Strong Monogamy and Genuine Multipartite Entanglement

Strong Monogamy and Genuine Multipartite Entanglement


Gerardo Adesso

Informal Quantum Information Informal Quantum Information

GatheringGathering 20072007Informal Quantum Information Informal Quantum Information

GatheringGathering 20072007

The Gaussian Case The Gaussian Case

Quantum Theory groupUniversity of

Salerno

G. Adesso Strong Monogamy and Genuine Multipartite Entanglement


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ContentsContents

What we know on entanglement What we know on entanglement monogamymonogamy Distributed bipartite entanglement: CKW

• Qubits (spin chains)• Gaussian states (harmonic lattices)

A A strongerstronger monogamy constraint… monogamy constraint… Addressing shared bipartite and multipartite

entanglement• Genuine N-party entanglement of symmetric N-mode

Gaussian states Scale invarianceScale invariance and and molecular molecular

entanglement hierarchyentanglement hierarchyBased on recently published joint works with: F. Illuminati (Salerno), A. Serafini (London), M. Ericsson (Cambridge), T. Hiroshima (Tokyo); and especially on: G. Adesso & F. Illuminati, quant-ph/0703277



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Entanglement is monogamousEntanglement is monogamous

Suppose that A-B and A-C are both maximally entangled…

AliceBob Charlie

…then Alice could exploit both channels simultaneously to achieve perfect 12 telecloning, violating no-cloning theorem

B. M. Terhal, IBM J. Res. & Dev. 48, 71 (2004); G. Adesso & F. Illuminati, Int. J. Quant. Inf. 4, 383 (2006)



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CKW monogamy inequalityCKW monogamy inequality

No-sharing for maximal entanglement, but nonmaximal one can be shared… under some constraints

While monogamy is a fundamental quantum property, fulfillment of the above inequality depends on the entanglement measure

[CKW] Coffman, Kundu, Wootters, Phys. Rev. A (2000)

≥ +

EAj(BC) ¸ EAjBÁC +EAjCÁBEAj(BC) ¸ EAjBÁC +EAjCÁBEAj(BC) ¸ EAjBÁC +EAjCÁBEAj(BC) ¸ EAjBÁC +EAjCÁB

• it was originally proven for 3 qubits using the tangle (squared concurrence)

The difference between LHS and RHS yields the residual three-way shared entanglement

Dür, Vidal, Cirac, Phys. Rev. A (2000)



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‘Generalized’ monogamy inequality‘Generalized’ monogamy inequality

…

≥…

…

…+ +…

+

E s1 j(s2 ;:::;sN ) ¸NX

j =2

E s1 jsjE s1 j(s2 ;:::;sN ) ¸NX

j =2

E s1 jsj

Conjectured by CKW (2000); proven for N qubits by Osborne and Verstraete, Phys. Rev. Lett. (2006)

The difference between LHS and RHS yields not the genuine N-way shared

entanglement, but all the quantum correlations not stored in couplewise

form



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Continuous variable systemsContinuous variable systems

Quantum systems such as material particles (motional degrees of freedom), harmonic oscillators, light modes, or bosonic fields

Infinite-dimensional Hilbert spaces for N modes

Quadrature operators

H =N N

i=1 H iH =N N

i=1 H i

X = (q1; p1; :: : ; qN ; pN )X = (q1; p1; :: : ; qN ; pN )

states whose Wigner function is Gaussian

Gaussian statesGaussian states

can be realized experimentally with current technology (e.g. coherent, squeezed states) successfully implemented in CV quantum information processes (teleportation, QKD, …)

qj = aj + ayj ; pj = (aj ¡ ay

j )=iqj = aj + ayj ; pj = (aj ¡ ay

j )=i

G. Adesso and F. Illuminati, review article, quant-ph/0701221, J. Phys. A in press

Vector of first moments (arbitrarily adjustable by local displacements: we

will set them to 0) Covariance Matrix (CM) σ (real, symmetric, 2N x 2N) of the second

moments ¾i j ´ hX i X j +X j X i i=2¡ hX i ihX j i¾i j ´ hX i X j +X j X i i=2¡ hX i ihX j i¾i j ´ hX i X j +X j X i i=2¡ hX i ihX j i¾i j ´ hX i X j +X j X i i=2¡ hX i ihX j i

fully determined by



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Monogamy of Gaussian entanglementMonogamy of Gaussian entanglement

Monogamy inequality for 3-mode Gaussian

states

NJP 2006, PRA 2006

GLOCC monotonicity of

three-way residual entanglement

NJP 2006

---(didn’t check!)

Monogamy inequality for fully symmetric N-mode

Gaussian states

NJP 2006

Monogamy inequality for all (pure and mixed) N-mode Gaussian

states

numerical evidence

NJP 2006

full analytical prooffull analytical proof

PRL 2007

implies

implies

G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 73, 032345 (2006)T. Hiroshima, G. Adesso, and F. Illuminati, Phys. Rev. Lett. 98, 050503 (2007)G. Adesso and F. Illuminati, review article, quant-ph/0701221, J. Phys. A in press

Gaussian contangle[GCR* of (log-neg)2]

Gaussian tangle[GCR* of negativity2]

traditional

*GCR = “Gaussian convex roof” minimum of

the average pure-state entanglement over all decompositions of the

mixed Gaussian state into pure Gaussian states



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A stronger monogamy constraint…A stronger monogamy constraint…

Consider a general quantum system multipartitioned in N subsystems

each comprising, in principle, one or more elementar units (qubit, mode, …)Does a Does a stronger, general monogamy stronger, general monogamy

inequalityinequality a prioria priori dictated by quantum dictated by quantum mechanics exist, which constrains mechanics exist, which constrains on the on the same groundsame ground the distribution of bipartite the distribution of bipartite andand multipartite entanglement?multipartite entanglement?

Can we provide a suitable Can we provide a suitable generalization of generalization of the tripartite analysisthe tripartite analysis to arbitrary to arbitrary NN, such that , such that a a genuine genuine NN-partite entanglement-partite entanglement quantifier is quantifier is naturally derived?naturally derived?



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Ep1j(p2:::pN ) =P N

j =2 Ep1jpj +P N

k>j =2 Ep1jpj jpk

+ :::+Ep1jp2j:::jpN

Ep1j(p2:::pN ) =P N

j =2 Ep1jpj +P N

k>j =2 Ep1jpj jpk

+ :::+Ep1jp2j:::jpN

Decomposing the block entanglementDecomposing the block entanglement

… ≥∑j=2

N11 2 3 4 2 3 4 NN 11 jj

+∑k>j

N 1 1 jj kk

+··· +11 2 3 4 2 3 4 NN

…

=?

iterative structure

• Each K-partite entanglement (K<N) is

obtained by nesting the same decomposition at orders 3,

…,K

• The N-partite entanglement is then implicitly defined by

difference

……

……

……

……

…Genuine Genuine NN-partite -partite entanglemenentanglementt

=minfocus

EEpp11|p|p22|…|p|…|pNN



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Strong monogamy inequalityStrong monogamy inequality

fundamental requirement

• implies the traditional ‘generalized’ monogamy

inequality (in which only the two-party entanglements are

subtracted)

• extremely hard to prove in general!

EEpp11|p|p22|…|p|…|pNN

≥ ≥ 00??

… =∑j=2

N11 2 3 4 2 3 4 NN 11 jj

+∑k>j

N 1 1 jj kk

+··· +11 2 3 4 2 3 4 NN

…

?… =∑j=2

N11 2 3 4 2 3 4 NN 11 jj

+∑k>j

N 1 1 jj kk

+∑k>j

N 1 1 jj kk

+··· +11 2 3 4 2 3 4 NN

…11 2 3 4 2 3 4 NN

……

? ……

……

……

……

…

= minfocus

EEpp11|p|p22||……|p|pNN

……

……

……

……

…

……

……

……

……

…

= minfocus

EEpp11|p|p22||……|p|pNN



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Permutation-invariant quantum statesPermutation-invariant quantum states

Why consider such instances Main testgrounds for theoretical investigations of multipartite

entanglement (structure, scaling, etc…) Main practical resources for multiparty quantum information

& communication protocols (teleportation networks, secret sharing, …)

What matters to our construction Entanglement contributions are independent of mode indexes No minimization required over focus party We can use combinatorics

(N-1) N-1 K

N-1 K

N=any

3 3N=4

22N=3genuine N-party

entanglement



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Resolving the recursionResolving the recursion

(N-1) N-1 K

N-1 K

N=any

3 3N=4

22N=3genuine N-party

entanglement

EEpp11jjpp22jj::::::jjppNN ==NN ¡¡ 11XX

KK ==11

¡¡NN ¡¡ 11KK

¢¢((¡¡ 11))KK ++NN ++11EE pp11jj((pp22::::::ppKK ++ 11))EEpp11jjpp22jj::::::jjppNN ==

NN ¡¡ 11XX

KK ==11

¡¡NN ¡¡ 11KK

¢¢((¡¡ 11))KK ++NN ++11EE pp11jj((pp22::::::ppKK ++ 11))

conjectured general expression for the genuine N-partite entanglement of permutation-invariant quantum states

(for a proper monotone E)alternating finite sum involving only bipartite entanglements

between one party and a block of K other parties



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van Loock & Braunstein, Phys. Rev. Lett. (2000) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, 220504 (2004)G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005)

mom-sqz r1pos-sqz r2

pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //mom-sqz r1

pos-sqz r2

pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2

mom-sqz r1pos-sqz r2

pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

CMCMσ(N)=

det¾(N )K =

2K 2 ¡ 2NK +2(N ¡ K )cosh(4¹r)K +N2

N2det¾(N )K =

2K 2 ¡ 2NK +2(N ¡ K )cosh(4¹r)K +N2

N2

General instance (one mode per party)mixed N-mode state obtained from pure (N+M)-mode state by tracing over M modes

NM

Validating the conjecture…Validating the conjecture…… in permutation-invariant Gaussian states… in permutation-invariant Gaussian states



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det¾(N )K =

2K 2 ¡ 2NK +2(N ¡ K )cosh(4¹r)K +N2

N2det¾(N )K =

2K 2 ¡ 2NK +2(N ¡ K )cosh(4¹r)K +N2

N2

IngredientsIngredients

Unitary localizability LK ULUK L+K-2

LxK 1x1 entanglement

Localized 1x1 state is a GLEMS (min-uncertainty mixed state) fully specified by global (two-mode) and local (single-mode) determinants, --.i.e. det σL

(N+M) , det σK(N+M) , and det σL+K

(N+M)

Gaussian entanglement measures (including contangle) are known for GLEMS

G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, 220504 (2004); A. Serafini, G. Adesso, and F. Illuminati, Phys. Rev. A 71, 023249 (2005).G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 92, 087901 (2004); Phys. Rev. A 70, 022318 (2005).G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006); Phys. Rev. A 72, 032334 (2005).

N-party contangleN-party contangleEp1jp2j:::jpN =N¡ 1X

K =1

¡N¡ 1K

¢(¡ 1)K +N+1Ep1j(p2:::pK +1)Ep1jp2j:::jpN =

N¡ 1X

K =1

¡N¡ 1K

¢(¡ 1)K +N+1Ep1j(p2:::pK +1)

CMCMσ(N)=

GresNNM

j0

N11jN 1j arcsinh2

2

j N 1sinh2rM N j 4rj M N



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N-party contangleN-party contangle

GresNNM

j0

N11jN 1j arcsinh2

2

j N 1sinh2rM N j 4rj M N

pure states (M=0)ALWAYS POSITIVEincreases with the average squeezing rdecreases with the number of modes N(for mixed states) decreases with mixedness Mgoes to zero in the field limit (N+M)



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Entanglement is strongly monogamous…Entanglement is strongly monogamous…… in permutation-invariant Gaussian

states… in permutation-invariant Gaussian states

van Loock & Braunstein, Phys. Rev. Lett. (2000) G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005)Yonezawa, Aoki, Furusawa, Nature (2004) [experimental demonstration for N=3]

Nopt

2N r2rN 24r 2N 4r 1

N 24r 2

12

23

48

20

50

23

48

20

50

Genuine N-partite entanglement is monotonic in the optimal

teleportation-network fidelity (in turn, given by the localizable

entanglement), and exhibits the same scaling with N

There’s a There’s a uniqueunique, theoretically and , theoretically and operationally motivated, operationally motivated, entanglement entanglement

quantificationquantification in symmetric Gaussian states in symmetric Gaussian states (generalizing the known cases (generalizing the known cases NN=2, 3)=2, 3)



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Scale invariance of entanglementScale invariance of entanglement

det σ(mN)mK is independent of m …

Strong monogamy constrains also multipartite entanglement of molecules each made by more than one mode

N-partite entanglement of N molecules (each of the same size m) is independent of m, i.e. it is scale invariant

== ==



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…a smaller number of larger molecules share strictly more entanglement than a larger number of smaller molecules

(but all forms of multipartite entanglement are nonzero…)

Molecular entanglement hierarchyMolecular entanglement hierarchy

At fixed N…

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0

Gres¿

¹r

(N=

12

)



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PromiscuityPromiscuity

• ‘Promiscuity’ means that bipartite and genuine multipartite entanglement are increasing functions of each other, and the genuine multipartite entanglement is enhanced by the simultaneous presence of the bipartite one, while typically in low-dimensional systems like qubits only the opposite behaviour is compatible with monogamy (cfr. GHZ and W states of qubits)• This was pointed out in our first Gaussian monogamy paper (NJP 2006) in the subcase N=3 of permutation-invariant Gaussian states, there re-baptised CV GHZ/W states

: the general picture: the general picture

Michael PerezCreative threesome

Michael PerezCreative threesome

Michael PerezFire

Michael PerezFire

in in NN-mode permutation-invariant -mode permutation-invariant Gaussian states, Gaussian states, allall possible forms of possible forms of

bipartite and multipartite bipartite and multipartite entanglement coexist and are entanglement coexist and are

mutually enhanced for any nonzero mutually enhanced for any nonzero squeezingsqueezingunlimited promiscuity also occurs in some

four-mode nonsymmetric Gaussian states, where strong monogamy can be proven to hold as well…

Despite entanglement is

Despite entanglement is stronglystrongly monogamous, monogamous,

there’s infinitely more freedom when addressing


distributed correlations in harmonic CV systems, as


compared to qubit systems

compared to qubit systemsDespite entanglement is

Despite entanglement is stronglystrongly monogamous, monogamous,









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Summary and concluding remarksSummary and concluding remarks

We recalled the current state-of-the-art on entanglement sharing including our recent conclusive results on monogamy of Gaussian states

We proposed a novel, general approach to monogamy, wherein a stronger a priori constraint imposes trade-offs on both bipartite and multipartite entanglement on the same ground

We demonstrated that approach to be successful when dealing with permutation-invariant Gaussian states of an arbitrary N-mode continuous variable system: namely, I derived an analytic expression for the genuine N-partite entanglement of such states, and operationally connected it to the optimal fidelity of teleportation networks employing such resources

Scale invariance of continuous variable entanglement has been demonstrated in symmetric Gaussian states, and a hierarchy of molecular entanglements has been established, from which the promiscuous structure of distributed entanglement arises naturallyG. Adesso and F. Illuminati, quant-ph/0703277



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Some interesting open issuesSome interesting open issues

Checking the strong monogamy in systems of 4 or more qubits (work in progress with M. Piani), and/or trying to prove strong monogamy in general for states of arbitrary dimension (e.g. using the squashed entanglement)

Further investigating interdependence relationships between monogamy constraints, entanglement frustration effects and De Finetti-like bounds (for symmetric states)

Investigating the origin of promiscuity, e.g. by considering dimension-dependent qudit families and studying the onset of shareability towards the continuous variable limit

Devising new protocols to take advantage of promiscuous entanglement for communication and computation purposes



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Strong Monogamy and Genuine Multipartite Entanglement