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Informal Quantum Information Gathering 2007. Strong Monogamy and Genuine Multipartite Entanglement. The Gaussian Case. Gerardo Adesso. Quantum Theory group University of Salerno. Contents. What we know on entanglement monogamy Distributed bipartite entanglement: CKW Qubits (spin chains) - PowerPoint PPT Presentation
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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
Strong Monogamy and Genuine Multipartite Entanglement
2
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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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)
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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)
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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?
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite 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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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)
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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)
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
== ==
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
)
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
there’s infinitely more freedom when addressing
distributed correlations in harmonic CV systems, as
distributed correlations in harmonic CV systems, as
compared to qubit systems
compared to qubit systemsDespite entanglement is
Despite entanglement is stronglystrongly monogamous, monogamous,
there’s infinitely more freedom when addressing
there’s infinitely more freedom when addressing
distributed correlations in harmonic CV systems, as
distributed correlations in harmonic CV systems, as
compared to qubit systems
compared to qubit systems
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
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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
G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
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G. Adesso Strong Monogamy and Genuine Multipartite Entanglement
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