Quantum Theory GroupQuantum Theory Group

Qualification and quantification of entanglement in continuous variable systemsStatics and dynamics of information in quantum spin systemsProduction of entangled states of atomic samples and multiphoton systems

G. Adesso, F. Dell’Anno, S. De Siena, A. Di Lisi, S. M. Giampaolo, F. Illuminati, G. Mazzarella

former members: A. Albus and A. Serafini

Dipartimento di Fisica Dipartimento di Fisica ““E.R. CaianielloE.R. Caianiello””UniversitUniversitàà di Salernodi Salerno

Main lines of researchMain lines of research

Entanglement Scaling, Localization and Sharing in Continuous Variable SystemsEntanglement Scaling, Localization and Sharing in Continuous Variable Systems

Fabrizio IlluminatiFabrizio Illuminati

in collaboration with

Gerardo AdessoAlessio Serafini

PISA, December 16, 2004

OutlineOutline

Gaussian statesof continuous variable (CV) systemsEntanglement and puritiesUnitary localization and scalingof multimode bipartite entanglementGenuine multipartite entanglement: the continuous variable tangleSharing (polygamy) of CV entanglementOptimal use of entanglement for CV teleportation

Continuous variable systemsContinuous variable systems

Quantum systems such as harmonic oscillators, light modes, or cold bosonic gases

Infinite-dimensional Hilbert spaces for N modes

Quadrature operators

Canonical commutation relations

Described in phase space by quasiprobability distributions, such as Wigner function, Glauber P-function, Husimi Q-function

H =NN

i=1HiH =NN

i=1Hi

X = (q1, p1, . . . , qN , pN )X = (q1, p1, . . . , qN , pN )qj = aj + a

†j , pj = (aj − a†j)/i

[Xi, Xj ] iΩij , Ω[Xi, Xj ] iΩij , Ω=[Xi, Xj ] iΩij , Ω[Xi, Xj ] iΩij , Ω[Xi, Xj ] iΩij , Ω[Xi, Xj ] iΩij , Ω=2

Gaussian statesGaussian statesstates whose Wigner function is Gaussian

• Vector of first moments(arbitrarily adjustable by local displacements)

• Second moments encoded in the Covariance Matrix (CM) σ(real, symmetric, 2N x 2N)

X ≡ (hX1i, . . . , hXN i)X ≡ (hX1i, . . . , hXN i)

σij ≡ hXiXj + XjXii/2− hXiihXjiσij ≡ hXiXj + XjXii/2− hXiihXjiσij ≡ hXiXj + XjXii/2− hXiihXjiσij ≡ hXiXj + XjXii/2− hXiihXji

fully determined by

can be realized experimentally with current technologythermal, coherent, squeezed states are Gaussianimplemented in CV quantum information processes

Robertson-Schrödinger uncertainty principle

σ + iΩ ≥ 0σ + iΩ ≥ 0σ + iΩ ≥ 0σ + iΩ ≥ 0 (bona fide condition for any physical CM)

Phase space and symplecticsPhase space and symplecticsHilbert space H Phase space Γ

Unitary operations U Symplectic operations S

…so we move into phase space…

Symplectic ‘Williamson’ diagonalization of a CM: normal mode decomposition

Density matrix ρ Covariance matrix σ

1 12 1

12 2 2

1 2

N

TN

T TN N N

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

the ’s are thesymplectic

eigenvalues

νiνi1122

NN

νν

ννν

ν

S

0

0

determined by N symplectic invariants, including…

Determinant

SeralianDetσ =

Qi ν

2i

∆ =Pi ν

2i

Detσ =Qi ν

2iDetσ =

Qi ν

2i

∆ =Pi ν

2i∆ =

Pi ν

2i

Detσ =Qi ν

2i

∆ =Pi ν

2i

Detσ =Qi ν

2iDetσ =

Qi ν

2i

∆ =Pi ν

2i∆ =

Pi ν

2i (sum of 2x2 sub-determinants)

(Purity = [Det σ ] -1/2 )

computable as the standard eigenvalues of the matrix |i Ωσ|

Entanglement propertiesEntanglement properties(Simon 2000) Transposition Time reversal in phase space

Partial transposition Inversion of the p operator of a modeσ → σσ → σ

PPT: separable iff bona fide i.e.σσ σσ σ + iΩ ≥ 0σ + iΩ ≥ 0PPT: separable iff bona fide i.e.σσ σσ σ + iΩ ≥ 0σ + iΩ ≥ 0PPT: separable iff bona fide i.e.σσ σσ σ + iΩ ≥ 0σ + iΩ ≥ 0PPT: separable iff bona fide i.e.σσ σσ σ + iΩ ≥ 0σ + iΩ ≥ 0 for 1xNpartitions

physical statefull saturation: pure statepartial saturation: minimum-uncertainty mixed state

separable state(only for M x N, M>1)

violation: entanglement

νi ≥ 1⇔ νi ≥ 1⇔⇐

νi ≥ 1⇔ νi ≥ 1⇔νi ≥ 1⇔νi ≥ 1⇔ νi ≥ 1⇔νi ≥ 1⇔⇐⇐

We can compute the logarithmic negativity to quantify the entanglement

EN (σ) =

½0, νi ≥ 1 ∀ i ;

−Pi: νi<1log νi , else .

EN (σ) =

½0, νi ≥ 1 ∀ i ;

−Pi: νi<1log νi , else .

The EoF is computable* for 1x1 symmetric states and it is completely equivalent*Giedke et al., PRL 2003

and are the symplectic eigenvalues of andνi νi σσ

Unitary localizationUnitary localizationBisymmetric (M+N)-mode states

T

T

T

T

M

N

M

M

N

N

νν

νν

SM SN

0

0

T

PPT criterion holds: no bisymmetric bound entanglement

Logarithmic negativity (also EoF if α’=β’) computableReversible multimode/two-mode entanglement switch

1 1.5 2 2.5 3 3.5 4

b0

0.5

1

1.5

2

2.5

3

Eb

k »b01-k

k=1k=2k=3k=5

k=1k=2k=3

k=5

Entanglement scalingEntanglement scalingWe exploit the two-mode equivalence to investigate multimode entanglement. Example: fully symmetric N-mode states

1 2 3 4 5 6 7 8b ª 1êm b ~ squeezing

0

0.5

1

1.5

2

2.5

3

1äK

tnemelgnatne

Eb

»bK

K=1K=3K=5K=7K=8

K=9

Entanglement 1xK (K≤N) Entanglement Kx(N-K) (K≤N/2)

Best localization strategy: equal splitting between two parties

1 2 3 4 5 6 7 8 9 10 11 12

n

0

0.2

0.4

0.6

0.8

1

1.2

1.4

EF

Scaling with the number of modes• Bipartite two-mode entanglement (original)

goes to zero• Bipartite two-mode entanglement (localized)= bipartite multimode entanglement

increases (diverges only in pure states)

PURE STATE

mixed states

PURE STATES

mixed states

Localized PURE

Localized mixed

The Continuous TangleThe Continuous TangleThe hierarchy of unitarily localizable bipartite entanglements gives a hint on the structure of the multipartite entanglement

what about the GENUINE 1x1x…1 entanglement?

For 3 qubits: T[A(BC)] ≥ T[AB] + T[AC], with T: Tangle (CKW 2000)

Continuous Variable TangleContangle EEττ ≡≡ ((EENN ))22

Could the same hold for Gaussian states?... What measure?

analogy with discrete systems

DV CV

bipartite

multipart

C EN

E2NC2

1 2 3 4 5b

0

1

2

3

4

5

E t1ä1ä

…ä1

õúúúúúúúúúúù

ûúúúúúúúú

N

N=2N=3N=4N=5N=9

Multiparty entanglementMultiparty entanglementStructure of multipartite entanglement(example: fully symmetric pure N-mode states)

22N=3

3 3N=4

N NKNKNK

N K

N=any

genuine N-party 1x1x…1 contangle

E1×Nτ =NXK=1

µN

K

¶E

K+1z | 1×1×...×1τE1×Nτ =

NXK=1

µN

K

¶E

K+1z | 1×1×...×1τ

Contangle in generic statesContangle in generic states

min

beyond the symmetry…

Generic three-mode pure statesonly parametrized by the 3 local single-mode purities 1/a, 1/b, 1/c, with (|a-b| + 1) ≤ c ≤ (a+b-1) [triangle ineq]

a

a

c

b

Tripartite Contangle

Polygamous entanglementPolygamous entanglementMonogamy of quantum entanglement

… when there is an ‘harem’ of infinitely many degrees of freedom available for the entanglement, its

monogamy inevitably fails !

3 qubits: two inequivalent families of tripartite entangled states• GHZ states: no 1x1, max any 1x2 max 1x1x1 three-tangle• W states: max 1x1 between any couple (1x2)=2(1x1)

zero 1x1x1 three-tangle

CV finite-squeezing analogy: Gaussian fully symmetric 3-mode states, based on the same bipartite properties…

W states: max 1x1, max 1x2 AND max 1x1x1 !!! (GHZ states: lower 1x1x1)

…the more two-party, the more three-party…

PolygamyPolygamy of CV systemsof CV systems

Rewind/1: W & GHZ statesRewind/1: W & GHZ statesCV entanglement is polygamously shareablethis follows by comparing the tri-contangle in the CV GHZ and W states: the latter maximize tripartite and any reduced bipartite entanglement.

How can these states be produced?

BS1:2

BS1:1

TRIT

TER

IN

OUT

mom-sq (r)posit-sq (r)posit-sq (r)

W states

mom-sq (r)therm (n[r])therm (n[r])

GHZ states

Rewind/2: there’s a multipartyRewind/2: there’s a multipartyThe Contangle is a measure of genuine multipartite entanglement it can be measured e.g. in three-mode pure states by measurements of local purities (diagonal elements of CM)

The multimode entanglement under symmetry can be computedIts scaling can be investigated, and the MxN entanglement can be reversibly converted into 1x1 (‘localized’) by optical means.

PPT criterion is necessary and sufficient for separability of MxN symmetric and bisymmetric Gaussian states

Rewind/3: unitarily localizingRewind/3: unitarily localizing

When you cut the head

When you cut the head of a basset houndof a basset hound…………it will grow again!

it will grow again!

W

ReferencesReferencesTwo-mode entanglement vs purity & entropic measures

G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 92, 087901 (2004)G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 70, 022318 (2004)

1xN and MxN multimode entanglementG. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, 220504 (2004)A. Serafini, G. Adesso and F. Illuminati, quant-ph/0411109 (2004)

Genuine multipartite entanglementG. Adesso and F. Illuminati, quant-ph/0410050 (2004)

Three-mode entanglement production and characterizationin preparation…

See also the poster by Gerardo AdessoOptimal use of multipartite entanglement for continuous variable teleportation

Storing massive information - 1Storing massive information - 1

( )N N

zi i 1 i

i 1 i 1

H S S H.C B S+ −+

= =

= −λ + +∑ ∑

Quantum spin system on a ring with periodic boundary condition

N Ni zNi i 1 i

i 1 i 1

H e S S H.C B Sφ

+ −+

= =

⎛ ⎞= −λ + +⎜ ⎟⎝ ⎠

∑ ∑

H

A

H

A

Linked magnetic flux

Local perturbation (Spin Flip)

H

A

H

A

φ constant in time φ modulated: ( )t TNφ

= α + πθ −⎡ ⎤⎣ ⎦

Physical situation after a time t=2T

Storing massive information - 2Storing massive information - 2

See also the poster by S. M. Giampaolo & A. Di LisiStorage of massive logical memory in a quantum spin ring with modulated magnetic flux

Two-mode Gaussian statesTwo-mode Gaussian statesStandard form: 4 parameters 4 symplectic invariants

µ1 =1

a, µ2 =

1

b,

1

µ2= Detσ=(ab)2−ab(c2++c2−)+(c+c−)2 ,

∆ = a2+b2+2c+c− .

µ1 =1

a, µ2 =

1

b,

1

µ2= Detσ=(ab)2−ab(c2++c2−)+(c+c−)2 ,

∆ = a2+b2+2c+c− .

local purities

global purity

seralian

σsf≡

⎛⎜⎜⎝a 0 c+ 00 a 0 c−c+ 0 b 00 c− 0 b

⎞⎟⎟⎠σsf≡

⎛⎜⎜⎝a 0 c+ 00 a 0 c−c+ 0 b 00 c− 0 b

⎞⎟⎟⎠Partial transposition flips the sign of c– ∆=a2+b2−2c+c−=−∆+2/µ21+2/µ22∆=a2+b2−2c+c−=−∆+2/µ21+2/µ22

Symplectic eigenvalues : 2ν2∓ =∆∓r∆2− 4

µ2, 2ν2∓ = ∆∓

r∆2− 4

µ2.2ν2∓ =∆∓

r∆2− 4

µ2, 2ν2∓ = ∆∓

r∆2− 4

µ2.

The entanglement is fully determined by !ν−ν−

Logarithmic negativity EN = max 0,− log ν−EN = max 0,− log ν−

Symplectic parametrizationSymplectic parametrizationWe choose this parametrization: a, b, c+, c−→ µ1, µ2, µ, ∆a, b, c+, c−→ µ1, µ2, µ, ∆

we know the purities, but who is ∆ ???

the seralian regulates the entanglement of a generic Gaussian state with given purities

∂ ν2−∂ ∆

¯µ1, µ2, µ

> 0 ⇒∂ ν2−∂ ∆

¯µ1, µ2, µ

> 0 ⇒

Maximally and minimally entangled Gaussian statesfor fixed global and marginal degrees of purity

µ1µ2 ≤

≤ 1 + 1

µ2

µ1µ2 ≤µ1µ2 ≤

≤ 1 + 1

µ2≤ 1 + 1

µ2

µ

∆

µµ

∆∆

≤ µ1µ2µ1µ2 + µ1 − µ2

2

µ+(µ1 − µ2)2µ21µ

22

≤

≤ µ1µ2µ1µ2 + µ1 − µ2

≤ µ1µ2µ1µ2 + µ1 − µ2

2

µ+(µ1 − µ2)2µ21µ

22

≤2

µ+(µ1 − µ2)2µ21µ

22

≤

We have some constraints on the symplectic invariants…

Extremal entanglementExtremal entanglement

GLEMSGaussian Least Entangled Mixed States

one-mode squeezed beam

one-mode thermal state

BS 50:50

OPO / OPA

two-m

ode

GLEMS

minimum-uncertaintymixed states

GMEMSGaussian Max. Entangled Mixed States

two-mode squeezedthermal states

squeezing parametertanh2r = 2(µ1µ2 − µ21µ22/µ)1/2/(µ1 + µ2)tanh2r = 2(µ1µ2 − µ21µ22/µ)1/2/(µ1 + µ2)

pure GMEMS are a goodapproximation of EPR beams

(infinitely entangled)

Degrees of Purity Separability

µ < µ1µ2 unphysical regionµ1µ2 ≤ µ ≤ µ1µ2

µ1+µ2−µ1µ2 separable statesµ1µ2

µ1+µ2−µ1µ2 < µ ≤µ1µ2√

µ21+µ22−µ21µ22

coexistence region

µ1µ2√µ21+µ

22−µ21µ22

< µ ≤ µ1µ2µ1µ2+µ1−µ2 entangled states

µ > µ1µ2µ1µ2+µ1−µ2 unphysical region

Degrees of Purity Separability

µ < µ1µ2 unphysical regionµ1µ2 ≤ µ ≤ µ1µ2

µ1+µ2−µ1µ2 separable statesµ1µ2

µ1+µ2−µ1µ2 < µ ≤µ1µ2√

µ21+µ22−µ21µ22

coexistence region

µ1µ2√µ21+µ

22−µ21µ22

< µ ≤ µ1µ2µ1µ2+µ1−µ2 entangled states

µ > µ1µ2µ1µ2+µ1−µ2 unphysical region

Degrees of Purity Separability

µ < µ1µ2 unphysical regionµ1µ2 ≤ µ ≤ µ1µ2

µ1+µ2−µ1µ2 separable statesµ1µ2

µ1+µ2−µ1µ2 < µ ≤µ1µ2√

µ21+µ22−µ21µ22

coexistence region

µ1µ2√µ21+µ

22−µ21µ22

< µ ≤ µ1µ2µ1µ2+µ1−µ2 entangled states

µ > µ1µ2µ1µ2+µ1−µ2 unphysical region

Degrees of Purity Separability

µ < µ1µ2 unphysical regionµ1µ2 ≤ µ ≤ µ1µ2

µ1+µ2−µ1µ2 separable statesµ1µ2

µ1+µ2−µ1µ2 < µ ≤µ1µ2√

µ21+µ22−µ21µ22

coexistence region

µ1µ2√µ21+µ

22−µ21µ22

< µ ≤ µ1µ2µ1µ2+µ1−µ2 entangled states

µ > µ1µ2µ1µ2+µ1−µ2 unphysical region

The separability is completelyqualified by the puritiesexcept for a narrowcoexistence region

Entanglement vs puritiesEntanglement vs purities

GMEMSGMEMS

GLEMSGLEMS

A more quantitative look…

We can estimate entanglementby measurements of purity

EN(µ1,2, µ) ≡ENmax(µ1,2, µ) +ENmin(µ1,2, µ)

2EN(µ1,2, µ) ≡

ENmax(µ1,2, µ) +ENmin(µ1,2, µ)2

‘Average Logarithmic Negativity’

δEN(µ1,2,µ)≡ENmax(µ1,2,µ)−ENmin(µ1,2,µ)ENmax(µ1,2,µ)+ENmin(µ1,2,µ)

δEN(µ1,2,µ)≡ENmax(µ1,2,µ)−ENmin(µ1,2,µ)ENmax(µ1,2,µ)+ENmin(µ1,2,µ)

Relative error on the estimate

decreasesdecreasesexponentiallyexponentiallyδδ ¯EENNδδ ¯EENN

…the estimate is reliable !

Entanglement estimationEntanglement estimation