Upload
others
View
16
Download
0
Embed Size (px)
Citation preview
From Hudson’s theoremto generalized uncertainty relations
E. Karpov, A. Mandilara, and N. Cerf
Quantum Information and Communication, Université Libre de Bruxelles
Mathematical Modeling and Computational Physics 2009Dubna, Moscow region, Russia
July 9, 2009
MMOTIVATIONOTIVATION : : QUANTUM INFORMATIONQUANTUM INFORMATION
Using « CONTINUOUS VARIABLESCONTINUOUS VARIABLES » instead of qubits
… quantum superposition and entanglement
► Quantum information has been developed originally with qubits (cryptography, coding, teleportation, algorithms, etc.)
► BUT several advantages of CV quantum information carriers !
Bit Qubit
►discrete degree of freedom (polarization of single photon)►continuous degrees of freedom (amplitude of quadratures)
E = X cos(ω t) + P sin(ω t)Quantum optics: X and P are non-commuting observables
Advantages :
● use standard telecommunication techniques● no need for single-photon sources/detectors● high rates & efficiencies with homodyne detection
Successful achievements with Gaussian statesGaussian states :
● CV quantum teleportation (theory 1997, experiment 1998)
● CV entanglement of distant atomic ensembles● CV quantum key distribution● CV quantum memory of light
… etc …
QQUANTUM UANTUM CV CV IINFORMATION NFORMATION PPROCESSESROCESSES
… first experimental ”Schrödinger cat” states of lightWigner function
GGAUSSIAN AUSSIAN SSTATES TATES OOFTEN FTEN DDO O NNOT OT SSUFFICEUFFICE
P. Grangier et al., Science (2006)
■ No-Go theorem for Gaussian entanglement purification■ No violation of Bell inequality with positive Wigner functions■ No gain of quantum computation with Gaussian states/operations
( ) ( ) dyeyxyxpxW ipy h
h/2,1, ∫
∞
∞−−+≡ ρ
π
… joint quasi-probabilitydistribution of (x,p)
What about mixed states with W(x,p)>0 ???
A pure state has a non-negative Wigner function iff it is a Gaussian state
Not everywhere positive W(x,p) ↔ « non-classicality »↔ non-Gaussian states,
e.g. Schrödinger cats
•Gaussian states (e.g. thermal states)•Convex mixtures of Gaussian states•States with positive Wigner function•All states
T. Bröcker, R. F. Werner (1995)
Fixed purity space
HHUDSON UDSON –– PPIQUET IQUET TTHEOREM (1974)HEOREM (1974)
∑ ≥=i
iGi pxWwpxW 0),(),(non-Gaussian 0,1 ≥=∑ ii i wwwith
0≥
Take a reference reference Gaussian state determined by the covariance matrix γ and displacement vector d. Its purity is
Consider all all states with the same γ
purity
… that have a positive Wigner function
Purity
Purity
line of states with same covariance matrix γand displacement vector d
[ ] ( )2GG Tr ρρµ =
How much « non-Gaussian » can be a mixed state with positive Wigner functiongiven its purity and the purity of the corresponding Gaussian state ?How much « non-Gaussian » can be a mixed state with positive Wigner functiongiven its purity and the purity of the corresponding Gaussian state ?[ ]Gρµ[ ]ρµ
[ ] ( )
[ ] ( )2
2
ρρµ
ρρµ
Tr
Tr GG
=
=
[ ] ( ) [ ]GTr ρµρρµ ≠= 2
Gρ
ρ
pxpxW ,0),( ∀≥ρ
2 (distinct) parameters
( )( ) ][2/2 ρµρρδ GTr −=nonnon--GaussianityGaussianity
Bound on nonBound on non--Gaussianity for mixed states Gaussianity for mixed states with positive Wigner function with positive Wigner function
Method: Lagrange multipliers – minimization of Tr(ρρG) and Cauchy-Schwartz inequalityConstrains: Normalization, Positivity, Continuity, Fixed Purity µ[ρ]
Is this bound tight and physical?
Cauchy-Schwartz
Lagrange multipliers
µGµ
δ
1
0
0
0.5
A. Mandilara , E. Karpov, N. Cerf PRA (2009)
Now, looking at purities and regardless of ][ρµ ][ Gρµ ],[ Gρρδ
Gaussian mixed states
mixedstates
Gaussianpure states
][ Gρµ
][ρµ
],[ Gρρδ0 ][ Gρµ1
0
1GAP
Gap indicates that the bound is not physicalThere may be states that are more mixed than Gaussian states !!!
Are they physical ?
Hudson
][ρµ
Lower bound on the purity µ=8/9 µ[ρG]is not tight because it is not physical
!!!But there exist physical states more mixed than Gaussian states
HudsonConvex mixtureof squeezed states
µ [ρ] = 8µ [ρG] / (9 − µ [ρG]2)
Exact bound for all quantum statesafter Dodonov and Manko 1989
Tentative explanation: purity = quantum Renyi entropy (of parameter α=2)classical Renyi entropy is maximized by
Student distribution(Gaussian distribution if α=1)
( )αα ρ
αρ Tr
11)(−
=S
Better bound on “non-Gaussianity” ? – Another view on the problem
( )δµσσσ ,2
2 Fxpppxxh
≥−
22/1 xpppxxG σσσµ −=
( )δµµ ,/1 FG h≥
Bound on µ[ρG]
Uncertainty relation !
µ98
22 h≥− xpppxx σσσM.J. Bastiaans, JOSA A (1983)
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−=
−=∆
ppyx
xyxx
xx xx
xxx
σσσσ
γ
σ
2
22
22
( )2ρµ Tr=
Review of the Uncertainty relationsReview of the Uncertainty relations
HeisenbergHeisenberg--Kenard (1927)Kenard (1927)
ScrhScrhöödingerdinger--Robertson (1930)Robertson (1930)
HudsonHudson--Piquet (1974Piquet (1974) Pure Gaussian states ) Pure Gaussian states Positive Wigner functionPositive Wigner function
DodonovDodonov--ManMan’’ko (1989)ko (1989)
Our proposal (2009)Our proposal (2009)
Minimized by:Coherent state
Coherent(Squeezed) states= Gaussian states
Non Gaussian statesPositive Wigner function
All statesPositive Wigner Functions
( )
( )δµσσσ
µσσσ
σσσ
,2
2
2
2
2
2
2
F
px
xpppxx
xpppxx
xpppxx
h
h
h
h
≥−
Φ≥−
≥−
≥∆∆
Purity & nonPurity & non--Gaussianity bounded Gaussianity bounded Uncertainty relation for mixed statesUncertainty relation for mixed states
( )δµσσσ ,2
2 Fxpppxxh
≥−
Steps of the derivation:• It is minimized by states with phase-independent Wigner function• Lagrange multipliers
nnn
n∑= λρ
Pure states
n=0
n=1
n=2
n=3
( )µσσσ Φ≥−2
2 hxpppxx
( )δσσσ ,12
2 Fxpppxxh
≥−
22 xpppxx σσσ −22 xpppxx σσσ −
µ
Gµ
δ
Purity & nonPurity & non--Gaussianity bounded Gaussianity bounded Uncertainty relation and HudsonUncertainty relation and Hudson’’s theorems theorem
µ
Gµ
δ
µ
Gµ
δ
µ
Gµ
δ
All classical continuous distributions Part of the bound covered bystates with positive Wigner function (not tight for quantum states)
All quantum mixed states(tight bound)
( )δµσσσ ,2
2 Fxpppxxh
≥−
ConclusionGeneralized uncertainty relation for mixed states as a bound on the “non-Gaussianity” the state
New uncertainty relation saturated by number states
Partial generalization of the Hudson’s theorem to mixed states is given by a part of new bound
OutlookFurther generalization of the Hudson’s theorem:• Finding the rest of the bound for mixed states with positive
Wigner function• Multimode states
Applications ?