Quantum Theory Group

Transcription

1 Quantum Theory Group Dipartimento di Fisica E.R. Caianiello Università di Salerno 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 Main lines of research 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

2 PISA, December 16, 2004 Entanglement Scaling, Localization and Sharing in Continuous Variable Systems Fabrizio Illuminati in collaboration with Gerardo Adesso Alessio Serafini

3 Outline Gaussian states of continuous variable (CV) systems Entanglement and purities Unitary localization and scaling of multimode bipartite entanglement Genuine multipartite entanglement: the continuous variable tangle Sharing (polygamy) of CV entanglement Optimal use of entanglement for CV teleportation

4 Continuous variable systems Quantum systems such as harmonic oscillators, light modes, or cold bosonic gases Infinite-dimensional Hilbert spaces H = N N Quadrature operators ˆX =(ˆq 1, ˆp 1,...,ˆq N, ˆp N ) Canonical commutation relations i=1 H i ˆq j =â j +â j, ˆp j =(â j â j )/i [ ˆXi, ˆXj] =2 i Ωij, Ω for N modes Described in phase space by quasiprobability distributions, such as Wigner function, Glauber P-function, Husimi Q-function

5 Gaussian states states whose Wigner function is Gaussian fully determined by Vector of first moments X (h ˆX 1 i,...,h ˆX N i) (arbitrarily adjustable by local displacements) Second moments encoded in the Covariance Matrix (CM) σ σ ij h ˆX i ˆXj + ˆX j ˆXi i/2 h ˆX i ih ˆX j i Robertson-Schrödinger uncertainty principle σ + iω 0 (real, symmetric, 2N x 2N) (bona fide condition for any physical CM) can be realized experimentally with current technology thermal, coherent, squeezed states are Gaussian implemented in CV quantum information processes

6 Phase space and symplectics Hilbert space H Density matrix ρ Unitary operations U Phase space Γ Covariance matrix σ Symplectic operations S so we move into phase space Symplectic Williamson diagonalization of a CM: normal mode decomposition N T N T T 1N 2N N S ν 1 ν1 ν 2 ν ν N ν N ν i the s are the symplectic eigenvalues computable as the standard eigenvalues of the matrix i Ωσ determined by N symplectic invariants, including Determinant Det σ Seralian = Q i ν2 i = P i ν2 i (Purity = [Det σ ] -1/2 ) (sum of 2x2 sub-determinants)

7 Entanglement properties (Simon 2000) Transposition Time reversal in phase space Partial transposition σ σ Inversion of the p operator of a mode PPT: σ separable iff σ σ bona fide i.e. σ + iω 0 for 1xN partitions {ν i } and { ν i } are the symplectic eigenvalues of σ and σ ν i 1 physical state full saturation: pure state partial saturation: minimum-uncertainty mixed state ν i 1 (only separable state for M x N, M>1) violation: entanglement We can compute the logarithmic negativity to quantify the entanglement ½ 0, ν E N (σ) = i 1 i ; P i: ν i <1 log ν i, else. The EoF is computable* for 1x1 symmetric states and it is completely equivalent *Giedke et al., PRL 2003

8 Unitary localization Bisymmetric (M+N)-mode states M T T T T S M S N N PPT criterion holds: no bisymmetric bound entanglement Logarithmic negativity (also EoF if α =β ) computable Reversible multimode/two-mode entanglement switch ν M 0 ν M T ν N 0 ν N

9 Entanglement scaling b» b 1äK e ntanglemen t E K We exploit the two-mode equivalence to investigate multimode entanglement. Example: fully symmetric N-mode states Entanglement 1xK (K N) Entanglement Kx(N-K) (K N/2) PURE STATE mixed states b ª 1êm b ~ squeezing K=9 K=8 K=7 K=5 K=3 K=1 b E k» b 1 0 -k PURE STATES mixed states b Best localization strategy: equal splitting between two parties k=5 k=3 k=2 k=1 k=5 k=3 k=2 k=1 E F Localized PURE Localized mixed n 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)

10 The Continuous Tangle The 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? Continuous Contangle Variable Tangle E τ (E N ) 2 analogy with discrete systems bipartite multipart DV C C 2 CV E N E 2 N

11 Multiparty entanglement Structure of multipartite entanglement (example: fully symmetric pure N-mode states) N=3 2 N=4 3 3 genuine N-party 1x1x 1 contangle N=any N N K N K õúúúúúúú ún úúùúúúú úúú ú û 1ä1ä ä1 E t N =2 N =3 N =4 N =5 N =9 E τ 1 N = K+1 NX µ N z } { E τ K K= b

12 Contangle in generic states beyond the symmetry min c Tripartite Contangle Generic three-mode pure states a a b only parametrized by the 3 local single-mode purities 1/a, 1/b, 1/c, with ( a-b + 1) c (a+b-1) [triangle ineq]

13 Polygamous entanglement Monogamy of quantum entanglement 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 Polygamy of CV systems Polygamy when there is an harem of infinitely many degrees of freedom available for the entanglement, its monogamy inevitably fails!

14 Rewind/1: W & GHZ states CV entanglement is polygamously shareable this follows by comparing the tri-contangle in the CV GHZ and W states: the latter maximize tripartite and any reduced bipartite entanglement. W states OUT TRITTER BS 1:1 GHZ states How can these states be produced? IN mom-sq (r) posit-sq (r) posit-sq (r) BS 1:2 mom-sq (r) therm (n[r]) therm (n[r])

15 Rewind/2: there s a multiparty The 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 computed Its 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

16 Rewind/3: unitarily localizing Rewind/3: unitarily localizing When you cut the head of a basset hound it will grow again! W

17 References Two-mode entanglement vs purity & entropic measures G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 92, (2004) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 70, (2004) 1xN and MxN multimode entanglement G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, (2004) A. Serafini, G. Adesso and F. Illuminati, quant-ph/ (2004) Genuine multipartite entanglement G. Adesso and F. Illuminati, quant-ph/ (2004) Three-mode entanglement production and characterization in preparation See also the poster by Gerardo Adesso Optimal use of multipartite entanglement for continuous variable teleportation

18 Storing massive information - 1 Quantum spin system on a ring with periodic boundary condition Linked magnetic flux H A N φ ( S + i N + H= λ e S + H.C) + B S + S + H.C + B S + N N z z i i i1 i 1 i i i = 1 i= 1 i= 1 Local perturbation (Spin Flip) H A

19 Storing massive information - 2 φ = α+πθ N φ constant in time φ modulated: ( t T) Physical situation after a time t=2t H H A See also the poster by S. M. Giampaolo & A. Di Lisi Storage of massive logical memory in a quantum spin ring with modulated magnetic flux A

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21 Two-mode Gaussian states Standard form: 4 parameters a 0 c + 0 σ sf 0 a 0 c c + 0 b 0 0 c 0 b local purities global purity seralian 4 symplectic invariants µ 1 = 1 a, µ 2 = 1 b, 1 µ 2 = Detσ =(ab) 2 ab(c 2 + +c 2 )+(c + c ) 2, = a 2 +b 2 +2c + c. Partial transposition flips the sign of c = a 2 +b 2 2c + c = +2/µ /µ2 2 r Symplectic eigenvalues : 2 2ν2 = 2 4 r µ 2, 2 ν2 = 2 4 µ 2. The entanglement is fully determined by ν! Logarithmic negativity E N =max{0, log ν }

22 Symplectic parametrization We choose this parametrization: {a, b, c +,c } {µ 1,µ 2,µ, } we know the purities, but who is??? ν 2 > 0 µ1,µ 2,µ the seralian regulates the entanglement of a generic Gaussian state with given purities We have some constraints on the symplectic invariants µ µ 1 µ 2 µ 1 µ 2 µ 1 µ 2 + µ 1 µ 2 2 µ + (µ 1 µ 2 ) 2 µ 2 1 µ µ 2 Maximally and minimally entangled Gaussian states for fixed global and marginal degrees of purity

23 Extremal entanglement GMEMS Gaussian Max. Entangled Mixed States two-mode squeezed thermal states GLEMS Gaussian Least Entangled Mixed States minimum-uncertainty mixed states squeezing parameter tanh 2r =2(µ 1 µ 2 µ 2 1µ 2 2/µ) 1/2 /(µ 1 + µ 2 ) OPO / OPA one-mode squeezed beam pure GMEMS are a good approximation of EPR beams (infinitely entangled) one-mode thermal state BS 50:50 two-mode GLEMS

24 Entanglement vs purities The separability is completely qualified by the purities except for a narrow coexistence region Degrees of Purity Separability µ<µ 1 µ 2 unphysical region µ µ 1 µ 2 µ 1 µ 2 µ 1 +µ 2 µ 1 µ 2 separable states µ 1 µ 2 µ µ 1 +µ 2 µ 1 µ 2 <µ 1 µ 2 coexistence region µ 1 µ 2 µ 2 1 +µ 2 2 µ2 1 µ2 2 µ 2 1 +µ 2 2 µ2 1 µ2 2 µ 1 µ 2 <µ µ 1 µ 2 +µ 1 µ 2 entangled states µ µ> 1 µ 2 µ 1 µ 2 +µ 1 µ 2 unphysical region

25 Entanglement estimation A more quantitative look Average Logarithmic Negativity Ē N (µ 1,2,µ) E Nmax(µ 1,2,µ)+E Nmin (µ 1,2,µ) 2 Relative error on the estimate δēn(µ 1,2,µ) E Nmax(µ 1,2,µ) E Nmin (µ 1,2,µ) E Nmax (µ 1,2,µ)+E Nmin (µ 1,2,µ) GLEMS GMEMS δēe N decreases exponentially We can estimate entanglement by measurements of purity the estimate is reliable!