High dimension and symmetries in quantum information theory

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High dimension and symmetries in quantum information theory Grande dimension et symétries en théorie quantique de l’information

Cécilia Lancien

Université Claude Bernard Lyon 1 & Universitat Autónoma de Barcelona

PhD defense - June 9^th2016

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Motivations

•One-particle quantum system :pure state=unit vector in a Hilbert space.

•Multi-particle quantum system :Hilbert space=tensor product of the individual ones.

→Dimension grows exponentially with the number of subsystems.

One way around this curse of dimensionality : make use of extra a priori information on the state of the considered system.

Indeed : invariances⇒reduction in the effective number of degrees of freedom.

Example :nindistinguishable particles→a pure state of such system is in the symmetric subspace of the global Hilbert space, whose dimension is polynomial, not exponential, inn. However, having to deal with high dimensional objects is not necessarily a misfortune : as the dimension grows, universal behaviours may emerge.

Two main goals served by asymptotic geometric analysis in quantum information theory :

•Determine some expected features of big quantum systems.

•Exhibit objects having a given property (paradigm : constructing the latter explicitly might be harder than asserting that suitable random ones will do with overwhelming probability). Two-sided objective in this thesis :

How to reduce the study of large dimensional situations to that of lower dimensional ones ? What typical aspects arise precisely as the dimension of the studied object grows ?

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Motivations

Example :nindistinguishable particles→a pure state of such system is in the symmetric subspace of the global Hilbert space, whose dimension is polynomial, not exponential, inn.

However, having to deal with high dimensional objects is not necessarily a misfortune : as the dimension grows, universal behaviours may emerge.

•Exhibit objects having a given property (paradigm : constructing the latter explicitly might be harder than asserting that suitable random ones will do with overwhelming probability). Two-sided objective in this thesis :

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Motivations

•Exhibit objects having a given property (paradigm : constructing the latter explicitly might be harder than asserting that suitable random ones will do with overwhelming probability).

Two-sided objective in this thesis :

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Motivations

•Exhibit objects having a given property (paradigm : constructing the latter explicitly might be harder than asserting that suitable random ones will do with overwhelming probability).

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The mathematical formalism of quantum physics in a nutshell

State of a quantum system :Positive and trace 1 operatorρ(density operator) on a complex Hilbert spaceH(state space).

Here : finite (but large) dimensional systems→H≡Cⁿwithn1.

Separability vs Entanglement :A bipartite quantum stateρ_ABonA⊗Bisseparableif it may be written as a convex combination of product states, i.e.ρAB=∑xp_xσ^x_A⊗τ^x_B. Otherwise, it isentangled.

Measurement on a quantum system :Positive operatorsM= (M_i)_i_∈_Isumming to the identity operatorIdonH(Positive Operator-Valued Measure, POVM).

→If the system is in stateρbefore the measurement, the latter yields outcomei∈Iwith probability Tr(M_iρ).

Evolution of a quantum system :Completely Positive and Trace Preserving (CPTP) map

N

^:ρ7→Tr_E(VρV^∗)for some isometryV:H,→E⊗H⁰(quantum channel).

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Outline

1 Complexity reduction in quantum information theory

2 Some aspects of generic entanglement in high dimension

3 Making use of permutation-symmetry to tackle multiplicativity issues

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Outline

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General setting

Natural wonder :Given an ideal process, with many degrees of freedom, is it possible to approximate it by a more realistic one, i.e. one which can be described with a reasonable number of parameters ?

→By just allowing some small error, can we execute a task which potentially requires a lot of resources with much less resources ?

Two examples of such problems :

Compressing a quantum channel into one with an environment as small as possible. Sparsifying a quantum measurement into one with as few outcomes as possible. In both cases, closeness between original and approximating transformations is defined by : closeness between their two outputs, whatever the input.

→Goal :Propose universal (and optimal) ways of achieving that... based on random constructions.

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General setting

Natural wonder :Given an ideal process, with many degrees of freedom, is it possible to approximate it by a more realistic one, i.e. one which can be described with a reasonable number of parameters ?

→By just allowing some small error, can we execute a task which potentially requires a lot of resources with much less resources ?

Two examples of such problems :

Compressing a quantum channel into one with an environment as small as possible.

Sparsifying a quantum measurement into one with as few outcomes as possible.

In both cases, closeness between original and approximating transformations is defined by : closeness between their two outputs, whatever the input.

→Goal :Propose universal (and optimal) ways of achieving that... based on random constructions.

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Outline

Sparsification of quantum measurements and approximating zonoids by zonotopes

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State discrimination and distinguishability norms

State discrimination problem :System that can be in two quantum states,ρorσ, with equal prior probabilities 1/2.

Task :Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVMM= (M_i)i∈Iperformed on it.

Minimal probability of error (applying “maximum likelihood rule”) :

P_error=¹ 2 1−¹

2

∑

i∈I

Tr

M_i(ρ−σ)

!

=:¹ 2

1−¹

2kρ−σkM

.

→Distinguishability normk · kMassociated with the POVMM(Matthews/Wehner/Winter).

Goal :Given a POVMMwith potentially many outcomes, find a POVMM⁰with a minimal number of outcomes, which still achieves almost the same distinguishing performance asMon any pair of states (i.e. is s.t.(1−ε)k · kM6k · kM⁰6(1+ε)k · kM).

From a convex geometry viewpoint :LetM= (M_i)i∈Ibe a POVM onC^d.

•Associated norm : for any Hermitian∆onC^d,k∆kM:=∑i∈I

Tr M_i∆ .

•Associated convex body :K_Mdual of the unit ball fork · kM.

Observation :For two POVMsM,N,ck · kM6k · kN6^Ck · kM ⇔ c K_M⊂K_N⊂C K_M.

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State discrimination and distinguishability norms

P_error=¹ 2 1−¹

2

∑

i∈I

Tr

M_i(ρ−σ)

!

=:¹ 2

1−¹

2kρ−σkM

.

Tr M_i∆ .

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State discrimination and distinguishability norms

P_error=¹ 2 1−¹

2

∑

i∈I

Tr

M_i(ρ−σ)

!

=:¹ 2

1−¹

2kρ−σkM

.

Tr M_i∆ .

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Sparsifying any POVM

Question :What “kind” of convex bodies are associated to POVMs ? Proposition

Mis a POVM onC^diffK_Mis a symmetric zonoid in the set of Hermitians onC^dsatisfying

±Id∈KM⊂[−Id,Id].

Zonoid :convex body which can be approximated by zonotopes.

Zonotope :convex body which can be written as a finite Minkowski sum of segments.

Cube inR³:[−u₁,u₁] + [−u₂,u₂] + [−u₃,u₃] u₃

u₂ u₁

Approximating zonoids by zonotopes :K⊂Rⁿa symmetric zonoid. There exists a zonotope Zwhich is the sum ofC_εnlognsegments s.t.(1−ε)K⊂Z⊂(1+ε)K (Talagrand).

Corollary

For any POVMMonC^d and any 0<ε<1, there exists a sub-POVMM⁰with less than C_εd²logdoutcomes s.t.(1−ε)k · kM6k · kM⁰6(1+ε)k · kM.

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Sparsifying any POVM

Question :What “kind” of convex bodies are associated to POVMs ? Proposition

Mis a POVM onC^diffK_Mis a symmetric zonoid in the set of Hermitians onC^dsatisfying

±Id∈KM⊂[−Id,Id].

Zonoid :convex body which can be approximated by zonotopes.

Zonotope :convex body which can be written as a finite Minkowski sum of segments.

Cube inR³:[−u₁,u₁] + [−u₂,u₂] + [−u₃,u₃] u₃

u₂ u₁

Approximating zonoids by zonotopes :K⊂Rⁿa symmetric zonoid. There exists a zonotope Zwhich is the sum ofC_εnlognsegments s.t.(1−ε)K⊂Z⊂(1+ε)K (Talagrand).

Corollary

For any POVMMonC^d and any 0<ε<1, there exists a sub-POVMM⁰with less than C_εd²logdoutcomes s.t.(1−ε)k · kM6k · kM⁰6(1+ε)k · kM.

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Refinements in the case of the uniform POVM

Most symmetric POVM : continuous POVMUwhere all rank-1 projectors play the same role.

→“Most efficient” POVM :k · kUis dimension-independently equivalent to a Euclidean norm.

Strategy to sparsifyU:P₁, . . . ,P_nrank-1 projectors onC^d, drawn independently and uniformly. SetS:=∑ⁿ_k₌₁PkandP:= Pek:=S⁻¹^/²PkS⁻¹^/²

16k6n. Pis a random POVM onC^d withnoutcomes s.t.k∆kP= ∑ⁿ

k=1

Tr(eP_k∆) . Theorem

Let 0<ε<1. There existc_ε,C_ε>0 s.t. ifn>^Cεd², then with probability greater than 1−e⁻^c^ε^d, (1−ε)k · kU6k · kP6(1+ε)k · kU.

Remark :Optimal dimensional order of magnitude becausek · kPis not a true norm ifPhas less thand²outcomes.

Main steps in the proof :

•Large deviation probability of ^d_n∑ⁿ_k₌₁

Tr(P_k∆)

fromk∆kU?

•Large deviation probability of∑ⁿ_k=1

Tr(eP_k∆)

from^d_n∑ⁿ_k=1

Tr(P_k∆) ?

→Twice : (i) Individual deviation probability : Bernstein-type tail bound for sums of i.i.d. centered sub-exponential r.v.’s (ii) Individual to global deviation probability : Net argument.

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Refinements in the case of the uniform POVM

Strategy to sparsifyU:P₁, . . . ,P_nrank-1 projectors onC^d, drawn independently and uniformly.

SetS:=∑ⁿ_k₌₁P_kandP:= Pe_k:=S⁻¹^/²P_kS⁻¹^/²

k=1

•Large deviation probability of ^d_n∑ⁿ_k₌₁

Tr(P_k∆)

fromk∆kU?

•Large deviation probability of∑ⁿ_k=1

Tr(eP_k∆)

from^d_n∑ⁿ_k=1

Tr(P_k∆) ?

→Twice : (i) Individual deviation probability : Bernstein-type tail bound for sums of i.i.d. centered sub-exponential r.v.’s (ii) Individual to global deviation probability : Net argument.

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Refinements in the case of the uniform POVM

Strategy to sparsifyU:P₁, . . . ,P_nrank-1 projectors onC^d, drawn independently and uniformly.

SetS:=∑ⁿ_k₌₁P_kandP:= Pe_k:=S⁻¹^/²P_kS⁻¹^/²

k=1

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Some open problems

Universal constructions to reduce complexity :optimal in the most symmetric situations but maybe not in less balanced ones.

→What about more adaptive solutions that would take into account the specific geometry of the process to compress ?

Random constructions to reduce complexity :require a lot of resources.

→What about full or partial derandomization ?

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Outline

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General setting

If a compound system is in a separable global state, there are no “intrinsically quantum”

correlations between its local constituents.

→The dichotomy entangled vs (close to) separable is crucial in quantum physics.

Problem :Deciding between one or the other is known to be, in general, a hard task.

Solution :Find necessary conditions for separability that have a simple mathematical description and that may be checked efficiently on a computer (e.g. by a semi-definite programme).

Kind of problems that one may consider :

How do various levels of locality constraints typically affect observers’ ability to distinguish global states ?

How well do easily implementable tests for detecting entanglement typically perform ? General answer :As the dimensions of the subsystems grow, any too simple necessary condition for separability is doomed to be very rough.

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General setting

If a compound system is in a separable global state, there are no “intrinsically quantum”

correlations between its local constituents.

→The dichotomy entangled vs (close to) separable is crucial in quantum physics.

Problem :Deciding between one or the other is known to be, in general, a hard task.

Solution :Find necessary conditions for separability that have a simple mathematical description and that may be checked efficiently on a computer (e.g. by a semi-definite programme).

Kind of problems that one may consider :

How do various levels of locality constraints typically affect observers’ ability to distinguish global states ?

How well do easily implementable tests for detecting entanglement typically perform ?

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Outline

2 Some aspects of generic entanglement in high dimension k-extendibility of high dimensional bipartite quantum states

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The

k

-extendibility criterion for separability

Definition[k-extendibility]

Letk∈N. A stateρABonA⊗Bisk-extendible if there exists a stateρ_AB^k onA⊗B^⊗^kwhich is invariant under permutation of theBsubsystems and s.t.ρAB=Tr_Bk−1

ρ_AB^k .

Theorem[NSC for separability(Doherty/Parrilo/Spedalieri)]

On a bipartite Hilbert spaceA⊗B, a state is separable iff it isk-extendible for allk∈N.

Remarks :

•“ρABseparable⇒ρABk-extendible for allk” : obvious sinceσA⊗τB=Tr_Bk−1

σA⊗τ^⊗_B^k .

•“ρABk-extendible for allk⇒ρABseparable” : can be shown thanks to the quantum de Finetti theorem(Christandl/König/Mitchison/Renner), and can also be seen as a consequence of monogamy of entanglement(Brandao/Christandl/Yard).

•ρAB k-extendible⇒ ρAB k⁰-extendible fork⁰6^k.

→Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point (but one cannot tell when a priori).

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The

k

-extendibility criterion for separability

Definition[k-extendibility]

Letk∈N. A stateρABonA⊗Bisk-extendible if there exists a stateρ_AB^k onA⊗B^⊗^kwhich is invariant under permutation of theBsubsystems and s.t.ρAB=Tr_Bk−1

ρ_AB^k .

Theorem[NSC for separability(Doherty/Parrilo/Spedalieri)]

On a bipartite Hilbert spaceA⊗B, a state is separable iff it isk-extendible for allk∈N.

Remarks :

•“ρABseparable⇒ρABk-extendible for allk” : obvious sinceσA⊗τB=Tr_Bk−1

σA⊗τ^⊗_B^k .

•“ρABk-extendible for allk⇒ρABseparable” : can be shown thanks to the quantum de Finetti theorem(Christandl/König/Mitchison/Renner), and can also be seen as a consequence of monogamy of entanglement(Brandao/Christandl/Yard).

•ρAB k-extendible⇒ ρAB k⁰-extendible fork⁰6^k.

→Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point (but one cannot tell when a priori).

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Quantifying the strength of the

k

-extendibility criterion

Problem :For a fixedk∈N, how rough is the approximation of separability byk-extendibility ? Known :There exist states which arek-extendible, and nevertheless “very” entangled (i.e. far away from the set of separable states in some standard or operational distance measure).

→Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours ?

Two possible quantitative strategies :

Estimate the average size of the set of states being eitherk-extendible or indeed separable.

→Information on how much bigger than the separable set thek-extendible set is.

Characterize when certain random states are with high probability either nonk-extendible or indeed entangled.

→Information on how powerful thek-extendibility test is to detect entanglement.

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Quantifying the strength of the

k

-extendibility criterion

Problem :For a fixedk∈N, how rough is the approximation of separability byk-extendibility ? Known :There exist states which arek-extendible, and nevertheless “very” entangled (i.e. far away from the set of separable states in some standard or operational distance measure).

→Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours ?

Two possible quantitative strategies :

Estimate the average size of the set of states being eitherk-extendible or indeed separable.

→Information on how much bigger than the separable set thek-extendible set is.

Characterize when certain random states are with high probability either nonk-extendible or indeed entangled.

→Information on how powerful thek-extendibility test is to detect entanglement.

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Random matrix theory interlude : asymptotic spectrum of GUE matrices

Definitions[Gaussian Unitary Ensemble and semi-circular distributions]

•n×nmatrix from the Gaussian Unitary Ensemble (GUE) :G= (H+H^∗)/√

2 withHan×n matrix having independent complex normal entries.

•Centered semi-circular distribution with variance parameterσ²:

dµ_SC_(σ²₎(x) = ¹ 2πσ²

p4σ²−x²1_[−₂_σ,₂_σ](x)dx.

Density ofµ_SC₍₁₎: For any Hermitian M on Cⁿ, define its spectral distribution as

N_M=¹ n

n

∑

i=1

δ_λ_i(M).

Wigner’s semi-circle law : if (G_n)_n∈N is

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Mean width of a set of states

Definitions

LetKbe a convex set of states onCⁿcontainingId/n(maximally mixed state).

•For∆an×nHermitian s.t.k∆k₂=1, thewidth of K in the direction∆is w(K,∆) =sup_σ∈_KTr(∆(σ−Id/n)).

•Themean width of Kis the average ofw(K,·)over the Hilbert-Schmidt unit sphere ofn×n Hermitians, equipped with the uniform probability measure.

It is equivalently defined asw(K) =Ew(K,G)/γ_n, whereGis an×nGUE matrix and γn=EkGk2∼n→+∞n.

→The mean width of a set of states is a certain measure of its size (for any “reasonable”K,w(K)'vrad(K), where vrad(K)is the volume radiusofK, i.e. the radius of the Euclidean ball with same volume asK).

Observation : Esup_σ_stateTr(G(σ−Id/n)) =Eλmax(G). So by Wigner’s semi-circle law, the mean width of the set of all states onCⁿis asymptotically 2√

n/γ_n, i.e. 2/√ n.

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Mean width of a set of states

Definitions

LetKbe a convex set of states onCⁿcontainingId/n(maximally mixed state).

•For∆an×nHermitian s.t.k∆k₂=1, thewidth of K in the direction∆is w(K,∆) =sup_σ∈_KTr(∆(σ−Id/n)).

•Themean width of Kis the average ofw(K,·)over the Hilbert-Schmidt unit sphere ofn×n Hermitians, equipped with the uniform probability measure.

It is equivalently defined asw(K) =Ew(K,G)/γ_n, whereGis an×nGUE matrix and γn=EkGk2∼n→+∞n.

→The mean width of a set of states is a certain measure of its size (for any “reasonable”K,w(K)'vrad(K), where vrad(K)is the volume radiusofK, i.e. the radius of the Euclidean ball with same volume asK).

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Mean widths of the set of separable vs

k

-extendible states

Theorem

Fixk∈N. OnC^d⊗C^d, the set

E

kofk-extendible states satisfies w(

E

k) ∼

d→+∞

√2 k

1 d.

In comparison :OnC^d⊗C^d, the mean width of the set of separable states is of order 1/d³^/²

(Aubrun/Szarek), hence much smaller than the mean widths of the sets of all or ofk-extendible states, which are both of order 1/d.

→On high dimensional bipartite systems, the set of separable states forms a very small fraction of the set of all states, and the set ofk-extendible states is a very rough approximation of it.

Partial generalization to the case wherekgrows withd:k&^dis needed so thatw(

E

k)'w(

S

⁾^.

•ExpressEsup_σk-ext stateTr(G(σ−Id/d²))asEλmax Ge

, for some “modified” GUE matrixG.e

•EstimateEλmax Ge

by computing allp-order momentsETrGe^p, and thus identifying the limiting spectral distribution. It is (after rescaling byd) a centered semi-circular distributionµ_SC₍₁_/_k₎, whose support has 2/√

kas upper edge (+ extreme eigenvalues are indeed not isolated).

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Mean widths of the set of separable vs

k

-extendible states

Theorem

Fixk∈N. OnC^d⊗C^d, the set

E

kofk-extendible states satisfies w(

E

k) ∼

d→+∞

√2 k

1 d.

In comparison :OnC^d⊗C^d, the mean width of the set of separable states is of order 1/d³^/²

(Aubrun/Szarek), hence much smaller than the mean widths of the sets of all or ofk-extendible states, which are both of order 1/d.

→On high dimensional bipartite systems, the set of separable states forms a very small fraction of the set of all states, and the set ofk-extendible states is a very rough approximation of it.

Partial generalization to the case wherekgrows withd:k&^dis needed so thatw(

E

k)'w(

S

⁾^.

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Outline

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General setting

Ubiquitous issue in quantum information theory : determine whether certain quantities have a multiplicative behaviour.

Perfect multiplicativity :no way of combining two copies of a device which would allow performing better than when using them independently.

Not what usually happens : clever use of correlations does help.

→Asymptotic performances are much harder to quantify than single-copy ones.

However, whenever the study of the multi-copy case can be reduced to that of i.i.d. cases, the analysis becomes easy again.

→Motivation behind de Finetti type statements : use the permutation-invariance of the considered problem to do so.

Types of problems which may be tackled in that way :

Multiplicative behaviour under tensoring of the support function of sets of quantum states. Exponential decay under parallel repetition of the winning probability in multi-player non-local games.

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General setting

Ubiquitous issue in quantum information theory : determine whether certain quantities have a multiplicative behaviour.

Perfect multiplicativity :no way of combining two copies of a device which would allow performing better than when using them independently.

Not what usually happens : clever use of correlations does help.

→Asymptotic performances are much harder to quantify than single-copy ones.

However, whenever the study of the multi-copy case can be reduced to that of i.i.d. cases, the analysis becomes easy again.

→Motivation behind de Finetti type statements : use the permutation-invariance of the considered problem to do so.

Types of problems which may be tackled in that way :

Multiplicative behaviour under tensoring of the support function of sets of quantum states.

Exponential decay under parallel repetition of the winning probability in multi-player non-local games.

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Outline

3 Making use of permutation-symmetry to tackle multiplicativity issues Flexible constrained de Finetti reductions

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De Finetti reductions

Motivation :Reduce the study of a permutation-invariant scenario to that of i.i.d. ones.

Observation :In several applications, being able to upper bound a permutation-invariant object by product ones is enough.

Theorem[Universal de Finetti reduction for quantum states(Christandl/König/Renner)] Letρ⁽ⁿ⁾be a permutation-invariant state on(C^d)^⊗ⁿ. Then,

ρ⁽ⁿ⁾6(n+1)^d² Z

σ

σ^⊗ⁿdµ(σ), µ: uniform p.d. over the set of states onC^d.

Drawback :All permutation-invariant states are upper bounded by the same mixture of tensor power states.→Any other information is lost.

Theorem[Flexible de Finetti reduction for quantum states] Letρ⁽ⁿ⁾be a permutation-invariant state on(C^d)^⊗ⁿ. Then,

ρ⁽ⁿ⁾6(n+1)^3d² Z

σ F

ρ⁽ⁿ⁾,σ^⊗ⁿ2

Fidelity between statesρ,σ:F(ρ,σ) =k√ ρ√

σk161, with equality iffρ=σ.

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De Finetti reductions

Motivation :Reduce the study of a permutation-invariant scenario to that of i.i.d. ones.

Observation :In several applications, being able to upper bound a permutation-invariant object by product ones is enough.

Theorem[Universal de Finetti reduction for quantum states(Christandl/König/Renner)] Letρ⁽ⁿ⁾be a permutation-invariant state on(C^d)^⊗ⁿ. Then,

ρ⁽ⁿ⁾6(n+1)^d² Z

σ

Drawback :All permutation-invariant states are upper bounded by the same mixture of tensor power states.→Any other information is lost.

Theorem[Flexible de Finetti reduction for quantum states] Letρ⁽ⁿ⁾be a permutation-invariant state on(C^d)^⊗ⁿ. Then,

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What is the flexible de Finetti reduction good for ?

ρ⁽ⁿ⁾6^poly(n) Z

σ F

σ^⊗ⁿdµ(σ)

State-dependent upper bound : amongst statesσ^⊗ⁿ, only those which have a high fidelity with the state of interestρ⁽ⁿ⁾get an important weight.

→Useful when one knows thatρ⁽ⁿ⁾satisfies some additional property : only statesσ^⊗ⁿ approximately satisfying this same property should have a non-negligible fidelity weight.

Example :ρ⁽ⁿ⁾fixed point of

N

^⊗ⁿ^{, for}

N

a quantum-classical channel.

→Derive a classical flexible d.F. reduction from the quantum flexible d.F. reduction. Corollary[Flexible de Finetti reduction for probability distributions]

Let

X

be finite andP⁽ⁿ⁾be a permutation-invariant p.d. on

X

ⁿ. Then, P⁽ⁿ⁾6^poly(n)

Z

Q

F

P⁽ⁿ⁾,Q^⊗ⁿ 2

Q^⊗ⁿdQ, µ: p.d. over the set of p.d.’s on

X

^.

Utility :Apply it to the optimal strategy of players in a non-local game to show that, if they can win 1 instance of a game with probability at most 1−δ, then they can win simultaneouslyninstances of this game with probability at most(1−δ⁰)ⁿ.

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What is the flexible de Finetti reduction good for ?

ρ⁽ⁿ⁾6^poly(n) Z

σ F

σ^⊗ⁿdµ(σ)

State-dependent upper bound : amongst statesσ^⊗ⁿ, only those which have a high fidelity with the state of interestρ⁽ⁿ⁾get an important weight.

→Useful when one knows thatρ⁽ⁿ⁾satisfies some additional property : only statesσ^⊗ⁿ approximately satisfying this same property should have a non-negligible fidelity weight.

Example :ρ⁽ⁿ⁾fixed point of

N

^⊗ⁿ^{, for}

N

a quantum-classical channel.

→Derive a classical flexible d.F. reduction from the quantum flexible d.F. reduction.

Corollary[Flexible de Finetti reduction for probability distributions] Let

X

be finite andP⁽ⁿ⁾be a permutation-invariant p.d. on

X

ⁿ. Then,

P⁽ⁿ⁾6^poly(n) Z

Q

F

P⁽ⁿ⁾,Q^⊗ⁿ 2

Q^⊗ⁿdQ, µ: p.d. over the set of p.d.’s on

X

^.

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Another approach to multiplicativity issues

An important example amongst multiplicativity questions :

Dimension-independent multiplicative behaviour under tensoring of the support function of the set of separable states ?

To show such result, a more “local” approach than the “global” de Finetti reduction one is possible : successive conditioning and bound on the disturbance it induces.

To make it go through, we would be in need of a “magical” measure of entanglement.

Indeed, it would have to combine features which usually exclude one another, namely monogamy-type and strong faithfulness properties(Adesso/Di Martino/Huber/Lancien/Piani/Winter).