Multipartite entanglement detection via projective tensor norms

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Multipartite entanglement detection via projective tensor norms

Based on joint work with Maria Anastasia Jivulescu and Ion Nechita, available at arXiv:2010.06365

Cécilia Lancien

Institut Fourier Grenoble & CNRS

Workshop GOQI, MFO – October 7 2021

Cécilia Lancien Multipartite entanglement detection via projective tensor norms Workshop GOQI, MFO – October 7 2021 1

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Motivations

A stateρonC^d¹⊗ · · · ⊗C^d^mis calledseparableif it is a convex combination of product states:

positive semidefinite operator with trace 1 onC^d¹⊗ · · · ⊗C^d^m

ρ=

r

∑

k=1

λkρ^k₁⊗ · · · ⊗ρ^k_m,with (

λk>⁰,16^k6^r,∑^r_k=1λk=1 ρ^k_i state onC^dⁱ,16^k6^r,16ⁱ6^m . Otherwise it is calledentangled.

[Note: Ifρis a pure state, i.e.ρ=|ϕihϕ|for some unit vectorϕ∈C^d¹⊗ · · · ⊗C^d^m, thenρis separable iffϕ=ϕ1⊗ · · · ⊗ϕmfor some unit vectorsϕi∈C^dⁱ, 16ⁱ6^m.]

−→If a quantum system is in a separable state, there is no intrinsically quantum correlation between its subsystems. It thus does not provide any advantage over a classical system in information processing tasks.

Known:The problem of deciding whether a given multipartite quantum state is entangled or separable (and even just approximate versions of it) is computationally hard(Gharibian).

−→Solution in practice: Look for necessary conditions to separability, which are easier to check than separability itself, akaentanglement criteria.

Our goal:Design and study a class of entanglement criteria, based on the idea of applyinglocal contractionsto an input multipartite state and computing a suitabletensor normof the output.

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Motivations

ρ=

r

∑

k=1

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Motivations

ρ=

r

∑

k=1

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Motivations

ρ=

r

∑

k=1

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Plan

1 Tensor norms and entanglement

2 Detecting entanglement with testers: definitions and first examples

3 Entanglement testers in the bipartite setting

4 Entanglement testers in the multipartite setting

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Tensor norms in Banach spaces

LetA₁, . . . ,A_mbe Banach spaces. Givenx∈A₁⊗ · · · ⊗A_m, itsprojective tensor normis kxk_A₁⊗π···⊗πAm:= inf

( _r

∑

k=1

|α_k|:a^k_i ∈A_i,ka^k_ik_A_i 6¹,x=

r

∑

k=1

α_ka^k₁⊗ · · · ⊗a^k_m,r∈N )

,

and itsinjective tensor normis kxkA1⊗ε···⊗εAm:= sup

|hb₁⊗ · · · ⊗b_m|xi|:b_i∈A^∗_i,kb_ik_A^∗_i 6¹ .

These norms are dual to one another: for allx∈A₁⊗ · · · ⊗A_m, kxkA1⊗π···⊗πAm= sup

|hy|xi|:kyk_A^∗

1⊗ε···⊗εA^∗m6¹ , kxkA1⊗ε···⊗εAm= sup

|hy|xi|:kyk_A^∗₁⊗π···⊗πA^∗m6¹ .

The projective and injective norms are examples oftensor norms: for alla1∈A1, . . . ,am∈Am, ka₁⊗ · · · ⊗a_mk_A₁⊗π···⊗πAm=ka₁⊗ · · · ⊗a_mk_A₁⊗ε···⊗εAm=ka₁k_A₁· · · ka_mk_A_m. And they are extremal among such norms: for any other tensor normk · konA₁⊗ · · · ⊗A_m,

k · kA1⊗ε···⊗εAm6k · k6k · kA1⊗π···⊗πAm.

Note: The unit ball fork · k_A₁⊗π···⊗πAmisconv{a₁⊗ · · · ⊗a_m:a_i∈A_i,ka_ik_A_i 6¹}.

projective tensor productof the unit balls fork · kA_i

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Tensor norms in Banach spaces

( _r

∑

k=1

r

∑

k=1

α_ka^k₁⊗ · · · ⊗a^k_m,r∈N )

,

|hb₁⊗ · · · ⊗b_m|xi|:b_i∈A^∗_i,kb_ik_A^∗_i 6¹ . These norms are dual to one another: for allx∈A1⊗ · · · ⊗Am,

kxk_A₁⊗π···⊗πAm= sup

|hy|xi|:kyk_A^∗

|hy|xi|:kyk_A^∗₁⊗π···⊗πA^∗m6¹ .

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Tensor norms in Banach spaces

( _r

∑

k=1

r

∑

k=1

α_ka^k₁⊗ · · · ⊗a^k_m,r∈N )

,

|hy|xi|:kyk_A^∗

|hy|xi|:kyk_A^∗₁⊗π···⊗πA^∗m6¹ .

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Tensor norms in Banach spaces

( _r

∑

k=1

r

∑

k=1

α_ka^k₁⊗ · · · ⊗a^k_m,r∈N )

,

|hy|xi|:kyk_A^∗

|hy|xi|:kyk_A^∗₁⊗π···⊗πA^∗m6¹ .

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Characterizing entanglement through tensor norms

•Pure state entanglement:

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^m.

Notation:∀x∈C^d,kxk_`^d

2:= ∑^d_k₌₁|x_k|²1/2

vector 2-norm A pure stateϕ∈C^d¹⊗ · · · ⊗C^d^mis s.t.kϕk_`^d1···dm

2

=1.

Sincek · k_`d1···dm 2

is a tensor norm, this implieskϕk_`d1

2⊗ε···⊗ε`^dm₂ 6^{1 and}kϕk_`d1

2⊗π···⊗π`^dm₂ >^1.

Andϕis separable iffkϕk_`d1

2⊗ε···⊗ε`^dm₂ =kϕk_`d1

2⊗π···⊗π`^dm₂ =1, where kϕk_`^d1

2⊗ε···⊗ε`^dm₂ := supn

|hχ1⊗ · · · ⊗χm|ϕi|:χi∈C^dⁱ,kχik_`di 2

=1 o

, kϕk_`^d1

2⊗π···⊗π`^dm₂ := inf _r

∑

k=1

|αk|:φ^k_i ∈C^dⁱ,kφ^k_ik_`di 2

=1,ϕ=

r

∑

k=1

αkφ^k₁⊗ · · · ⊗φ^k_m

.

•Mixed state entanglement: Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^m.

Notation:∀X∈

M

d(C),kXk_Sd

1:= Tr|X|

matrix 1-norm A mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dm(C)is s.t.ρ>^{0 and}kρk

S^d1¹^···^dm

=1. Sincek · k_S^d1···dm

1

is a tensor norm, this implieskρk_S^d1

1⊗π···⊗πS₁^dm >^1. Andρis separable iffkρk_S^d1

1⊗π···⊗πS₁^dm=1(Rudolph, Pérez-García), where kρk_S^d1

1 ⊗π···⊗πS^dm₁ := inf _r

∑

k=1

|αk|:τ^k_i ∈

M

di(C),kτ^k_ik_Sdi 1

=1,ρ= ∑^r

k=1

αkτ^k₁⊗ · · · ⊗τ^k_m

.

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Characterizing entanglement through tensor norms

•Pure state entanglement:

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^m.

Notation:∀x∈C^d,kxk_`^d

2:= ∑^d_k₌₁|x_k|²1/2

vector 2-norm A pure stateϕ∈C^d¹⊗ · · · ⊗C^d^mis s.t.kϕk_`^d1···dm

2

=1.

Sincek · k_`d1···dm 2

is a tensor norm, this implieskϕk_`d1

2⊗ε···⊗ε`^dm₂ 6^{1 and}kϕk_`d1

2⊗π···⊗π`^dm₂ >^1.

Andϕis separable iffkϕk_`d1

2⊗ε···⊗ε`^dm₂ =kϕk_`d1

2⊗π···⊗π`^dm₂ =1, where kϕk_`^d1

2⊗ε···⊗ε`^dm₂ := supn

|hχ1⊗ · · · ⊗χm|ϕi|:χi∈C^dⁱ,kχik_`di 2

=1 o

, kϕk_`^d1

2⊗π···⊗π`^dm₂ := inf _r

∑

k=1

|αk|:φ^k_i ∈C^dⁱ,kφ^k_ik_`di 2

=1,ϕ=

r

∑

k=1

αkφ^k₁⊗ · · · ⊗φ^k_m

.

•Mixed state entanglement:

Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^m.

Notation:∀X∈

M

d(C),kXk_Sd

1:= Tr|X|

matrix 1-norm A mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dm(C)is s.t.ρ>^{0 and}kρk

S^d1¹^···^dm

=1.

Sincek · k_S^d1···dm 1

is a tensor norm, this implieskρk_S^d1

1⊗π···⊗πS₁^dm >^1.

Andρis separable iffkρk_S^d1

1⊗π···⊗πS₁^dm=1(Rudolph, Pérez-García), where kρk_S^d1

1 ⊗π···⊗πS^dm₁ := inf _r

∑

k=1

|αk|:τ^k_i ∈

M

di(C),kτ^k_ik_Sdi 1

=1,ρ= ∑^r

k=1

αkτ^k₁⊗ · · · ⊗τ^k_m

.

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Quantifying entanglement through tensor norms (1)

Ifkϕk_`^d1

2⊗ε···⊗ε`^dm₂ 1 orkϕk_`^d1

2⊗π···⊗π`^dm₂ 1, thenϕis ‘very’ entangled.

Ifkρk_S^d1

1 ⊗π···⊗πS^dm₁ 1, thenρis ‘very’ entangled.

Question:Can this be made quantitative?

Definition [Geometric measure of entanglement(Shimony, Wei/Goldbart, Zhu/Chen/Hayashi)] G(ϕ) :=−log sup

n

| hχ1⊗ · · · ⊗χm|ϕi |²:χi∈C^dⁱ,kχik_`di 2

=1 o

=−2logkϕk_`^d1

2⊗ε···⊗ε`^dm₂ . G(ϕ) =0 iffϕis separable. How large canG(ϕ)be forϕentangled?

Gis afaithful entanglement measurefor pure states

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Quantifying entanglement through tensor norms (1)

Ifkϕk_`^d1

2⊗ε···⊗ε`^dm₂ 1 orkϕk_`^d1

2⊗π···⊗π`^dm₂ 1, thenϕis ‘very’ entangled.

Ifkρk_S^d1

1 ⊗π···⊗πS^dm₁ 1, thenρis ‘very’ entangled.

Question:Can this be made quantitative?

Definition [Geometric measure of entanglement(Shimony, Wei/Goldbart, Zhu/Chen/Hayashi)] G(ϕ) :=−log supn

| hχ1⊗ · · · ⊗χm|ϕi |²:χi∈C^dⁱ,kχik_`di 2

=1o

=−2logkϕk_`^d1

2⊗ε···⊗ε`^dm₂ . G(ϕ) =0 iffϕis separable. How large canG(ϕ)be forϕentangled?

Gis afaithful entanglement measurefor pure states

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Quantifying entanglement through tensor norms (2)

Observation:`^d₂⊗_ε`^d₂≡S^d_∞and`^d₂⊗_π`^d₂≡S₁^d.

Indeed, identifying|xi=∑^d_k,l=1x_kl|kli ∈C^d⊗C^d withX=∑^d_k,l=1x_kl|kihl| ∈

M

d(C), the Schmidt decompositionofxcorresponds to thesingular value decompositionofX:

|xi=

r

∑

k=1

p

λk|e_kf_ki ←→X=

r

∑

k=1

p

λk|e_kihf_k|,withr6^d^and (

λk>0,16^k6^r

{e_k}^r_k₌₁,{f_k}^r_k₌₁o.n.f. inC^d . This identification preserves the Euclidean norm:kxk_`d2

2

= (∑^r_k=1λk)¹^/²=kXk_S^d

2. Whilekxk_`^d

2⊗ε`^d₂= max₁₆_k₆_r√

λk=kXk_S^d

∞andkxk_`^d

2⊗π`^d₂=∑^r_k=1

√

λk=kXk_S^d

1.

−→Ifkxk_`d2 2

=1, then^√¹

d 6kxk_`d

2⊗ε`^d₂6^{1 and 1}6kxk_`d

2⊗π`^d₂6√

d(tight bounds).

More generally:Assume thatd₁6· · ·6^dmand setD:=d₁× · · · ×d_m₋₁.

•For any pure stateϕ∈C^d¹⊗ · · · ⊗C^d^m,kϕk_`d1

2⊗ε···⊗ε`^dm₂ >^√¹_D^andkϕk_`d1

2⊗π···⊗π`^dm₂ 6√ D. Proof idea:Recursive argument from bipartite case.

•For any mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dm(C),kρk_S^d1

1⊗π···⊗πS₁^dm6^D. Proof idea:Pure states are extremal andk|ϕihϕ|k_S^d1

1⊗π···⊗πS₁^dm=kϕk²

`^d₂¹⊗π···⊗π`^dm₂ .

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Quantifying entanglement through tensor norms (2)

Observation:`^d₂⊗_ε`^d₂≡S^d_∞and`^d₂⊗_π`^d₂≡S₁^d.

Indeed, identifying|xi=∑^d_k,l=1x_kl|kli ∈C^d⊗C^d withX=∑^d_k,l=1x_kl|kihl| ∈

M

d(C), the Schmidt decompositionofxcorresponds to thesingular value decompositionofX:

|xi=

r

∑

k=1

p

λk|e_kf_ki ←→X=

r

∑

k=1

p

λk|e_kihf_k|,withr6^d^and (

λk>0,16^k6^r

{e_k}^r_k₌₁,{f_k}^r_k₌₁o.n.f. inC^d . This identification preserves the Euclidean norm:kxk_`d2

2

= (∑^r_k=1λk)¹^/²=kXk_S^d

2. Whilekxk_`^d

2⊗ε`^d₂= max₁₆_k₆_r√

λk=kXk_S^d

∞andkxk_`^d

2⊗π`^d₂=∑^r_k=1

√

λk=kXk_S^d

1.

−→Ifkxk_`d2 2

=1, then^√¹

d 6kxk_`d

2⊗ε`^d₂6^{1 and 1}6kxk_`d

2⊗π`^d₂6√

d(tight bounds).

More generally:Assume thatd₁6· · ·6^dmand setD:=d₁× · · · ×d_m₋₁.

•For any pure stateϕ∈C^d¹⊗ · · · ⊗C^d^m,kϕk_`d1

2⊗ε···⊗ε`^dm₂ >^√¹_D^andkϕk_`d1

2⊗π···⊗π`^dm₂ 6√ D.

Proof idea:Recursive argument from bipartite case.

•For any mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dm(C),kρk_S^d1

1⊗π···⊗πS₁^dm6^D.

Proof idea:Pure states are extremal andk|ϕihϕ|k_S^d1

1⊗π···⊗πS₁^dm=kϕk²

`^d₂¹⊗π···⊗π`^dm₂ .

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Plan

1 Tensor norms and entanglement

2 Detecting entanglement with testers: definitions and first examples

3 Entanglement testers in the bipartite setting

4 Entanglement testers in the multipartite setting

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Entanglement testers

LetE₁, . . . ,E_n∈

M

d(C)and let{|1i, . . . ,|ni}be an o.n.b. ofCⁿ. Define:

E

:X∈

M

d(C)7→

n

∑

k=1

Tr(E_k^∗X)|ki ∈Cⁿ. If

E

^{is s.t}k

E

k_S^d

1→`ⁿ₂:= max n

k

E

(X)k`ⁿ₂ :kXk_S^d

1=1 o

=1, we call it an(entanglement) tester.

Observation:LetT_E:= ∑ⁿ

k=1

E_k⊗E_k^∗be thetest operatorassociated to

E

^.

Then,k

E

k_Sd

1→`ⁿ₂= maxn

(Tr(T_E^∗X⊗X^∗))¹^/²:kXk_Sd 1=1

o .

A tester

E

^:

M

d(C)→Cⁿ: E

E X

Its associated test operatorT_E:C^d⊗C^d→C^d⊗C^d:

E E^∗

TE =

4 3

2 1

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Detecting entanglement with testers

Theorem [Multipartite entanglement criterion based on testers]

Let

E

i:

M

di(C)→Cⁿⁱ, 16ⁱ6m, be testers. For anyX∈

M

d1(C)⊗ · · · ⊗

M

dm(C), we have k

E

1⊗ · · · ⊗

E

m(X)k_`ⁿ1

2⊗π···⊗π`^nm₂ 6kXk_S^d1

1⊗π···⊗πS₁^dm. Hence, for any stateρonC^d¹⊗ · · · ⊗C^d^m,

k

E

1⊗ · · · ⊗

E

m(ρ)k_`ⁿ1

2⊗π···⊗π`^nm₂ >1 =⇒ kρk_S^d1

1 ⊗π···⊗πS^dm₁ >1 ⇐⇒ ρentangled. Proof idea:k

E

1⊗ · · · ⊗

E

mk_S^d1

1⊗π···⊗πS₁^dm→`ⁿ₂¹⊗π···⊗π`^nm₂ =k

E

1k_S^d1

1→`ⁿ₂¹· · · k

E

mk_Sdm

1 →`^nm₂ =1.

factorization property The action of testers

E

i:

M

di(C)→Cⁿⁱ, 16ⁱ6^m,

on a stateρ∈

M

d1(C)⊗ · · · ⊗

M

dm(C):

ρ d1 d1 d2 d2

dm dm E1

E2

Em n1

n2

nm

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Practical interest of tester-based entanglement criteria

Entanglement criterion based on reducing the study of mixed state entanglement (checking if an (S^d₁)^⊗^π^mnorm is>1) to that of pure state entanglement (checking if an(`ⁿ₂)^⊗^π^mnorm is>1).

•Bipartite case:Checking if testers

E

^,

F

^:

M

d(C)→Cⁿdetect the entanglement of a stateρ onC^d⊗C^dconsists in computing the`ⁿ₂⊗π`ⁿ₂norm of

E

⊗

F

^(ρ)∈Cⁿ⊗Cⁿ, i.e. itsSⁿ₁norm if seen as an element of

M

n(C).

−→This is much easier than computing theS₁^d⊗_πS₁^dnorm ofρ.

•Multipartite case:The computation of an(S₁^d)^⊗^π^mnorm, i.e. of an(`^d₂⊗_π`^d₂)^⊗^π^m≡(`^d₂)^⊗^π^2m norm, is reduced to the computation of an(`ⁿ₂)^⊗^π^mnorm. associativity of⊗_π

−→Reduction by a factor 2 of the number of factors (at the cost of potentially increasing the dimension of each of them fromdton>^d...)

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Practical interest of tester-based entanglement criteria

E

^,

F

^:

M

d(C)→Cⁿdetect the entanglement of a stateρ onC^d⊗C^d consists in computing the`ⁿ₂⊗π`ⁿ₂norm of

E

⊗

F

M

n(C).

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Practical interest of tester-based entanglement criteria

E

^,

F

^:

M

d(C)→Cⁿdetect the entanglement of a stateρ onC^d⊗C^d consists in computing the`ⁿ₂⊗π`ⁿ₂norm of

E

⊗

F

M

n(C).

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Important examples of testers

•Maps defined from matrix bases:

Let{G₁, . . . ,G_d2}be an o.n.b. of

M

d(C)and define:

G

:X∈

M

d(C)7→

d²

∑

k=1

Tr(G^∗_kX)|ki ∈C^d². Clearlyk

G

(X)k_`d2

2

=kXk_S^d

26kXk_S^d

1. So

G

is indeed a tester.

Example:

R

^:^X∈

M

d(C)7→

d

∑

k,k⁰=1

Tr(R^∗_kk0X)|kk⁰i ∈C^d², whereR_kk⁰:=|kihk⁰|, 16^k,k⁰6^d.

Associated test operator:T_R=

d

∑

k,k⁰=1

R_kk⁰⊗R_kk^∗0=

d

∑

k,k⁰=1

|kihk⁰| ⊗ |k⁰ihk|=F.

•Maps defined from vector 2-designs:

Let{|x₁i, . . . ,|x_d2i}be a symmetric spherical 2-design ofC^d, i.e._d¹2

d²

∑

k=1

|x_kihx_k|^⊗²=_d₍^I_d⁺₊^F₁₎.

SetS_k:= qd+1

2d |x_kihx_k|, 16^k6^d², and define:

S

^:^X∈

M

d(C)7→

d²

∑

k=1

Tr(S_k^∗X)|ki ∈C^d². We havek

S

⁽^X)k_`d2

2

= ¹₂(|TrX|²

| {z }

6kXk²

Sd1

+ Tr|X|²

| {z }

=kXk²

Sd2

6kXk²

Sd1

)1/2

6kXk_S^d

1. So

S

is indeed a tester.

Associated test operator:T_S=

d²

∑

k=1

S_k⊗S_k^∗=^d+1 2d

d²

∑

k=1

|x_kihx_k|^⊗²=^I+F 2 .