Characterizing expansion, classically and quantumly

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Characterizing expansion, classically and quantumly

Perspective on “Quasirandom quantum channels”

T. Bannink, J. Briët, F. Labib, H. Maassen

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

QIT Journal Club – 18 Novembre 2020

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Motivations

A graph isexpandingif it is simultaneouslysparse(i.e. ‘economical’) andhighly connected (i.e. modeling a network which is robust against perturbations, converges fast to equilibrium etc).

−→Important structural feature in many contexts.

Question:Given a regular graph, even if it is sparse, how ‘close’ is it to the complete graph from an ‘operational’ point of view?

Obvious properties of the complete graph:

1 fast mixing of a random walk supported on it,

2 uniformity of edge density between subsets of vertices.

−→Is measuring closeness to the complete graph in the sense of property (1) or (2) equivalent? Known:

Random regular graphs exhibit both (1) and (2) with high probability.

For any graph, (1) implies (2)(Hoory/Linial/Widgerson). The opposite implication holds only for dense graphs(Chung/Graham/Wilson)or vertex-transitive sparse graphs(Conlon/Zhao).

−→For those, the two notions of ‘quasi-randomness’ defined by (1) and (2) are equivalent. What about a quantum analogue of the problem?

1 Walk on a regular graph→Evolution under iterative application of a unital quantum channel.

2 Graph with uniform edge density→Quantum channel with uniform output directions.

−→Are these two ‘random looking’ properties of unital quantum channels in fact equivalent?

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Motivations

−→Is measuring closeness to the complete graph in the sense of property (1) or (2) equivalent?

Known:

−→For those, the two notions of ‘quasi-randomness’ defined by (1) and (2) are equivalent. What about a quantum analogue of the problem?

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Motivations

Known:

−→For those, the two notions of ‘quasi-randomness’ defined by (1) and (2) are equivalent.

What about a quantum analogue of the problem?

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Motivations

Known:

−→For those, the two notions of ‘quasi-randomness’ defined by (1) and (2) are equivalent.

What about a quantum analogue of the problem?

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Spectral expansion and uniformity for regular graphs

G= (V,E)ad-regular graph, with vertex setVand edge setE(dedges at each vertex).

Aits normalized adjacency matrix, i.e. the|V| × |V|matrix s.t.A_uv=e(u,v)

| {z }

number of edges between verticesuandv

/dfor eachu,v∈V.

•Property (1) is related tospectral expansion:

λ1(A),λ2(A), . . .eigenvalues ofA, with|λ₁(A)|>|λ₂(A)|>· · ·. SinceGis regular,λ1(A) =1 (all-one vector as associated eigenvector), and the spectral expansion parameter ofGis

λ(G) :=|λ2(A)|; the smallestλ(G),the most spectrally expandingG.

•Property (2) is calleduniformity, and is an edge expansion property: The uniformity parameterε(G)ofGis the smallestεs.t.

∀S,T⊂V,

∑

u∈S,v∈T

A_uv−|S||T|

|V|

6ε|V|; the smallestε(G),the most uniformG.

−→Two ways of defining and quantifying expansion of a regular graphG: spectral perspective (throughλ(G)) and geometrical perspective (throughε(G)).

Known:If a graphGis spectrally expanding (i.e. ifλ(G)is small), then it is necessarily geometrically expanding (i.e.ε(G)is small as well). The converse (i.e.ε(G)small impliesλ(G) small) is known to hold only for some classes of graphs.

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Spectral expansion and uniformity for regular graphs

| {z }

/dfor eachu,v∈V.

•Property (2) is calleduniformity, and is an edge expansion property: The uniformity parameterε(G)ofGis the smallestεs.t.

∀S,T⊂V,

∑

u∈S,v∈T

A_uv−|S||T|

|V|

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Spectral expansion and uniformity for regular graphs

| {z }

/dfor eachu,v∈V.

•Property (2) is calleduniformity, and is an edge expansion property:

The uniformity parameterε(G)ofGis the smallestεs.t.

∀S,T⊂V,

∑

u∈S,v∈T

A_uv−|S||T|

|V|

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Spectral expansion and uniformity for regular graphs

| {z }

/dfor eachu,v∈V.

∀S,T⊂V,

∑

u∈S,v∈T

A_uv−|S||T|

|V|

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Spectral expansion and uniformity for regular graphs

| {z }

/dfor eachu,v∈V.

∀S,T⊂V,

∑

u∈S,v∈T

A_uv−|S||T|

|V|

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Quantum analogues

Adjacency matrixAof a graph: transition matrix (maps probability vectors to probability vectors).

−→Quantum analogue: quantum channelΦ(maps density operators to density operators).

Regularity condition onG:Aleaves the all-one vector invariant.

−→Quantum analogue:Φleaves the identity matrix invariant, i.e.Φis unital.

Φa unital quantum channel, acting on operators onCⁿ. It hasspectral expansionparameter λ(Φ) :=|λ2(Φ)|,

anduniformityparameterε(Φ), the smallestεs.t.

∀V,W ⊂Cⁿ,

Tr(P_VΦ(P_W))−Tr(P_V)Tr(P_W) n

6εn.

•Quantum spectral expansion: first introduced to study entanglement in quantum many-body 1D systems(Hastings)and complexity of quantum entropy estimation(Ben-Aroya/Schwartz/Ta-Shma). Since then, great interest forquantum expandersin various fields: randomness reduction, quantum cryptography, quantum error-correction, quantum many-body physics etc.

•Quantum uniformity: formalized only recently(Bannink/Briët/Labib/Maassen).

−→Proving equivalence between these two notions of expansion could open the way to new ways of constructing quantum expanders.

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Quantum analogues

∀V,W ⊂Cⁿ,

6εn.

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Quantum analogues

∀V,W ⊂Cⁿ,

6εn.

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Tensor norms on commutative and non-commutative Banach spaces

•Given a vectorx∈Cⁿ, its (normalized)`_pnorm is:kxk`p:= ¹ n

n

∑

i=1

|x_i|^p

!1/p

. Given a linear operatorM:Cⁿ→Cⁿ, its`_p→`_qnorm is:

kMk_`_p_→`_q:= max 1

n|hy|M|xi|,x,y∈Cⁿ,kxk_`_p=1,kyk_`

q0 =1

. And the cut norm ofMis:kMkcut:= max

₁

n|hy|M|xi|,x,y∈ {0,1}ⁿ

.

•Given a matrixX∈

M

n(C), its (normalized)S_pnorm is:kXkSp:= 1

nTr(|X|^p) 1/p

. Given a linear operatorΨ :

M

n(C)→

M

n(C), itsS_p→S_qnorm is:

kΨkSp→Sq:= max 1

n|Tr(Y^∗Ψ(X))|,X,Y∈

M

n(C),kXkSp=1,kYkS_q0 =1

. And the cut norm ofΨiskΨkcut:= max

1

n|Tr(Y^∗Ψ(X))|,X,Yprojectors onCⁿ

. Fact:The commutative, resp. non-commutative, cut-norm is equivalent to the`_∞→`₁, resp.S_∞→S₁, norm, with domination constants independent from the dimension. Specifically:

∀M:Cⁿ→Cⁿ,kMkcut6kMk`∞→`16π²kMkcut(Conlon/Zhao).

∀Ψ :

M

n(C)→

M

n(C),kΨk_cut6kΨk_S_∞_→_S₁6π²kΨk_cut(Bannink/Briët/Labib/Maassen).

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Tensor norms on commutative and non-commutative Banach spaces

n

∑

i=1

|x_i|^p

!1/p

q0 =1

₁

.

M

n(C), its (normalized)S_pnorm is:kXkSp:=

1 nTr(|X|^p)

1/p

M

n(C)→

M

kΨkSp→Sq:= max 1

M

1

.

Fact:The commutative, resp. non-commutative, cut-norm is equivalent to the`_∞→`₁, resp.S_∞→S₁, norm, with domination constants independent from the dimension. Specifically:

∀Ψ :

M

n(C)→

M

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Tensor norms on commutative and non-commutative Banach spaces

n

∑

i=1

|x_i|^p

!1/p

q0 =1

₁

.

M

n(C), its (normalized)S_pnorm is:kXkSp:=

1 nTr(|X|^p)

1/p

M

n(C)→

M

kΨkSp→Sq:= max 1

M

1

. Fact:The commutative, resp. non-commutative, cut-norm is equivalent to the`_∞→`₁, resp.S_∞→S₁, norm, with domination constants independent from the dimension. Specifically:

∀Ψ :

M

n(C)→

M

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Proof of the inequality relating the cut norm and the

S_∞

→

S1

norm

•Since projectors haveS_∞norm equal to 1, we have:

max 1

n|Tr(Y^∗Ψ(X))|,kXkS∞=kYkS∞=1

>

1

n|Tr(Y^∗Ψ(X))|,X,Yprojectors

. By definition, this means thatkΨkS∞→S1>kΨkcut.

•The set of matrices withS_∞norm at most 1 is the convex hull of unitary matrices. So there exist unitariesX,Ys.t.kΨk_S_∞_→_S₁=¹

n|Tr(Y^∗Ψ(X))|. WriteX=UAU^∗andY=VBV^∗, withA,Bdiagonal andU,Vunitary. For anyw∈Cs.t.|w|=1, setA⁰(w)ii=1_{_R₍_A

iiw¯)},B⁰(w)ii=1_{_R₍_B

iiw¯)}, 16ⁱ6^n. Observe that, for anyz∈Cs.t.|z|=1,z=πE_w_∈_C_,|_w_|=₁

w1_{_R₍_z_w_¯_)}

. Hence,X=πE_u(u UA⁰(u)U^∗)andY=πE_v(v VB⁰(v)V^∗).

Now,UA⁰(u)U^∗andVB⁰(u)V^∗are projectors for allu,v∈Cs.t.|u|=|v|=1, and therefore kΨk_S_∞→S1=¹

nTr(Y^∗Ψ(X))

=π² n

E_u_,_v

¯

v uTr(V^∗B⁰(v)^∗VΨ(UA⁰(u)U^∗))

6π² nE_u_,_v

Tr(V^∗B⁰(v)^∗VΨ(UA⁰(u)U^∗))

6π²kΨk_cut

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Proof of the inequality relating the cut norm and the

S_∞

→

S1

norm

•Since projectors haveS_∞norm equal to 1, we have:

max 1

n|Tr(Y^∗Ψ(X))|,kXkS∞=kYkS∞=1

>

1

n|Tr(Y^∗Ψ(X))|,X,Yprojectors

. By definition, this means thatkΨkS∞→S1>kΨkcut.

•The set of matrices withS_∞norm at most 1 is the convex hull of unitary matrices.

So there exist unitariesX,Ys.t.kΨk_S_∞_→_S₁=¹

n|Tr(Y^∗Ψ(X))|. WriteX=UAU^∗andY=VBV^∗, withA,Bdiagonal andU,Vunitary.

For anyw∈Cs.t.|w|=1, setA⁰(w)ii=1_{_R₍_A

iiw¯)},B⁰(w)ii=1_{_R₍_B

iiw¯)}, 16ⁱ6^n.

Observe that, for anyz∈Cs.t.|z|=1,z=πE_w_∈_C_,|_w_|=₁

w1_{_R₍_z_w_¯_)}

. Hence,X=πE_u(u UA⁰(u)U^∗)andY=πE_v(v VB⁰(v)V^∗).

Now,UA⁰(u)U^∗andVB⁰(u)V^∗are projectors for allu,v∈Cs.t.|u|=|v|=1, and therefore kΨk_S_∞→S1=¹

nTr(Y^∗Ψ(X))

=π² n E_u_,_v

¯

v uTr(V^∗B⁰(v)^∗VΨ(UA⁰(u)U^∗))

6π² nE_u_,_v

Tr(V^∗B⁰(v)^∗VΨ(UA⁰(u)U^∗))

6π²kΨk_cut

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Characterizing spectral expansion and uniformity through tensor norms

Observation:The parametersλandεcan be re-expressed in terms oftensor norms.

•For any regular graphGonnvertices, with adjacency matrixA, λ(G) =kA−Jk_`

2→`2 andε(G) =kA−Jk_cut,

whereJis the adjacency matrix of the complete graph, i.e. the matrix with all entries equal to 1/n.

•For any unital quantum channelΦonn×nmatrices, λ(Φ) =kΦ−Πk_S

2→S2 andε(Φ) =kΦ−Πk_cut, whereΠis the completely randomizing quantum channel, i.e.Π :X7→Tr(X)I/n.

−→Comparing parametersλandε, for graphs or quantum channels, boils down to comparing

`₂→`₂and`_∞→`₁norms orS₂→S₂andS_∞→S₁norms.

Fact:By ordering of the`pandS_pnorms,k · k`2→`2>k · k`∞→`1andk · kS2→S2>k · kS∞→S1. So for any regular graphGand unital quantum channelΦ, spectral expansion implies uniformity:

λ(G) =kA−Jk_`

2→`2>kA−Jk_`

∞→`1>kA−Jk_cut=ε(G), λ(Φ) =kΦ−Πk_S

2→S2>kΦ−Πk_S

∞→S1>kΦ−Πk_cut=ε(Φ).

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Characterizing spectral expansion and uniformity through tensor norms

λ(G) =kA−Jk_`

2→`2>kA−Jk_`

2→S2>kΦ−Πk_S

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Characterizing spectral expansion and uniformity through tensor norms

λ(G) =kA−Jk_`

2→`2>kA−Jk_`

2→S2>kΦ−Πk_S

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Characterizing spectral expansion and uniformity through tensor norms

λ(G) =kA−Jk_`

2→`2>kA−Jk_`

2→S2>kΦ−Πk_S

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Proof of tensor norm expressions for λ and ε

Note that theS₂→S₂norm ofΦ−Πis simply its operator norm, i.e. its largest eigenvalue.

Πhas eigenvalues 1,0, . . . ,0, withIas eigenvector associated to 1.

Φhas eigenvalues 1,λ₂(Φ), . . . ,λn(Φ), withIas eigenvector associated to 1.

So we indeed have:|λ1(Φ−Π)|=|λ2(Φ)|=λ(Φ).

For any projectorsX,YonCⁿ, we have: 1

nTr(Y^∗(Φ−Π)(X)) = ¹

nTr(Y^∗Φ(X))− ¹

n²Tr(Y^∗)Tr(X).

By definition, the maximum of the l.h.s. iskΦ−Πkcutand the maximum of the r.h.s. isε(Φ).

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Proof of tensor norm expressions for λ and ε

Note that theS₂→S₂norm ofΦ−Πis simply its operator norm, i.e. its largest eigenvalue.

Πhas eigenvalues 1,0, . . . ,0, withIas eigenvector associated to 1.

Φhas eigenvalues 1,λ₂(Φ), . . . ,λn(Φ), withIas eigenvector associated to 1.

So we indeed have:|λ1(Φ−Π)|=|λ2(Φ)|=λ(Φ). For any projectorsX,YonCⁿ, we have:

1

nTr(Y^∗(Φ−Π)(X)) = ¹

nTr(Y^∗Φ(X))− ¹

n²Tr(Y^∗)Tr(X).

By definition, the maximum of the l.h.s. iskΦ−Πkcutand the maximum of the r.h.s. isε(Φ).

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Equivalence between spectral expansion and uniformity in the dense case

•Classical case:Gad-regular graph onnvertices,δ:=d/nitsedge density.

λ(G)6 2ε(G)

δ² 1/4

, so ifδ= Ω(1), thenλ(G)6^Cε(G)¹^/⁴(Chung/Graham/Wilson).

−→For a dense regular graph, uniformity implies spectral expansion.

•Quantum case:Φa unital quantum channel,η:=kΦ−ΠkS1→S∞ itsrandomizing parameter. λ(Φ)6 π²η³ε(G)1/4

, so ifη=O(1), thenλ(G)6^Cε(G)¹^/⁴(Bannink/Briët/Labib/Maassen).

−→For a randomizing unital channel, uniformity implies spectral expansion.

Observation:In the classical case,η:=kA−Jk`1→`∞=1/δ−1, so the conditionδ= Ω(1)is equivalent to the conditionη=O(1).

−→The quantum statement can be seen as similar to the classical one.

In fact, the inequality involvingηrather thanδholds classically exactly as quantumly, and is better than the original one in the regime whereηis small enough (i.e.δis large enough).

Main technical tool in the proof:Clever use of Cauchy-Schwartz inequality (thus valid for`_p norms as forSpnorms).

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Equivalence between spectral expansion and uniformity in the dense case

λ(G)6 2ε(G)

δ² 1/4

•Quantum case:Φa unital quantum channel,η:=kΦ−ΠkS1→S∞ itsrandomizing parameter.

λ(Φ)6 π²η³ε(G)1/4

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Equivalence between spectral expansion and uniformity in the dense case

λ(G)6 2ε(G)

δ² 1/4

λ(Φ)6 π²η³ε(G)1/4

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Equivalence between spectral expansion and uniformity in the dense case

λ(G)6 2ε(G)

δ² 1/4

λ(Φ)6 π²η³ε(G)1/4

Main technical tool in the proof:Clever use of Cauchy-Schwartz inequality (thus valid for`_p norms as forS_pnorms).

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Equivalence between spectral expansion and uniformity in the symmetric case

•Classical case:Gavertex-transitiveregular graph onnvertices.

This means: There existsΓa transitive subgroup of the group of permutations ofnelements, s.t.∀γ∈Γ,M_γAM_γ⁻¹=A, whereM_γis then×npermutation matrix associated toγ. By a factorization version of thecommutative Grothendieck inequality(Pisier):

kA−Jk`2→`26^KkA−Jk`∞→`1,with 1.33<K<1.41 the complex Grothendieck constant.

−→For any vertex-transitive regular graphG,

λ(G) =kA−Jk`2→`26^KkA−Jk`∞→`16^Kπ²kA−Jkcut=Kπ²ε(G).

•Quantum case:Φanirreducibly covariantunital quantum channel.

This means: There exists a compact groupΓwith irreducible unitary representations γ∈Γ7→U_γ∈U(n)andγ∈Γ7→V_γ∈U(n)s.t.∀γ∈Γ,Φ(U_γ·U_γ⁻¹) =V_γΦ(·)V_γ⁻¹. By a factorization version of thenon-commutative Grothendieck inequality(Haagerup):

kΦ−ΠkS2→S26²kΦ−ΠkS∞→S1.

−→For any irreducibly covariant unital quantum channelΦ,

λ(Φ) =kΦ−Πk_S₂_→_S₂6²kΦ−Πk_S_∞_→_S₁6²π²kΦ−Πk_cut=2π²ε(Φ).

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Equivalence between spectral expansion and uniformity in the symmetric case

•Classical case:Gavertex-transitiveregular graph onnvertices.

This means: There existsΓa transitive subgroup of the group of permutations ofnelements, s.t.∀γ∈Γ,M_γAM_γ⁻¹=A, whereM_γis then×npermutation matrix associated toγ. By a factorization version of thecommutative Grothendieck inequality(Pisier):

kA−Jk`2→`26^KkA−Jk`∞→`1,with 1.33<K<1.41 the complex Grothendieck constant.

−→For any vertex-transitive regular graphG,

λ(G) =kA−Jk`2→`26^KkA−Jk`∞→`16^Kπ²kA−Jkcut=Kπ²ε(G).

•Quantum case:Φanirreducibly covariantunital quantum channel.

This means: There exists a compact groupΓwith irreducible unitary representations γ∈Γ7→U_γ∈U(n)andγ∈Γ7→V_γ∈U(n)s.t.∀γ∈Γ,Φ(U_γ·U_γ⁻¹) =V_γΦ(·)V_γ⁻¹. By a factorization version of thenon-commutative Grothendieck inequality(Haagerup):

kΦ−ΠkS2→S26²kΦ−ΠkS∞→S1.

−→For any irreducibly covariant unital quantum channelΦ,

λ(Φ) =kΦ−Πk_S₂_→_S₂6²kΦ−Πk_S_∞_→_S₁6²π²kΦ−Πk_cut=2π²ε(Φ).

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Main ideas in the proof

Goal:Show that, ifψ:

M

n(C)→

M

n(C)is irreducibly covariant, thenkΨkS2→S26²kΨkS∞→S1.

Fact 1 (Factorized non-commutative Grothendieck inequality):

Letψ:

M

n(C)→

M

n(C). There exist statesρ,ρ⁰,σ,σ⁰onCⁿs.t., for allX,Y∈

M

n(C),

1

nTr(Y^∗Ψ(X))

6kΨkS∞→S1 Tr(ρXX^∗) +Tr(ρ⁰X^∗X)1/2

Tr(σYY^∗) +Tr(σ⁰Y^∗Y)1/2

.

Fact 2 (Characterization of irreducible unitary representations):

LetΓbe a compact group. A unitary representationγ∈Γ7→U_γ∈U(n)is irreducible iff

∀X∈

M

n(C), ¹

|Γ|

∑

γ∈Γ

U_γXU_γ^∗=Tr(X)^I n.

By assumption onΨ, 1

nTr(Y^∗Ψ(X))

= ¹

|Γ|

∑

γ∈Γ

1

nTr(Y_γ^∗Ψ(X_γ))

,X_γ:=U_γXU_γ^∗,Y_γ:=V_γXV_γ^∗. By Fact 1 and the AM-GM inequality, the r.h.s. above is upper bounded by:

1

2kΨk_S_∞_→_S₁ ¹

|Γ|

∑

γ∈Γ

Tr(ρX_γX_γ^∗) +Tr(ρ⁰X_γ^∗X_γ) +Tr(σY_γY_γ^∗) +Tr(σ⁰Y_γ^∗Y_γ) . By Fact 2, 1

|Γ|

∑

γ∈Γ

Tr(ρX_γX_γ^∗) =kXk²_S

2, and similarly for the other terms. Hence, for anyX,Y∈

M

n(C)s.t.kXkS2=kYkS2=1,

1

nTr(Y^∗Ψ(X))

6²kΨkS∞→S1.

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Main ideas in the proof

M

n(C)→

M

n(C)is irreducibly covariant, thenkΨkS2→S26²kΨkS∞→S1. Fact 1 (Factorized non-commutative Grothendieck inequality):

Letψ:

M

n(C)→

M

n(C),

1

nTr(Y^∗Ψ(X))

6kΨk_S_∞→S1 Tr(ρXX^∗) +Tr(ρ⁰X^∗X)1/2

.

∀X∈

M

n(C), ¹

|Γ|

∑

γ∈Γ

nTr(Y^∗Ψ(X))

= ¹

|Γ|

∑

γ∈Γ

1

2kΨk_S_∞_→_S₁ ¹

|Γ|

∑

γ∈Γ

|Γ|

∑

γ∈Γ

2, and similarly for the other terms. Hence, for anyX,Y∈

M

n(C)s.t.kXkS2=kYkS2=1,

1

nTr(Y^∗Ψ(X))

6²kΨkS∞→S1.

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Main ideas in the proof

M

n(C)→

M

n(C)is irreducibly covariant, thenkΨkS2→S26²kΨkS∞→S1. Fact 1 (Factorized non-commutative Grothendieck inequality):

Letψ:

M

n(C)→

M

n(C),

1

nTr(Y^∗Ψ(X))

6kΨk_S_∞→S1 Tr(ρXX^∗) +Tr(ρ⁰X^∗X)1/2

.

∀X∈

M

n(C), ¹

|Γ|

∑

γ∈Γ

nTr(Y^∗Ψ(X))

= ¹

|Γ|

∑

γ∈Γ

1

2kΨk_S_∞_→_S₁ ¹

|Γ|

∑

γ∈Γ

|Γ|

∑

γ∈Γ

2, and similarly for the other terms.

Hence, for anyX,Y∈

M

n(C)s.t.kXkS2=kYkS2=1, 1

nTr(Y^∗Ψ(X))

6²kΨkS∞→S1.

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Embedding the classical setting in the quantum one

GivenA:Cⁿ→Cⁿ, defineΦA:X∈

M

n(C)7→

n

∑

i,j=1

AijXjj|iihi| ∈

M

n(C).

−→Embedding of adjacency matrices of regular graphs into unital quantum channels.

Sanity check:The adjacency matrix of the complete graph is embedded into the completely randomizing quantum channel, i.e.Φ_J= Π.

Moreover:kΦ_A−Πk_S_p_→_S_q=kA−Jk_`_p_→`_qandkΦ_A−Πk_cut=kA−Jk_cut.

−→Spectral expansion and uniformity parameters are preserved under the embedding.

Preservation of density and symmetry properties under the embedding:

Φ_Airreducibly covariant quantum channel⇔Aadjacency matrix of a vertex-transitive graph. Φ_AO(1)-randomizing quantum channel⇔Aadjacency matrix of anΩ(1)-dense graph. Interest:There are examples of non-dense regular graphsGwhich are not vertex-transitive, and s.t.ε(G)is small butλ(G)is large. They provide examples of non-randomizing unital channelsΦ which are not irreducibly covariant, and s.t.ε(Φ)is small butλ(Φ)is large.

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Embedding the classical setting in the quantum one

GivenA:Cⁿ→Cⁿ, defineΦA:X∈

M

n(C)7→

n

∑

i,j=1

AijXjj|iihi| ∈

M

n(C).

−→Embedding of adjacency matrices of regular graphs into unital quantum channels.

Sanity check:The adjacency matrix of the complete graph is embedded into the completely randomizing quantum channel, i.e.Φ_J= Π.

Moreover:kΦ_A−Πk_S_p_→_S_q=kA−Jk_`_p_→`_qandkΦ_A−Πk_cut=kA−Jk_cut.

−→Spectral expansion and uniformity parameters are preserved under the embedding.

Preservation of density and symmetry properties under the embedding:

Φ_Airreducibly covariant quantum channel⇔Aadjacency matrix of a vertex-transitive graph.

ΦAO(1)-randomizing quantum channel⇔Aadjacency matrix of anΩ(1)-dense graph.

Interest:There are examples of non-dense regular graphsGwhich are not vertex-transitive, and s.t.ε(G)is small butλ(G)is large. They provide examples of non-randomizing unital channelsΦ which are not irreducibly covariant, and s.t.ε(Φ)is small butλ(Φ)is large.

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Final comments

Optimality 1:Being vertex-transitive, resp. irreducibly covariant, is a necessary condition for a non-dense regular graph, resp. non-randomizing unital channel, to be s.t. uniformity implies spectral expansion.

Optimality 2:In the classical case of a vertex-transitive regular graphG, the inequality λ(G)6^Kπ²ε(G)is optimal. In the quantum case of an irreducibly covariant unital channel Φ, it is not known whether the inequalityλ(Φ)6²π²ε(Φ)is optimal (each of the two steps in the proof is optimal, but their combination might not be).

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Final comments

Optimality 1:Being vertex-transitive, resp. irreducibly covariant, is a necessary condition for a non-dense regular graph, resp. non-randomizing unital channel, to be s.t. uniformity implies spectral expansion.

Optimality 2:In the classical case of a vertex-transitive regular graphG, the inequality λ(G)6^Kπ²ε(G)is optimal. In the quantum case of an irreducibly covariant unital channel Φ, it is not known whether the inequalityλ(Φ)6²π²ε(Φ)is optimal (each of the two steps in the proof is optimal, but their combination might not be).

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References

T. Bannink, J. Briët, F. Labib, H. Maassen.Quasirandom quantum channels. 2020.

A. Ben-Aroya, O. Schwartz, A. Ta-Shma.Quantum expanders: motivation and construction. 2010.

D. Conlon, Y. ZhaoQuasirandom Cayley graphs. 2017.

F.R.K. Chung, R.L. Graham, R.M. Wilson.Quasi-random graphs. 1988.

U. Haagerup.The Grothendieck inequality for bilinear forms on C*-algebras. 1985.

M.B. Hastings.Random unitaries give quantum expanders. 2007.

S. Hoory, N. Linial, A. Widgerson.Expander graphs and their application. 2006.

G. Pisier.Grothendieck’s theorem, past and present. 2011.