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• Continuous variable QST (Quantum state tomography)• Discrete variable QST– 1 qubit– Higher dimension
• Process tomography• Detector tomography – State reconstruction
• Detector calibration – State reconstruction
Wigner function - reminder
( ) ò¥
¥-+-= e ipyyqyqdypqW !
!/2|ˆ|1, r
p
• Quasi-probabilistic distribution
• Can be negative (only quantum state)
• Contains all the information about the state (including photon statistics)
Wigner function - definition
( ) ò¥
¥-+-= e ipyyqyqdypqW !
!/2|ˆ|1, r
p
( ) }ˆ)(ˆ{ rxx DTrW =
( ) ( ) ( )xxa
xaax 2dWW eò¥
¥-
- **
=
• Wigner function
• Characteristic function
• Wigner function ??
Wigner function - definition
( ) ò¥
¥-+-= e ipyyqyqdypqW !
!/2|ˆ|1, r
p
( ) }ˆ)(ˆ{ rxx DTrW =
( ) ( ) ( )ir
iirir dWW e riir xxxxaa
xaxa
ò¥
¥-
--=
2,,
• Wigner function
• Characteristic function
• Wigner function
Homodyne detection – measuring the Rotated quadratures
( )abbaiddccnn dc ˆˆˆˆˆˆˆˆˆˆ ++++ -=-=-
( ) 2/ˆˆˆ pq
qqbb +
-+ =-= Xaai ee ii
aaX ee ii ˆˆˆ qqq
-+ +=
• Rotated quadratures
( ) ( ) ( ) ( ) ( )( )ò¥
¥-+-= dppxpxWxW qqqq
pq cossin,sincos1,
!
Rotated quadratures
aaX ee ii ˆˆˆ qqq
-+ +=
( ) ( ) ( ) ( ) ( )( )ò¥
¥-+-= dppxpxWxW qqqq
pq cossin,sincos1,
!
( ) ( ) ( )qhqhrhrhqh qq cos,sin}ˆˆ{)}ˆˆ{exp(, -=== WeiDTrXiTrW i
( ) ( ) hqhqhdWxW e ix
ò¥
¥-
-= ,,
reconstruction of Wigner function
( ) ( )qhqhqh cos,sin, -=WW
( ) ( ) ( )ir
iirir dWW e riir xxxxaa
xaxa
ò¥
¥-
--=
2,,
( ) ( ) ( )qhhqhaa
qhaqha ddWW e iriir ò ò
+-=
sincos2,,
( ) ( ) ( ) dxddxWW e xiir
ir qhhqaaqaqah
ò ò ò-+-
=sincos,,
Reconstruction of Wigner function –squeezed vacuum state
G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).
Reconstruction of Wigner function –displaced single-photon
Zavatta A, Viciani S, Bellini M. Phys. Rev. A. 23 (2005).
Reconstruction of Wigner function –Cat states
S Deléglise et al. Nature 455, 510-514 (2008) doi:10.1038/nature07288
|lñjq
>y|
|hñ
|rñ
|pñ
|vñ
|mñ
S3
S1
S2
Logical qubit Polarization qubit
Ψ = # 0 + & 1 Ψ = # ℎ + & )
* = 12 ℎ + )
, = 12 ℎ − )
. = 12 ℎ + / )
0 = 12 / ℎ + )
Discrete QST - Single qubit
|lñjq
>y|
|hñ
|rñ
|pñ
|vñ
|mñ
S3
S1
S2
State reconstruction
å=
=3
0
ˆˆi
iiS sr
mp IIS -=1
rl IIS -=2vh IIS +=0
vh IIS -=3
SPD
PBS
!/#
!/$
N%&'( =D2
Discrete QST - Single qubit
1 0 0 0 1/ 2 1/ 2 1/ 2 / 2ˆ ˆ ˆ ˆ, , , .0 0 0 1 1/ 2 1/ 2 / 2 1/ 2H V P R
ii
r r r r-æ ö æ ö æ ö æ ö
= = = =ç ÷ ç ÷ ç ÷ ç ÷è ø è ø è ø è ø
1/ 2 0ˆ .0 1/ 2TotalyMixedr
æ ö= ç ÷è ø
|Hñ |Vñ
|lñjq
>y|
|hñ
|rñ
|pñ
|vñ
|mñ
S3
S1
S2
State reconstruction
SPD
PBS
!/#
!/$
N%&'( =D2
Discrete QST - two qubits
1 2 1 21 2
3,, 0
1ˆ ˆ ˆ4 i i i ii i
S=
r = s Ä så
Discrete QST – four qubits
HHHHVHHHHVHHVVHH
HHVHVHVHHVVHVVVH
HHHVVHHVHVHVVVHV
HHVVVHVVHVVVVVVV
HHHHVHHH
HVHHVVHH
HHVHVHVH
HVVHVVVHHHHV
VHHVHVHV
VVHVHHVV
VHVVHVVV
VVVV
0
0.1
0.2
0.3
0.4
0.5
twelve qubits
“12-photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairsfrom parametric down-conversion”Han-Sen Zhong, Yuan Li, Wei Li, Li-Chao Peng, Zu-En Su, Yi Hu, Yu-Ming He, Xing Ding, W.-J. Zhang, Hao Li,L. Zhang, Z. Wang, L.-X. You, Xi-Lin Wang, Xiao Jiang, Li Li, Yu-Ao Chen, Nai-Le Liu, Chao-Yang Lu, Jian-Wei PanarXiv:1810.04823 (Submitted on 11 Oct 2018)
“…and detected by 24 superconducting nanowire single-photon detectors…”
“…the 12-photon coincidence isabout one per hour.”
Quantum process
• Isotropic depolarization
S3
1
S1
1
1S2
•Amplitude damping (T1)
•Dephasing (T2)
( )ˆ ˆ ˆ ˆ ˆ( ) (1 )3 x x y y z zppe r r s rs s rs s rs= - + + +
† †0 0 1 1ˆ ˆ ˆ( ) E E E Ee r r r= +
0 1
1 0 0,0 1 0 0
E E gg
æ ö æ ö= =ç ÷ ç ÷ç ÷ç ÷- è øè ø
ˆ ˆ ˆ( ) 1 z ze r a r a s rs= × + - ×S3
1
S11
1S2
S3
1
S1
1
1S2
Quantum process
S3
S1
S2
ˆ 'rr̂ ( )ˆ ˆ'r e r=
S3
S1
S2
e
• The c(4x4) matrix uniquely describes a qubit process:
.• Quantum process tomography of a channel requires several
quantum state tomography measurements of different states.
( ) †ˆ ˆˆ ˆij i jijE Ee r c r=å
• Mapping of any arbitrary to : .
A. Shaham and H. S. Eisenberg, Phys. Rev. A 83, 022303 (2011).
Detector tomography (QDT)
• k photons hit PNRD, what is the measured statistics?
• QST -> known measurements, finding state• QDT-> known states, finding measurement
}ˆ{)( MTrmp!
r=
åå ==P=k
kknk
k
nkn kkTrnp rqrqr )()( ˆ}ˆˆ{)(kk
k
nkn å=P )(ˆ q
QDT – On-off detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
åå -==k
knk
kkk
nk e
knp
2
!)(
2)()( aa
qrq
QDT – On-off detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
åå -==k
knk
kkk
nk e
knp
2
!)(),(
2)()( aa
qarqa
QDT – On-off detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
å=k
kknknp )(),( )( arqa
11 ´´´ = FFNNp rqMFFNMNp ´´´ = rq
QDT – time-multiplexed detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
QDT – time-multiplexed detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
QDT – time-multiplexed detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
MFFNMNp ´´´ = rq
QDT – time-multiplexed detector
J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
State reconstruction
11 ´´´ = FFNNp rq 111
´´-´ = FNNF p rq
J. Renema, et. al. "Tomography and state reconstruction with superconducting single-photon detectors." Phys. Rev. A 86 062113 (2012).
QDT and state reconstruction
ConsProsKnown states requiredNo assumptions on
detector:Inversion problems1. Uniformity
2. Unknown non-lineareffects
Array detector - model
11 ´´´ = FFNNp rq
111
´´-´ = FNNF p rq
detection efficiency,
crosstalk,
X
X
dark counts,
finite size,
100 elements
x~
h
d
N
÷÷÷÷÷÷
ø
ö
çççççç
è
æ
=
!
!"
nP
PP
P1
0
real
÷÷÷÷÷÷÷÷
ø
ö
çççççççç
è
æ
=
N
s
P
P
PP
P
!
!"1
0
det
FN´q
Array detector - model
)()1()1(1)1(
1~11~),,,~,(
00 00
0
2
nPmn
jjk
kN
ddkpkN
Npx
Npx
psp
nNxdP
nn
mnm
m
k
j
jkmm
N
k
pNkp
N
p
spps
s
N åå åå
å¥
=
-¥
= =
-
=
--
=
--
-÷÷ø
öççè
æ-÷÷
ø
öççè
æ÷÷ø
öççè
æ-÷÷
ø
öççè
æ--
÷÷ø
öççè
æ÷øö
çèæ --÷÷
ø
öççè
æ÷øö
çèæ -÷÷
ø
öççè
æ-
=
hh
h
LC, Y. Pilnyak, D. Istrati, N. M. Studer, J. P. Dowling, and H. S. Eisenberg, ”Absolute self-calibration of single-photon and multiplexed photon-number-resolving detectors,” PhysRevA.98.013811 (2018).
x~
h
dN
Array detector - model
)()1()1(1)1(
1~11~),,,~,(
00 00
0
2
nPmn
jjk
kN
ddkpkN
Npx
Npx
psp
nNxdP
nn
mnm
m
k
j
jkmm
N
k
pNkp
N
p
spps
s
N åå åå
å¥
=
-¥
= =
-
=
--
=
--
-÷÷ø
öççè
æ-÷÷
ø
öççè
æ÷÷ø
öççè
æ-÷÷
ø
öççè
æ--
÷÷ø
öççè
æ÷øö
çèæ --÷÷
ø
öççè
æ÷øö
çèæ -÷÷
ø
öççè
æ-
=
hh
h x~
h
dN
SPD calibration
hd
)()1()1(1);,(
)()1()1();,(
01
00
nPdndp
nPdndp
nn
n
nn
n
å
å¥
=
¥
=
---=
--=
hh
hh
SPD calibration
01
2210
11
1
ppnn
dp
-=
-+-
=hhd
tntndppO
--+
==1
2221
0
1det
hh
BBOType IISPD
PBSH
TMSV
V
NDF
Variable transmission t?=h
hd
SPD calibration - results
0.0 0.2 0.4 0.6 0.8 1.00
0.5
1
1.5
2x10-3
Ode
t
Transmission
SPD1 SMSV SPD2 SMSV SPD1 TMSV SPD2 TMSV
x10-3
LC, Y. Pilnyak, D. Istrati, N. M. Studer, J. P. Dowling, and H. S. Eisenberg, ”Absolute self-calibration of single-photon and multiplexed photon-number-resolving detectors,” PhysRevA.98.013811 (2018).
dtntnd
ppO
--+
==1
2221
0
1det
hh
SPD# SV light Our method Klyshko’s method1 SMSV 11.3±1.1% 11.8±0.9%2 SMSV 7.4±0.9% 8.1±0.9% 1 TMSV 17.4±1.0% 17.3±0.8%2 TMSV 12.7±0.9% 11.7±0.8%
Detector calibration - PNRD
å÷÷÷
ø
ö
ççç
è
æ-
-÷÷ø
öççè
æ÷øö
çèæ -÷÷
ø
öççè
æ÷øö
çèæ --÷÷
ø
öççè
æ-÷÷
ø
öççè
æ-=
=
---
s
p
p
Nn
psspnN
s de
Npx
Npx
psp
pN
edP0
2
11
1~1~1)1(
h
h
0 0.15 0.3 0.45 0.6 0.75 0.90
5
10
15
20
Cou
nts
(kH
z)
Voltage (Volt)0 0.15 0.3 0.45 0.6 0.75 0.9
0.1
1
10
100
1000
10000
Cou
nts
(Hz)
Voltage (Volt)
state reconstruction - PNRD
111
´´-´ = FNNF p rq
1.06.7003.0048.0~
104.1
11
1~1~1)1(
5
0
2
±=±=
´»
÷÷÷
ø
ö
ççç
è
æ-
-÷÷ø
öççè
æ÷øö
çèæ -÷÷
ø
öççè
æ÷øö
çèæ --÷÷
ø
öççè
æ-÷÷
ø
öççè
æ-=
-
=
--- å
nxd
de
Npx
Npx
psp
pN
edPs
p
p
Nn
psspnN
s
h
h
h
QDT vs. detector modelingdetector modelingQDT
No assumptions on detector:1. Uniformity
2. Unknown non-lineareffectsKnown states required
4 parameters from experiment
Many parameters from experiment
Inversion problemInversion problems
reconstructionreconstruction