Quantum Theory Group Quantum Theory Group Qualification and quantification of entanglement in continuous variable systems Statics and dynamics of information in quantum spin systems Production 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. Caianiello E.R. Caianiello ” ” Universit Universit à à di Salerno di Salerno Main lines of research Main lines of research
25
Embed
Dipartimento di Fisica “E.R. Caianiello” Università ...crm.sns.it/media/course/552/Illuminati Pisa.pdf · Production of entangled states of atomic samples and multiphoton systems
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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)
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
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