Quantum Metrology with highly entangled states and realistic decoherence
FQRNT
Bill CoishDepartment of Physics, McGill University, Montréal QC
Work done with Maxime Hardy (co-op Université de Sherbrooke)
2012 June 11; IQC Colloquium, Waterloo, ON
Collaboration with Innsbruck: R. Blatt, T. Monz, P. Schindler, J. T. Barreiro, ...
General Philosophy
Abstract models are excellent for fast progress, well defined questions...
BUT: too many to choose from.
Physical considerations often show the way to go.
True dynamics/decoherence often more complex than initial models suggest.
Goal: Quantum Information Processing
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
Physical Implementation:
U = T exp½¡iZ t
0
dt0H(t0)
¾H 2 HS
Goal: Quantum Information Processing
Reality: Imperfections
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
~U = T exp½¡iZ t
0
dt0 (H(t0) + ±H(t0))
¾±H 2 HS HE
Qubit encoding: Single ions (40Ca+)
V(r)
r
¢E » 10eV
» 10¡10m
encoding:
R. Blatt and D. Wineland, Nature (2008)
jsi ! j0ijdi ! j1i
Collective dephasing
jÃ(0)i = 1p2(j0000:::i+ j1111:::i) (GHZ)
N qubits
N = 1
N = 2
N = 3N = 4
N = 6
deco
here
n ce
rate
1/T
2
Number of qubits, N
Uncorrelated noise
1
T2/ N
Expectation
deco
here
n ce
rate
1/T
2
Number of qubits, N
Correlated noise: “Superdecoherence”
1
T2/ N2
G. Palma et al., Proc. Roy. Soc. Lond. A (1996)
F (t) =DjhÃ(0)jÃ(t)ij2
Eav:
T. Monz et al., PRL (2011)
Collective dephasing
jÃ(0)i = 1p2(j0000:::i+ j1111:::i) (GHZ)
N qubits
N = 1
N = 2
N = 3N = 4
N = 6
deco
here
n ce
rate
1/T
2
Number of qubits, N
1
T2/ N
1
T2/ N2
Fits
something in between?
F (t) =DjhÃ(0)jÃ(t)ij2
Eav:
T. Monz et al., PRL (2011)
Dephasing: General
!(t)
j1i
j0i
_½ = ¡i [H(t); ½]H(t) = !(t)¾z=2
h¾+(t)i = eiÁ(t) h¾+(0)i Á(t) =
Z t
0
dt0!(t0)
!(t)
tprepare measure tprepare measure
Average over noise realizations:
h¾+(t)iav: =DeiÁ(t)
Eav:
h¾+(0)i
Dephasing: Generalh¾+(t)iav: =
DeiÁ(t)
Eav:
h¾+(0)i = e¡12 hÁ2(t)iav: h¾+(0)i
h±!(t)±!(0)iav:t
¿c
(Gaussian, stationary)Á2(t)
®av:=
Z t
0
dt0(t¡ t0) h±!(t0)±!(0)iav:
Reh¾+(t)i a
v:
¿c < ¿dec:
¿c ¿dec:
» e¡t=¿dec:
(Markovian)
Reh¾+(t)iav: ¿c > ¿dec:
¿dec: ¿c
» e¡(t=¿dec:)2
(Non-Markovian)
Sources of dephasing in ion traps
●Fluctuating global phase reference (laser stability, also slow)
●Global magnetic field fluctuations (slow)
s= j0i = j1i
d(orbital Zeeman)
AMO Physics: Usually assume fast, local noise.
Gaussian dephasing model:
Szk = (j0i h0jk ¡ j1i h1jk) =2
jÃ(0)i = 1p2(j0000:::i+ j1111:::i)
h±B(t)±B(0)i =±B2
®e¡t=¿c
(GHZ)
N qubits
H(t) = ±B(t)X
k
Szk
Gaussian dephasing model:
Szk = (j0i h0jk ¡ j1i h1jk) =2
jÃ(0)i = 1p2(j0000:::i+ j1111:::i)
h±B(t)±B(0)i =±B2
®e¡t=¿c
(GHZ)
N qubits
²(N; t) =N2
2
Z t
0
d¿(t ¡ ¿) h±B(t)±B(0)i
“Superdecoherence”
H(t) = ±B(t)X
k
Szk
F (t) =j hÃ(0)j Ã(t)i j2
®av:=1
2(1 + exp [¡2²(N; t)]) ' 1¡ ²(N; t)
N = 1
N = 2
N = 6
N = 3
N = 4
Revised noise model, accounting for a finite correlation time (non-Markovian)
Dominant noise source (B-field) identified; N extended to 14 qubits!
² » N2
T. Monz, ... WAC, ..., R. Blatt, PRL (2011)
Quantum MetrologyFrequency standards
Fundamental tests of gravitation
Mueller, Peters, Chu, Nature (2010)
Parameter estimation for Qm. Inf. Proc.
e.g., Rafal Demkowicz-Dobrzanski et al., arXiv (2012)
Quantum Metrology
Precision measurements?
j0i H HUÁ0 1
Á = !t
Repeat N times...
P = (1 + cos!t)=2j1i+ ei!t j0i
T: total experiment time
Classical:
±!class: =
pP (1¡ P )=N
jdP=d!j =1pNTt
/ 1pN
Quantum (GHZ state): j1111:::i+ eiN!t j0000:::i
±!quant: =1
NpT t
/ 1
N
Problem! DecoherenceMarkovian (exponential) dephasing, spatially uncorrelated noise:
S. F. Huelga et al., PRL (1997)
¿dec: = 1=(°N)
P = (1 + cosN!t)=2! (1 + e¡N°t cosN!t)=2
±!opt: / 1pNT
/ ±!class:
For (Markovian) spatially correlated noise:
¿dec: / 1=N2 ) ±!quant: / const:
Even worse!
U. Dorner, New J. Phys. (2012)
With dephasing (in general):
±!opt: / 1
Np¿dec:T
What kind of decoherence?
T. Monz, ..., WAC,..., R. Blatt PRL (2011)
non-exponential (long correlation time)
¿c > ¿dec:
“Superdecoherence”(long correlation length)
²(N)=²(1) / N2F = 1¡ ²
»c > L » N
Large N: dephasing becomes local
H(t) =X
k
±hk(t)Szk
L
»c
Generalized model:
Gaussian fluctuations:
Space Time
But: L / N (incr. with N) ¿dec: decr. with N!
Features of the environment(independent of N!)
h±hk(t)±hl(0)i =±h2(0)
®e¡jxk¡xlj=»c £ e¡t=¿c
Sz =1
2(j1i h1j ¡ j0i h0j)
Large N: Quantum advantage?
±!opt: / 1
Np¿dec:T
¿dec: /1pN
) ±! / 1
N3=4
Also see, e.g., Jones et al., Science (2009); Matsuzaki, Benjamin, Fitzsimons PRA (2011)
¿c > ¿dec:; »c < L :
Large N: Quantum advantage?
¿dec: = T1=N
I : ¿c < ¿dec:; L (» N) < »c
II : ¿c > ¿dec:; L (» N) < »c
OR ¿c < ¿dec:; L (» N) > »c
III : ¿c > ¿dec:; L (» N) > »c
If the qubit frequency fluctuates locally in space and is approximately constant in time, quantum wins.
local, non-Markovian
Model summary: Spatial and temporal correlations
¿c < ¿dec:
¿c ¿dec:
» e¡t=¿dec:
¿c > ¿dec:
¿dec: ¿c
» e¡(t=¿dec:)2
»c ¿ L
¿dec: / N¡1; ±! / N¡1=2
»c À L
¿dec: / N¡2; ±! = const:
»c À L
¿dec: / N¡1; ±! / N¡1=2
F (t) F (t)
»c ¿ L
¿dec: / N¡1=2; ±! / N¡3=4
See also: Matsuzaki, Benjamin, Fitzsimons, PRA (2011)
Scaling: Regimes
t t t t
»c » n0 = 7
¿c = 1
±!0 = 10¡4
Can we do better?
Measurement time long compared to correlation time!
1
N3=4
Jones et al., Science (2009)Matsuzaki et al., PRA (2011)
!(t)
tprepare measure tprepare measure ¿c
Parameter Estimation = “Instantaneous Measurement”
After many measurements, frequency still not precisely defined!
......!
±! = 1=pNtT !k = !0 + k¼=t
T ¿ ¿c“Instantaneous”: ½(!)
½(!) = PN+(1¡ P )N¡N+=N0
More realistic: Gaussian prior
!¾
½(!)½(!) = PN+(1¡ P )N¡N+½0(!)=N0
½0(!)
P = (1 + cos!t)=2
Measurement times t?
Advantage in performing measurements at short times, even if the standard formula suggests larger t is always better:
±! » 1ptT
Classical:
Quantum:
¢! =¼
t> ¾
¢! =¼
ntPeak spacing smaller: ¢! > ¾ ) t < 1=n¾
±!class: »1pNtT
»r
¾
NT
±!quant: »r
¾
NTThis protocol gives the same scaling! (not optimal?):
Improved measurement strategy
tMeasurement time
Measurement time t=2
Measurement time t=4
Measurement time overhead:
t0 =X
k
1
2kt = 2t
±!quant: =
p2
NptT
(GHZ)
Result:
T ¿ ¿c
±!class: =
p2p
NptT
Summary
A static random frequency can always be found with Heisenberg-limited precision using GHZ states, provided the prior distribution has finite width.
T ¿ ¿c
±!quant: »1
NpT t
This beats the ~1/N3/4 scaling found previously [Jones et al., Science (2009), Matsuzaki et al., PRA (2011)], even for Gaussian decay of P(t).
t ¿ ¿c ¿ T ?
In this regime, frequency drifts between measurements; problem still open?
Conclusions
●“Superdecoherence”: a short-term problem for ion-trap and other implementations.
●Quantum-enhanced precision measurements still possible in spite of dephasing.
How large can the quantum region be?
Physical dephasing mechanisms:
Charge traps (fluctating electric field)
Surface spins (magnetic field)
Power-law correlations in space/time
Open questions: