations on
Characteristics of
B;
• Vanishes iff
is classically
AB
1
A
correlated;
P   A B   inf F   A B ; H A 
• Reduces to an entanglement monotone
4 H
for pure states;
• It is analytically computable if A is a
qubit.
Examples for two qubits (obs: idAB = 4x4 identity matrix):
1) Werner states
A

W
AB
 f 
B e ll

AB

1 f
4
id A B ;  0  f  1  .
P
A
  AB  
2f
2
1 f
2) Bell diagonal states

BD
AB
1
  id A B 
4
3
C
ij
i , j 1
 iA  
jB



P
A
C
Details in ArXiv:1309.1472.
C
  AB  
2
2
2

2
C
2
 C
2

1 C
 c1  c 2  c 3 ,
2
2

2

 m ax c1 , c 2 , c 3 .
2
2
2
 2 det C
2

;
Suppose two families of states*:

Q
AB
1  p 2

1 0

4 0

 2p
0
1 p
0
2
0
1 p
0
0
0
2
2p 

0 
0 

2 
1 p 
D
A
A
   p ,
    log 1  p   p log 1  p 
Q
AB
A
Q
AB
2
4
   1
Q
AB
p
p
2
2
p
1
p
p
p
1
p
p
p
2
p 

p 
2
p 

1 
2
4
 p log 4  1  p  ,
U
1
classically correlated.
with quantum discord.
P

C
AB


1
 
4



tr   A B

Q ,C

2
  1 (1  p 2 ) 2 ;
 4
0  p  1.
1 p .
2
*K.
Modi et al. PRX 1, 021022 (2011).
Results:
Experiment
What shall we measure? What shall we test experimentally?
First: interferometric scheme
 A B  tw o fa m ilie s o f sta te s

UA e

 i  n 
A
,

†
 A B  U A  in U A
Second: check discord in the
initial states
P
A
  AB  
1
4


inf F  A B ; H A
HA

Third: verify the metrological
quantities



j
j ,



F  A B ; H A  tr  A B L ,
L 
l
j
2
j



1
2


L  L  

Compare and check if
discord can be seen as
a resourse for quantum
metrology!
NMR system:

Q
AB
Target: Prepare

@ CBPF
1  p 2

1 0

4 0

 2p
C
AB


1

4


1
p
2
0
0
1 p
2
0
1 p
0
0
p
2
0
p
1
p
p
p
1
p
p
p
Start preparing:
2
p 

p 
2
p 

1 
2
2p 

0 
0 

2
1  p 
After preparing state  A B  0 A 0 B 0 A 0 B , we implement the circuits below
to obtain the desired states. It is important to note that p  cos  .
 AB
C
p  0.5
 AB
Q
Fidelity above 99% for initial states!
How to implement unknown phase shift?
Setting the
estimated as
 ,C
 AB
 
phase
to
be

4
We can choose three directions
to rotate
 ,Q
 AB
Ok. But what is the (optimal) measurement?
We must measure  A B, s in the eigenbasis of the symmetric logarithm
derivative to obtain the maximum allowed precision.
Since:
 k ,s 
L


 k ,s 
lj
  k ,s 
  k ,s 
j
j
j

F 
s
AB
;H
k
A
  L
 k ,s 


2


 k ,s 
lj

2
 k ,s 
dj
j
 k ,s 
dj

 ,s
  k ,s 
 tr  A B  j
  k ,s 
j

s  C , Q ; k  1, 2, 3.
We can map the eigenvectors onto the
computational basis of two qubits.
Doing so, the ensemble expectation
values d   can be directly observed in
the diagonal elements of the density
matrix.
k ,s
j
But how?
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):
H A   zA
HA 
 xA   yA
2
H A   xA
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.
These guys are around here!
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)
Thanks for the attention!

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