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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
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