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 ?

The Belgian National Funds for Scientific Research

Prospective Research for Brussels

Thank you for your attention!