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 de Mathématiques de Toulouse & CNRS

Mathematical Quantum Physics & Algebra seminar, Innsbruck – March 25 2021

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Motivations

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

ρ=

r

∑

k=1

λkρ¹_k⊗ · · · ⊗ρ^m_k,with (

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

−→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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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, . . . ,Ambe Banach spaces. Givenx∈A1⊗ · · · ⊗Am, itsprojective tensor normis kxkA1⊗π···⊗πAm:= inf

( r

∑

k=1

|αk|:aⁱ_k∈A_i,kaⁱ_kk6¹,x=

r

∑

k=1

αka¹_k⊗ · · · ⊗a^m_k,r∈N )

,

and itsinjective tensor normis

kxkA1⊗ε···⊗εAm:= sup

b¹⊗ · · · ⊗b^m|x

:bⁱ∈A^∗_i,kbⁱk6¹ .

These norms are dual to one another: for allx∈A₁⊗ · · · ⊗A_m, kxk_A₁_⊗_π_···⊗_π_A_m= sup

hy|xi:kyk_A^∗

1⊗ε···⊗εA^∗_m6¹ , kxk_A₁⊗ε···⊗εAm= sup

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗m6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_m∈A_m, ka₁⊗ · · · ⊗a_mkA1⊗π···⊗πAm=ka₁⊗ · · · ⊗a_mkA1⊗ε···⊗εAm=ka₁k · · · ka_mk. And they are extremal among such norms: for any other tensor normk · konA1⊗ · · · ⊗Am,

k · k_A₁_⊗_ε_···⊗_ε_A_m6k · k6k · k_A₁_⊗_π_···⊗_π_A_m.

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Tensor norms in Banach spaces

( r

∑

k=1

r

∑

k=1

αka¹_k⊗ · · · ⊗a^m_k,r∈N )

,

b¹⊗ · · · ⊗b^m|x

:bⁱ∈A^∗_i,kbⁱk6¹ . These norms are dual to one another: for allx∈A₁⊗ · · · ⊗A_m,

kxk_A₁_⊗_π_···⊗_π_A_m= sup

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗m6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_m∈A_m, ka₁⊗ · · · ⊗a_mkA1⊗π···⊗πAm=ka₁⊗ · · · ⊗a_mkA1⊗ε···⊗εAm=ka₁k · · · ka_mk. And they are extremal among such norms: for any other tensor normk · konA1⊗ · · · ⊗Am,

k · k_A₁_⊗_ε_···⊗_ε_A_m6k · k6k · k_A₁_⊗_π_···⊗_π_A_m.

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Tensor norms in Banach spaces

( r

∑

k=1

r

∑

k=1

αka¹_k⊗ · · · ⊗a^m_k,r∈N )

,

b¹⊗ · · · ⊗b^m|x

:bⁱ∈A^∗_i,kbⁱk6¹ . These norms are dual to one another: for allx∈A₁⊗ · · · ⊗A_m,

kxk_A₁_⊗_π_···⊗_π_A_m= sup

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗m6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_m∈A_m, ka₁⊗ · · · ⊗a_mkA1⊗π···⊗πAm=ka₁⊗ · · · ⊗a_mkA1⊗ε···⊗εAm=ka₁k · · · ka_mk.

And they are extremal among such norms: for any other tensor normk · konA₁⊗ · · · ⊗A_m, k · k_A₁_⊗_ε_···⊗_ε_A_m6k · k6k · k_A₁_⊗_π_···⊗_π_A_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 stateψ∈C^d¹⊗ · · · ⊗C^d^mis s.t.kψk_`^d1···dm

2

=1.

And it is separable iffkψk_`^d1

2⊗ε···⊗ε`^dm₂ =kψk_`^d1

2⊗π···⊗π`^dm₂ =1, where by definition:

kψk_`^d1

2⊗ε···⊗ε`^dm₂ := supn

ϕ¹⊗ · · · ⊗ϕ^m|ψ

:ϕⁱ∈C^dⁱ,kϕⁱk_`di 2

=1 o

, kψk_`d1

2⊗π···⊗π`^dm₂ := inf _r

∑

k=1

|α_k|:χⁱ_k∈C^dⁱ,kχⁱ_kk_`di 2

=1,ψ= ∑^r

k=1

αkχ¹_k⊗ · · · ⊗χ^m_k

.

−→Geometric measure of entanglement(Shimony, Wei/Goldbart, Zhu/Chen/Hayashi): G(ψ) :=−log supn

|

ϕ¹⊗ · · · ⊗ϕ^m|ψ

|²:ϕⁱ∈C^dⁱ,kϕⁱk_`di 2

=1 o

=−2logkψk_`d1

2⊗ε···⊗ε`^dm₂ .

•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. And it is separable iffkρk_S^d1

1⊗π···⊗πS^dm₁ =1(Rudolph, Pérez-García), where by definition: kρkS1^d¹⊗π···⊗πS^dm1

:= inf _r

∑

k=1

|α_k|:τⁱ_k∈

M

di(C),kτⁱ_kk

S^di1

=1,ρ= ∑^r

k=1

αkτ¹_k⊗ · · · ⊗τ^m_k

.

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Characterizing entanglement through tensor norms

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^m.

2

=1.

2⊗ε···⊗ε`^dm₂ =kψk_`^d1

kψk_`^d1

2⊗ε···⊗ε`^dm₂ := supn

ϕ¹⊗ · · · ⊗ϕ^m|ψ

=1 o

, kψk_`d1

2⊗π···⊗π`^dm₂ := inf _r

∑

k=1

=1,ψ= ∑^r

k=1

αkχ¹_k⊗ · · · ⊗χ^m_k

.

|

ϕ¹⊗ · · · ⊗ϕ^m|ψ

=1 o

=−2logkψk_`d1

2⊗ε···⊗ε`^dm₂ .

•Mixed state entanglement: Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^m.

Notation:∀X∈

M

d(C),kXk_Sd

M

d1(C)⊗ · · · ⊗

M

S^d1¹^···^dm

=1. And it is separable iffkρk_S^d1

1⊗π···⊗πS^dm₁ =1(Rudolph, Pérez-García), where by definition: kρkS1^d¹⊗π···⊗πS^dm1

:= inf _r

∑

k=1

|α_k|:τⁱ_k∈

M

di(C),kτⁱ_kk

S^di1

=1,ρ= ∑^r

k=1

αkτ¹_k⊗ · · · ⊗τ^m_k

.

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Characterizing entanglement through tensor norms

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^m.

2

=1.

2⊗ε···⊗ε`^dm₂ =kψk_`^d1

kψk_`^d1

2⊗ε···⊗ε`^dm₂ := supn

ϕ¹⊗ · · · ⊗ϕ^m|ψ

=1 o

, kψk_`d1

2⊗π···⊗π`^dm₂ := inf _r

∑

k=1

=1,ψ= ∑^r

k=1

αkχ¹_k⊗ · · · ⊗χ^m_k

.

|

ϕ¹⊗ · · · ⊗ϕ^m|ψ

=1 o

=−2logkψk_`d1

2⊗ε···⊗ε`^dm₂ .

•Mixed state entanglement:

Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^m.

Notation:∀X∈

M

d(C),kXk_Sd

M

d1(C)⊗ · · · ⊗

M

S^d1¹^···^dm

=1.

And it is separable iffkρk_S^d1

1⊗π···⊗πS^dm₁ =1(Rudolph, Pérez-García), where by definition:

kρkS1^d¹⊗π···⊗πS^dm1

:= inf _r

∑

k=1

|α_k|:τⁱ_k∈

M

di(C),kτⁱ_kk

S^di1

=1,ρ= ∑^r

k=1

αkτ¹_k⊗ · · · ⊗τ^m_k

.

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Plan

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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.tk

E

k_Sd

1→`ⁿ₂:= maxn

k

E

(X)k_`ⁿ

2 :kXk_Sd 1=1

o

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

Observation:k

E

k_S^d

1→`ⁿ₂= max n

(Tr(T_E^∗X⊗X^∗))¹^/² :kXk_S^d

1 =1 o

=1.

T_E:= ∑ⁿ

k=1

E_k⊗E_k^∗,test operatorassociated to

E

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^d1¹⊗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

S1^d¹⊗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).

Observation:`ⁿ₂⊗_π`ⁿ₂≡S₁ⁿ. Identification|xi=

n

∑

k,l=1

x_kl|kli ∈Cⁿ⊗Cⁿ↔X=

n

∑

k,l=1

x_kl|kihl| ∈

M

n(C), s.t.kxk`ⁿ₂⊗π`ⁿ₂=kXk_Sⁿ₁ (Schmidt decomposition↔singular value decomposition).

•Bipartite case: Checking if testers

E

^,

F

^:

M

d(C)→Cⁿdetect the entanglement of a stateρon C^d⊗C^dconsists in computing the`ⁿ₂⊗_π`ⁿ₂norm of

E

⊗

F

(ρ)∈Cⁿ⊗Cⁿ, i.e. itsS₁ⁿnorm 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(S^d₁)^⊗^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

n

∑

k,l=1

n

∑

k,l=1

x_kl|kihl| ∈

M

E

^,

F

^:

M

E

⊗

F

M

n(C).

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Practical interest of tester-based entanglement criteria

n

∑

k,l=1

n

∑

k,l=1

x_kl|kihl| ∈

M

E

^,

F

^:

M

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 .

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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 .

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Realignment and SIC POVM entanglement criteria in the tester framework

•Let

R

^:^X^∈

M

d(C)7→

d

∑

k,k⁰=1

hk⁰|X|ki|kk⁰i ∈C^d²be the matrix unit tester.

For any stateρonC^d⊗C^d, we have:

R

^⊗²^{(ρ) =} ∑^d

k,k⁰,l,l⁰=1

hk⁰l⁰|ρ|kli|kk⁰i|ll⁰i. Seen as a matrix, this isρ^R, the realignment ofρ.

ρ^R

ρ^R = ρ

−→Computing the`^d₂²⊗_π`^d₂²norm of

R

^⊗²(ρ)is computing theS₁^d²norm ofρ^R. The tester criterion based on

R

is therealignment criterion(Chen/Wu, Rudolph).

•Let

S

:X∈

M

d(C)7→ rd+1

2d

d²

∑

k=1

hx_k|X|x_ki|ki ∈C^d²be the 2-design tester.

S

^⊗²^{(ρ) =}^d_2d⁺¹ ^d

2

∑

k,l=1

hx_kx_l|ρ|x_kx_li|ki|li. Seen as a matrix, this isρ^S, the SIC POVM transformation ofρ.

S

^⊗²^(ρ)is computing theS^d₁²norm ofρ^S. The tester criterion based on

S

^{is the}SIC POVM criterion(Shang/Asadian/Zhu/Gühne).

Interest:The realignment and SIC POVM criteria are originally defined only for bipartite states. Viewing them inside the tester framework allows for an immediate multipartite generalization.

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Realignment and SIC POVM entanglement criteria in the tester framework

•Let

R

^:^X^∈

M

d(C)7→

d

∑

k,k⁰=1

R

^⊗²^{(ρ) =} ∑^d

k,k⁰,l,l⁰=1

ρ^R

ρ^R = ρ

R

•Let

S

:X∈

M

d(C)7→

rd+1 2d

d²

∑

k=1

S

^⊗²^{(ρ) =}^d_2d⁺¹ ^d

2

∑

k,l=1

S

Interest:The realignment and SIC POVM criteria are originally defined only for bipartite states. Viewing them inside the tester framework allows for an immediate multipartite generalization.

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Realignment and SIC POVM entanglement criteria in the tester framework

•Let

R

^:^X^∈

M

d(C)7→

d

∑

k,k⁰=1

R

^⊗²^{(ρ) =} ∑^d

k,k⁰,l,l⁰=1

ρ^R

ρ^R = ρ

R

•Let

S

:X∈

M

d(C)7→

rd+1 2d

d²

∑

k=1

S

^⊗²^{(ρ) =}^d_2d⁺¹ ^d

2

∑

k,l=1

S

Interest:The realignment and SIC POVM criteria are originally defined only for bipartite states.

Viewing them inside the tester framework allows for an immediate multipartite generalization.

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Plan

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Detecting entanglement of bipartite pure states with symmetric testers

A tester

E

^:

M

d(C)→Cⁿis calledsymmetricif there existα,β∈Rs.t.T_E=αF+βI.

Fact:The set of admissible pairs(α,β)is{(α,1−α) :06α6¹} ∪ {(1,β) :−1/d6β6⁰}. Examples:

R

^and

S

are of this form, with(α,β) = (1,0)and(α,β) = (1/2,1/2), respectively.

[Note: A symmetric tester necessarily hasn>^d²^{. So}

R

^and

S

are ‘minimal’ symmetric testers.]

Theorem (Performance of symmetric testers on bipartite pure states)

Let

E

^:

M

d(C)→Cⁿbe a symmetric tester, with admissible parameters(α,β). For any pure stateϕ∈C^d⊗C^d, with Schmidt decomposition|ϕi=∑^r_k₌₁

√

λk|e_kf_ki, we have k

E

^⊗²(|ϕihϕ|)k`ⁿ₂⊗π`ⁿ₂=α+β+2α

∑

16k<l6r

p λkλl.

In particular: k

R

^⊗²^(|ϕihϕ|)k_`d2

2⊗π`^d₂²=1+2 ∑

16k<l6r

√

λkλlandk

S

^⊗²^(|ϕihϕ|)k_`d2

2⊗π`^d₂²=1+ ∑

16k<l6r

√ λkλl. Sok

R

^⊗²(|ϕihϕ|)k_`d2

2⊗π`^d₂²>k

S

^⊗²(|ϕihϕ|)k_`d2

2 ⊗π`^d₂²>1, with strict inequalities wheneverr>1.

−→

R

^and

S

detect all entangled pure states, and

R

always performs ‘better’ at it than

S

^.

In fact:k

R

^⊗²(|ϕihϕ|)k_`d2

2⊗π`^d₂²=k|ϕihϕ|k_S^d

1⊗πS^d₁. So

R

performs ‘perfectly’.