Should entanglement measures be monogamous or faithful?

Should entanglement measures be monogamous or faithful?

Based on Phys. Rev. Lett. 117, 060501 (2016).

Joint work with Gerardo Adesso, Sara Di Martino, Marcus Huber, Marco Piani, Andreas Winter

Cécilia Lancien

Universidad Complutense de Madrid

TQC - Berlin - September 27^th2016

Monogamy of entanglement : seminal observations

Key feature of entanglement :It cannot be shared unconditionally across many subsystems of a composite system(Terhal).

Sharpest manifestation : If two parties A and B share a maximally entangled state, then they cannot share any correlation (even classical ones) with a third party C.

In the more realistic scenario where A and B share a partially entangled (mixed) quantum state ρ_AB, they may share part of this entanglement with other parties, but restrictions remain.

For instance : A stateρABonA⊗Bis calledk-extendible if there exists a stateρ_AB^k onA⊗B^⊗k which is invariant under permutation of theBsubsytsems and s.t. Tr_Bk−1ρ_AB^k=ρAB.

IfρABis infinitely-extendible, it is separable(Christandl/König/Michison/Renner, Doherty/Parrilo/Spedalieri)

→How to formalize quantitatively this observation that entanglement is “monogamous” ? Natural idea :Given an entanglement measureE, show that, for any stateρABC,

E_A:BC(ρ_ABC)>^EA:B(ρ_AB) +E_A:C(ρ_AC).

Such inequality holds true for the squared concurrence(Coffman/Kundu/Wootters)or the squashed entanglement(Christandl/Winter), but fails for many other entanglement measures.

Introducing rescalings or mixing different measures may allow to recover inequalities of this type, but what about a lessad hoctreatment ?

Monogamy of entanglement : seminal observations

Key feature of entanglement :It cannot be shared unconditionally across many subsystems of a composite system(Terhal).

Sharpest manifestation : If two parties A and B share a maximally entangled state, then they cannot share any correlation (even classical ones) with a third party C.

In the more realistic scenario where A and B share a partially entangled (mixed) quantum state ρ_AB, they may share part of this entanglement with other parties, but restrictions remain.

For instance : A stateρABonA⊗Bis calledk-extendible if there exists a stateρ_AB^k onA⊗B^⊗k which is invariant under permutation of theBsubsytsems and s.t. Tr_Bk−1ρ_AB^k=ρAB.

IfρABis infinitely-extendible, it is separable(Christandl/König/Michison/Renner, Doherty/Parrilo/Spedalieri)

→How to formalize quantitatively this observation that entanglement is “monogamous” ? Natural idea :Given an entanglement measureE, show that, for any stateρABC,

E_A:BC(ρ_ABC)>^EA:B(ρ_AB) +E_A:C(ρ_AC).

Such inequality holds true for the squared concurrence(Coffman/Kundu/Wootters)or the squashed entanglement(Christandl/Winter), but fails for many other entanglement measures.

Introducing rescalings or mixing different measures may allow to recover inequalities of this type, but what about a lessad hoctreatment ?

How to define monogamy of an entanglement measure in general terms ?

Definition

[

]

An entanglement measureEismonogamousif there exists a non-trivial function f:R⁺×R⁺→R⁺s.t., for any stateρABCon any tripartite systemA⊗B⊗C,

E_A:BC(ρABC)>^{f E}A:B(ρAB),E_A:C(ρAC) .

Eis an entanglement monotone so w.l.o.g.f(x,y)>^max(x,y). Non-trivial constraint :f(x,y)>max(x,y)for at least somex,y.

→If the only possible choice isf(x,y) =max(x,y),E drastically fails monogamy : knowingz=EA:BC>0 does not constrain in any wayx=E_A:Bandy=E_A:Cin the interval[0,z].

Intuition :Eis monogamous if it obeys some trade-off between the values ofE_A:BandE_A:Cfor a givenE_A:BC.

CKW form of a monogamy relation :f(x,y) =x+y ^{EA:BC EA:B}

E_A:BC EA:C

0

Our results in a nutshell :Many entanglement measures (having the property of beingfaithful in a strong sense) cannot obey any non-trivial monogamy relation.

Monogamy can however be recovered by allowingfto be dimension-dependent.

How to define monogamy of an entanglement measure in general terms ?

Definition

[

]

An entanglement measureEismonogamousif there exists a non-trivial function f:R⁺×R⁺→R⁺s.t., for any stateρABCon any tripartite systemA⊗B⊗C,

E_A:BC(ρABC)>^{f E}A:B(ρAB),E_A:C(ρAC) .

Eis an entanglement monotone so w.l.o.g.f(x,y)>^max(x,y). Non-trivial constraint :f(x,y)>max(x,y)for at least somex,y.

→If the only possible choice isf(x,y) =max(x,y),E drastically fails monogamy : knowingz=EA:BC>0 does not constrain in any wayx=E_A:Bandy=E_A:Cin the interval[0,z].

Intuition :Eis monogamous if it obeys some trade-off between the values ofE_A:BandE_A:Cfor a givenE_A:BC.

CKW form of a monogamy relation :f(x,y) =x+y ^{EA:BC EA:B}

E_A:BC EA:C

0

Our results in a nutshell :Many entanglement measures (having the property of beingfaithful in a strong sense) cannot obey any non-trivial monogamy relation.

Monogamy can however be recovered by allowingfto be dimension-dependent.

Outline

1 Non-monogamy for the entanglement of formation and the relative entropy of entanglement

2 Non-monogamy for a whole class of additive entanglement measures

3 Recovering monogamy with non-universal relations

Definitions and preliminary results

E_F(entanglement of formation) and E_R(relative entropy of entanglement) :

Entropy :S(ρ) =−Tr(ρlogρ). Relative entropy :D(ρkσ) =−Tr(ρ[logρ−logσ]). E_F(ρ_A:B) :=inf

∑ip_iS Tr_B|ψiihψi|AB

: ∑ip_i|ψiihψi|AB=ρAB . E_R(ρA:B) :=inf

D ρABkσ_AB

: σAB∈

S

Random induced states :System of interestH. EnvironmentE.

→Random mixed state model onH:ρ=TrE|ψihψ|, where|ψiis a uniformly distributed pure state onH⊗E.

Note : If|E|6|H|,ρis uniformly distributed on the set of states onHwith rank at most|E|. Needed technical result (potentially of independent interest) :Estimating the typical value of E_FandE_Rfor random induced states(cf. Hayden/Leung/Winter).

LetρABbe a random state onA⊗B≡C^d⊗C^d, induced by some environmentE≡C^s, with Cd6^s6^d2. Then, for anyt>0,

P |E_F(ρAB)−logd|>t

6^{e−cd2t2/log2dandP}

E_R(ρAB)−logd² s

>t

6^e−c0st2.

Definitions and preliminary results

E_F(entanglement of formation) and E_R(relative entropy of entanglement) :

Entropy :S(ρ) =−Tr(ρlogρ). Relative entropy :D(ρkσ) =−Tr(ρ[logρ−logσ]). E_F(ρ_A:B) :=inf

∑ip_iS Tr_B|ψiihψi|AB

: ∑ip_i|ψiihψi|AB=ρAB . E_R(ρA:B) :=inf

D ρABkσ_AB

: σAB∈

S

Random induced states :System of interestH. EnvironmentE.

→Random mixed state model onH:ρ=TrE|ψihψ|, where|ψiis a uniformly distributed pure state onH⊗E.

Note : If|E|6|H|,ρis uniformly distributed on the set of states onHwith rank at most|E|.

Needed technical result (potentially of independent interest) :Estimating the typical value of E_FandE_Rfor random induced states(cf. Hayden/Leung/Winter).

LetρABbe a random state onA⊗B≡C^d⊗C^d, induced by some environmentE≡C^s, with Cd6^s6^d2. Then, for anyt>0,

P |E_F(ρAB)−logd|>t

6^{e−cd2t2/log2dandP}

E_R(ρAB)−logd² s

>t

6^e−c0st2.

Definitions and preliminary results

E_F(entanglement of formation) and E_R(relative entropy of entanglement) :

Entropy :S(ρ) =−Tr(ρlogρ). Relative entropy :D(ρkσ) =−Tr(ρ[logρ−logσ]). E_F(ρ_A:B) :=inf

∑ip_iS Tr_B|ψiihψi|AB

: ∑ip_i|ψiihψi|AB=ρAB . E_R(ρA:B) :=inf

D ρABkσ_AB

: σAB∈

S

Random induced states :System of interestH. EnvironmentE.

→Random mixed state model onH:ρ=TrE|ψihψ|, where|ψiis a uniformly distributed pure state onH⊗E.

Note : If|E|6|H|,ρis uniformly distributed on the set of states onHwith rank at most|E|. Needed technical result (potentially of independent interest) :Estimating the typical value of E_FandE_Rfor random induced states(cf. Hayden/Leung/Winter).

LetρABbe a random state onA⊗B≡C^d⊗C^d, induced by some environmentE≡C^s, with Cd6^s6^d2. Then, for anyt>0,

P |E_F(ρAB)−logd|>t

6^{e−cd2t2/log2dandP}

E_R(ρAB)−logd² s

>t

6^e−c0st2.

Non-monogamy for the entanglement of formation and the relative entropy of entanglement

Theorem

[

]

LetρABCbe a random state onA⊗B⊗C≡C^d⊗C^d⊗C^d, induced by some environment E≡C^s, withs'logd.

On the one hand,E_F(ρA:BC)6^logdandER(ρA:BC)6^logd.

While on the other hand, asdgrows, with probability going to 1 (exponentially), E_F(ρ_A:B),E_F(ρ_A:C) = (1−o(1))logdandE_R(ρ_A:B),E_R(ρ_A:C) = (1−o(1))logd.

Remark :Any valuezforE_A:BC(ρ_ABC)is indeed attainable, on systems of local dimension 2^z. Conclusion :E_FandE_Rare non-monogamous, in the most general sense. This feature even becomes generic for high-dimensional quantum states.

→Is this just a consequence of their subadditivity(Hastings, Vollbrecht/Werner)?

Non-monogamy for the entanglement of formation and the relative entropy of entanglement

Theorem

[

]

LetρABCbe a random state onA⊗B⊗C≡C^d⊗C^d⊗C^d, induced by some environment E≡C^s, withs'logd.

On the one hand,E_F(ρA:BC)6^logdandER(ρA:BC)6^logd.

While on the other hand, asdgrows, with probability going to 1 (exponentially), E_F(ρ_A:B),E_F(ρ_A:C) = (1−o(1))logdandE_R(ρ_A:B),E_R(ρ_A:C) = (1−o(1))logd.

Remark :Any valuezforE_A:BC(ρ_ABC)is indeed attainable, on systems of local dimension 2^z. Conclusion :E_FandE_Rare non-monogamous, in the most general sense. This feature even becomes generic for high-dimensional quantum states.

→Is this just a consequence of their subadditivity(Hastings, Vollbrecht/Werner)?

Outline

1 Non-monogamy for the entanglement of formation and the relative entropy of entanglement

2 Non-monogamy for a whole class of additive entanglement measures

3 Recovering monogamy with non-universal relations

Definition of the considered class of additive entanglement measures

Requirements on the considered entanglement measure E :

1 Normalization :For any stateρAB,EA:B(ρAB)6^min(log|A|,log|B|).

2 Lower boundedness on the anti-symmetric stateα:There existc,t>0 universal constants s.t.E_A:A0(αAA⁰)>^c/log^t|A|.

3 Additivity under tensor product :For any stateρAB,E_A^m_:B^m(ρ^⊗_AB^m) =m E_A:B(ρ_AB).

4 Linearity under locally orthogonal mixture :For any statesρAB,σABs.t. Tr(ρAσA) =0 and Tr(ρ_BσB) =0,E_A:B(λρ_AB+ (1−λ)σ_AB) =λE_A:B(ρ_AB) + (1−λ)E_A:B(σ_AB).

•Assumption(3)holds by construction for any regularized entanglement measure, i.e. one defined asE_A:B^∞ (ρ_AB) := lim

n→+∞ 1

nE_Aⁿ_:Bⁿ(ρ^⊗_ABⁿ).

•Assumption(2)is a faithfulness (or geometry-preserving) property : in 1-norm distanceαis dimension-independently separated from the set of separable states, so an entanglement measure which faithfully captures this geometrical feature should be dimension-independently (or weakly dimension-dependently) bounded away from 0 onα.

→Examples of entanglement measures fulfilling requirements(1−4): E_F^∞andE_R^∞, the regularized versions ofE_FandE_R.

Definition of the considered class of additive entanglement measures

Requirements on the considered entanglement measure E :

1 Normalization :For any stateρAB,EA:B(ρAB)6^min(log|A|,log|B|).

2 Lower boundedness on the anti-symmetric stateα:There existc,t>0 universal constants s.t.E_A:A0(αAA⁰)>^c/log^t|A|.

3 Additivity under tensor product :For any stateρAB,E_A^m_:B^m(ρ^⊗_AB^m) =m E_A:B(ρ_AB).

4 Linearity under locally orthogonal mixture :For any statesρAB,σABs.t. Tr(ρAσA) =0 and Tr(ρ_BσB) =0,E_A:B(λρ_AB+ (1−λ)σ_AB) =λE_A:B(ρ_AB) + (1−λ)E_A:B(σ_AB).

•Assumption(3)holds by construction for any regularized entanglement measure, i.e. one defined asE_A:B^∞ (ρ_AB) := lim

n→+∞

1

nE_Aⁿ_:Bⁿ(ρ^⊗_ABⁿ).

•Assumption(2)is a faithfulness (or geometry-preserving) property : in 1-norm distanceαis dimension-independently separated from the set of separable states, so an entanglement measure which faithfully captures this geometrical feature should be dimension-independently (or weakly dimension-dependently) bounded away from 0 onα.

Non-monogamy for a whole class of additive entanglement measures

Theorem

[

(

−

) ]

There exists a stateρABConA⊗B⊗C≡C^d⊗(C^d)^⊗2k⊗(C^d)^⊗2k, where 06^k6blogdc, s.t.

EA:B(ρAB),EA:C(ρAC)>(1−o(1))EA:BC(ρABC)asd→+∞.

Remark :By considering tensor products and mixtures, any valuezforEA:BC(ρ_ABC)is indeed attainable, on systems of suitably large local dimensions.

Conclusion :Additive entanglement measures may also be non-monogamous, in the most general sense, as soon as they are strongly faithful. Explicit construction of a counter-example, based on the anti-symmetric state.

→Can monogamy still be rescued in some way ? ! ?

Non-monogamy for a whole class of additive entanglement measures

Theorem

[

(

−

) ]

There exists a stateρABConA⊗B⊗C≡C^d⊗(C^d)^⊗2k⊗(C^d)^⊗2k, where 06^k6blogdc, s.t.

EA:B(ρAB),EA:C(ρAC)>(1−o(1))EA:BC(ρABC)asd→+∞.

Remark :By considering tensor products and mixtures, any valuezforEA:BC(ρ_ABC)is indeed attainable, on systems of suitably large local dimensions.

Conclusion :Additive entanglement measures may also be non-monogamous, in the most general sense, as soon as they are strongly faithful. Explicit construction of a counter-example, based on the anti-symmetric state.

→Can monogamy still be rescued in some way ? ! ?

Outline

1 Non-monogamy for the entanglement of formation and the relative entropy of entanglement

2 Non-monogamy for a whole class of additive entanglement measures

3 Recovering monogamy with non-universal relations

Dimension-dependent monogamy relations for non-(universally)-monogamous entanglement measures

Given an entanglement measureEand a tripartite systemA⊗B⊗C, does there exist a non-trivial functionf_A,B,C:R⁺×R⁺→R⁺s.t., for any stateρABConA⊗B⊗C,

E_A:BC(ρ_ABC)>^fA,B,C E_A:B(ρ_AB),E_A:C(ρ_AC)

?

Theorem

[

]

There exist universal constantsc,c⁰>0 s.t., for any stateρABConA⊗B⊗C≡C^d⊗C^d⊗C^d,

E_F(ρA:BC)>^max

E_F(ρA:B)+ ^c

d²log⁸dE_F(ρA:C)⁸,E_F(ρA:C)+ ^c

d²log⁸dE_F(ρA:B)⁸

, E_R^∞(ρ_A:BC)>^max

E_R^∞(ρ_A:B)+ ^c

0

d²log⁴dE_R^∞(ρ_A:C)⁴,E_R^∞(ρ_A:C)+ ^c

0

d²log⁴dE_R^∞(ρ_A:B)⁴

.

Remark :The proof is based on hybrid monogamy-type relations involving filtered through local measurements entanglement measures(Piani). Most likely, neither the dimensional pre-factors nor the powers 8 and 4 appearing in these relations are tight.

Conclusion :In any fixed finite dimension,E_FandE_R^∞can be regarded as monogamous. But the monogamy relations that they obey become trivial in the limit of infinite dimension.

Dimension-dependent monogamy relations for non-(universally)-monogamous entanglement measures

Given an entanglement measureEand a tripartite systemA⊗B⊗C, does there exist a non-trivial functionf_A,B,C:R⁺×R⁺→R⁺s.t., for any stateρABConA⊗B⊗C,

E_A:BC(ρ_ABC)>^fA,B,C E_A:B(ρ_AB),E_A:C(ρ_AC)

?

Theorem

[

]

There exist universal constantsc,c⁰>0 s.t., for any stateρ_ABConA⊗B⊗C≡C^d⊗C^d⊗C^d,

E_F(ρA:BC)>^max

E_F(ρA:B)+ ^c

d²log⁸dE_F(ρA:C)⁸,E_F(ρA:C)+ ^c

d²log⁸dE_F(ρA:B)⁸

, E_R^∞(ρ_A:BC)>^max

E_R^∞(ρ_A:B)+ ^c

0

d²log⁴dE_R^∞(ρ_A:C)⁴,E_R^∞(ρ_A:C)+ ^c

0

d²log⁴dE_R^∞(ρ_A:B)⁴

.

Remark :The proof is based on hybrid monogamy-type relations involving filtered through local measurements entanglement measures(Piani). Most likely, neither the dimensional pre-factors nor the powers 8 and 4 appearing in these relations are tight.

Conclusion :In any fixed finite dimension,E_FandE_R^∞can be regarded as monogamous. But the monogamy relations that they obey become trivial in the limit of infinite dimension.

Summary and perspectives

E_FandE_Rcannot satisfy any non-trivial monogamy relation.

In high dimensions, havingEA:BC'max(EA:B,EA:C)even becomes the typical behavior, forEbeing eitherE_ForE_R.

Additive entanglement measures which faithfully transcribe the distance of the anti-symmetric state to the set of separable states (e.g.E_F^∞andE_R^∞) cannot satisfy any non-trivial monogamy relation neither.

Nevertheless, some of these entanglement measures can be shown to obey non-trivial monogamy constraints on systems of any fixed dimensions.

Understand better the connections of the entanglement monogamy phenomenon with the quantum marginal problem and the question of quantum information

distribution/recoverability... ?

References

M. Christandl, R. König, G. Mitchison, R. Renner, “One-and-a-half quantum De Finetti theorems”.

M. Christandl, A. Winter, “Squashed entanglement : an additive entanglement measure”.

V. Coffman, J. Kundu, W.K. Wootters, “Distributed entanglement”.

A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, “A complete family of separability criteria”.

M.B. Hastings, “Superadditivity of communication capacity using entangled inputs”.

P. Hayden, D.W. Leung, A. Winter, “Aspects of generic entanglement”.

M. Koashi, A. Winter, “Monogamy of entanglement and other correlations”.

M. Piani, “Relative entropy of entanglement and restricted measurements”.

B. Terhal, “Is entanglement monogamous ?”.

K.G.H. Vollbrecht, R.F. Werner, “Entanglement measures under symmetry”.