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Random quantum graphs are asymmetric

Mateusz Wasilewski (joint work with Alexandru

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0 Outline

1 Quantum graphs

2 Random quantum graphs

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1 Outline

1 Quantum graphs

2 Random quantum graphs

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1 Origin story

Quantum graphs originated from quantum information theory in many ways:

I (Duan-Severini-Winter) Quantum confusability graphs associated to quantum channels;

I (Weaver, Kuperberg-Weaver) Symmetric, reflexive quantum rela- tions, motivated by the study of quantum metric spaces;

I (Musto-Reutter-Verdon) Finite dimensional C-algebras equipped with an analogue of an adjacency matrix, partially inspired by the graph homomorphism game of Manˇcinska and Roberson.

Point of view for today: quantum graphs are interesting mathematical objects in their own right.

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1 Origin story

Quantum graphs originated from quantum information theory in many ways:

I (Duan-Severini-Winter) Quantum confusability graphs associated to quantum channels;

I (Weaver, Kuperberg-Weaver) Symmetric, reflexive quantum rela- tions, motivated by the study of quantum metric spaces;

I (Musto-Reutter-Verdon) Finite dimensional C-algebras equipped with an analogue of an adjacency matrix, partially inspired by the graph homomorphism game of Manˇcinska and Roberson.

Point of view for today: quantum graphs are interesting mathematical objects in their own right.

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1 Origin story

Quantum graphs originated from quantum information theory in many ways:

I (Duan-Severini-Winter) Quantum confusability graphs associated to quantum channels;

I (Weaver, Kuperberg-Weaver) Symmetric, reflexive quantum rela- tions, motivated by the study of quantum metric spaces;

I (Musto-Reutter-Verdon) Finite dimensional C-algebras equipped with an analogue of an adjacency matrix, partially inspired by the graph homomorphism game of Manˇcinska and Roberson.

Point of view for today: quantum graphs are interesting mathematical objects in their own right.

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1 Origin story

Quantum graphs originated from quantum information theory in many ways:

I (Duan-Severini-Winter) Quantum confusability graphs associated to quantum channels;

I (Weaver, Kuperberg-Weaver) Symmetric, reflexive quantum rela- tions, motivated by the study of quantum metric spaces;

I (Musto-Reutter-Verdon) Finite dimensional C-algebras equipped with an analogue of an adjacency matrix, partially inspired by the graph homomorphism game of Manˇcinska and Roberson.

Point of view for today: quantum graphs are interesting mathematical objects in their own right.

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1 Origin story

Quantum graphs originated from quantum information theory in many ways:

I (Duan-Severini-Winter) Quantum confusability graphs associated to quantum channels;

I (Weaver, Kuperberg-Weaver) Symmetric, reflexive quantum rela- tions, motivated by the study of quantum metric spaces;

I (Musto-Reutter-Verdon) Finite dimensional C-algebras equipped with an analogue of an adjacency matrix, partially inspired by the graph homomorphism game of Manˇcinska and Roberson.

Point of view for today: quantum graphs are interesting mathematical objects in their own right.

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1 Quantum confusability graphs

We start from a quantum channel Φ : MnMm with a Kraus de- composition Φ(x) := PiSixSi, where PSiSi = 1, because Φ is trace preserving.

We associate to Φ its quantum confusability graph V := span{SiSj}. Why?

You cannot confuse two states iff they are orthogonal. Compute Tr(Φ(|vihv|)Φ(|wihw|)) =Pi,j|hSiSjv, wi|2.

It is equal to 0 iff |vihw|is orthogonal to V.

V is an operator systemindependent of the Kraus decomposition of Φ. Moreover, any operator subsystem of Mn arises in this way.

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1 Quantum confusability graphs

We start from a quantum channel Φ : MnMm with a Kraus de- composition Φ(x) := PiSixSi, where PSiSi = 1, because Φ is trace preserving.

We associate to Φ its quantum confusability graph V :=

span{SiSj}. Why?

You cannot confuse two states iff they are orthogonal. Compute Tr(Φ(|vihv|)Φ(|wihw|)) =Pi,j|hSiSjv, wi|2.

It is equal to 0 iff |vihw|is orthogonal to V.

V is an operator systemindependent of the Kraus decomposition of Φ. Moreover, any operator subsystem of Mn arises in this way.

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1 Quantum confusability graphs

We start from a quantum channel Φ : MnMm with a Kraus de- composition Φ(x) := PiSixSi, where PSiSi = 1, because Φ is trace preserving.

We associate to Φ its quantum confusability graph V :=

span{SiSj}. Why?

You cannot confuse two states iff they are orthogonal. Compute Tr(Φ(|vihv|)Φ(|wihw|)) =Pi,j|hSiSjv, wi|2.

It is equal to 0 iff|vihw|is orthogonal to V.

V is an operator systemindependent of the Kraus decomposition of Φ. Moreover, any operator subsystem of Mn arises in this way.

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1 Quantum confusability graphs

We start from a quantum channel Φ : MnMm with a Kraus de- composition Φ(x) := PiSixSi, where PSiSi = 1, because Φ is trace preserving.

We associate to Φ its quantum confusability graph V :=

span{SiSj}. Why?

You cannot confuse two states iff they are orthogonal. Compute Tr(Φ(|vihv|)Φ(|wihw|)) =Pi,j|hSiSjv, wi|2.

It is equal to 0 iff|vihw|is orthogonal toV.

V is an operator systemindependent of the Kraus decomposition of Φ. Moreover, any operator subsystem of Mn arises in this way.

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1 Quantum confusability graphs

We start from a quantum channel Φ : MnMm with a Kraus de- composition Φ(x) := PiSixSi, where PSiSi = 1, because Φ is trace preserving.

We associate to Φ its quantum confusability graph V :=

span{SiSj}. Why?

You cannot confuse two states iff they are orthogonal. Compute Tr(Φ(|vihv|)Φ(|wihw|)) =Pi,j|hSiSjv, wi|2.

It is equal to 0 iff|vihw|is orthogonal toV.

V is an operator systemindependent of the Kraus decomposition of Φ. Moreover, any operator subsystem of Mn arises in this way.

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1 Quantum relations – a naive approach A relation on a setXis just a subset ofX×X.

In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space.

Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V.

Aquantum graph is an operator subsystem VMn.

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1 Quantum relations – a naive approach

A relation on a setXis just a subset ofX×X. In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space.

Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V.

Aquantum graph is an operator subsystem VMn.

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1 Quantum relations – a naive approach

A relation on a setXis just a subset ofX×X. In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space.

Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V.

Aquantum graph is an operator subsystem VMn.

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1 Quantum relations – a naive approach

A relation on a setXis just a subset ofX×X. In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space. Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V.

Aquantum graph is an operator subsystem VMn.

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1 Quantum relations – a naive approach

A relation on a setXis just a subset ofX×X. In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space. Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V.

Aquantum graph is an operator subsystem VMn.

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1 Quantum relations – a naive approach

A relation on a setXis just a subset ofX×X. In the noncommutative case we replace X by a matrix algebra Mn, so a quantum relation should be a projection in MnMn.

It is actually better to take a projection PMnMnop. The reason is B(Mn)'MnMnop, whereMn is viewed as a Hilbert space. Thus pMnMnop can be identified with the image of a projection in B(Mn), i.e. a subspace ofMn.

Definition (Weaver)

A quantum relation onMn is an operator subspace VMn. It is symmetric ifV =V andreflexive if1∈V. Aquantum graph is an operator subsystem VMn.

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1 The quantum adjacency matrix

The usual adjacency matrix has entries 0 or 1. An abstract way to say it is that it is an idempotent with respect to the Schur product, i.e.

AA=A.

A more abstract way to do this: AB := m(AB)m, where m : Cn⊗Cn→Cn is the multiplication andm is its adjoint with respect to the usual inner product.

Easy to generalise to the noncommutative case.

Definition (Musto-Reutter-Verdon)

A quantum adjacency matrixA:MnMnis a completely positive map such that m(AA)m =A,A is self-adjoint and m(A⊗Id)m(1) =1.

The inner product on Mn ishA, Bi:=nTr(AB).

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1 The quantum adjacency matrix

The usual adjacency matrix has entries 0 or 1. An abstract way to say it is that it is an idempotent with respect to the Schur product, i.e.

AA=A.

A more abstract way to do this: AB := m(AB)m, where m : Cn⊗Cn→Cnis the multiplication andm is its adjoint with respect to the usual inner product.

Easy to generalise to the noncommutative case.

Definition (Musto-Reutter-Verdon)

A quantum adjacency matrixA:MnMnis a completely positive map such that m(AA)m =A,A is self-adjoint and m(A⊗Id)m(1) =1.

The inner product on Mn ishA, Bi:=nTr(AB).

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1 The quantum adjacency matrix

The usual adjacency matrix has entries 0 or 1. An abstract way to say it is that it is an idempotent with respect to the Schur product, i.e.

AA=A.

A more abstract way to do this: AB := m(AB)m, where m : Cn⊗Cn→Cnis the multiplication andm is its adjoint with respect to the usual inner product. Easy to generalise to the noncommutative case.

Definition (Musto-Reutter-Verdon)

A quantum adjacency matrixA:MnMnis a completely positive map such that m(AA)m =A,A is self-adjoint and m(A⊗Id)m(1) =1.

The inner product on Mn ishA, Bi:=nTr(AB).

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1 The quantum adjacency matrix

The usual adjacency matrix has entries 0 or 1. An abstract way to say it is that it is an idempotent with respect to the Schur product, i.e.

AA=A.

A more abstract way to do this: AB := m(AB)m, where m : Cn⊗Cn→Cnis the multiplication andm is its adjoint with respect to the usual inner product. Easy to generalise to the noncommutative case.

Definition (Musto-Reutter-Verdon)

A quantum adjacency matrixA:MnMnis a completely positive map such that m(AA)m =A,A is self-adjoint and m(A⊗Id)m(1) =1.

The inner product on Mn ishA, Bi:=nTr(AB).

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1 The quantum adjacency matrix

The usual adjacency matrix has entries 0 or 1. An abstract way to say it is that it is an idempotent with respect to the Schur product, i.e.

AA=A.

A more abstract way to do this: AB := m(AB)m, where m : Cn⊗Cn→Cnis the multiplication andm is its adjoint with respect to the usual inner product. Easy to generalise to the noncommutative case.

Definition (Musto-Reutter-Verdon)

A quantum adjacency matrixA:MnMnis a completely positive map such that m(AA)m =A,A is self-adjoint and m(A⊗Id)m(1) =1.

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V;

I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication; I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication; I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1. Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1. Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1. Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1.

Are these definitions equivalent?

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1 Summary

A quantum graphonMn is:

I an operator subsystemVMn, i.e. V =V and1∈V; I a projection PMnMnop, satisfying

σ(P) =P, whereσ:MnMnopMnMnop is the flip;

m(P) =1, where m:MnMnopMn is the multiplication;

I a completely positive map A:MnMn such that

m(AA)m=A;

Tr((Ax)y) = Tr(x(Ay)), i.e. Ais self-adjoint;

m(AId)m(1) =1. Are these definitions equivalent?

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1 The equivalences

Operator systems vs projections

Let p : MnV be the orthogonal projection wrt the trace. As B(HSn)'MnMnop, we get a corresponding PMnMnop.

V =V means that σ(P) =P and1∈V means that m(P) =1. Choi-Jamio lkowski

If PMnMnop is a projection then AP : MnMn given by AP(x) := (Id⊗nTr)(P(1⊗x))is cp and satisfiesm(APAP)m= AP.

If A :MnMn is cp and such that m(AA)m = A then PA := (A⊗Id)m(1), itsChoi matrix, is a projection inMnMnop.

(A⊗Id)m(1) = n1Pni,j=1A(eij)⊗eji

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1 The equivalences

Operator systems vs projections

Let p : MnV be the orthogonal projection wrt the trace. As B(HSn)'MnMnop, we get a corresponding PMnMnop. V =V means that σ(P) =P and1∈V means that m(P) =1.

Choi-Jamio lkowski

If PMnMnop is a projection then AP : MnMn given by AP(x) := (Id⊗nTr)(P(1⊗x))is cp and satisfiesm(APAP)m= AP.

If A :MnMn is cp and such that m(AA)m = A then PA := (A⊗Id)m(1), itsChoi matrix, is a projection inMnMnop.

(A⊗Id)m(1) = n1Pni,j=1A(eij)⊗eji

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1 The equivalences

Operator systems vs projections

Let p : MnV be the orthogonal projection wrt the trace. As B(HSn)'MnMnop, we get a corresponding PMnMnop. V =V means that σ(P) =P and1∈V means that m(P) =1. Choi-Jamio lkowski

If PMnMnop is a projection then AP : MnMn given by AP(x) := (Id⊗nTr)(P(1⊗x))is cp and satisfiesm(APAP)m = AP.

If A :MnMn is cp and such that m(AA)m = A then PA := (A⊗Id)m(1), itsChoi matrix, is a projection inMnMnop. (A⊗Id)m(1) = n1Pni,j=1A(eij)⊗eji

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1 The equivalences

Operator systems vs projections

Let p : MnV be the orthogonal projection wrt the trace. As B(HSn)'MnMnop, we get a corresponding PMnMnop. V =V means that σ(P) =P and1∈V means that m(P) =1. Choi-Jamio lkowski

If PMnMnop is a projection then AP : MnMn given by AP(x) := (Id⊗nTr)(P(1⊗x))is cp and satisfiesm(APAP)m = AP.

If A :MnMn is cp and such that m(AA)m = A then PA :=

(A⊗Id)m(1), itsChoi matrix, is a projection inMnMnop.

(A⊗Id)m(1) = n1Pni,j=1A(eij)⊗eji

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1 The equivalences

Operator systems vs projections

Let p : MnV be the orthogonal projection wrt the trace. As B(HSn)'MnMnop, we get a corresponding PMnMnop. V =V means that σ(P) =P and1∈V means that m(P) =1. Choi-Jamio lkowski

If PMnMnop is a projection then AP : MnMn given by AP(x) := (Id⊗nTr)(P(1⊗x))is cp and satisfiesm(APAP)m = AP.

If A :MnMn is cp and such that m(AA)m = A then PA :=

(A⊗Id)m(1), itsChoi matrix, is a projection inMnMnop. (A⊗Id)m(1) = n1Pni,j=1A(eij)⊗eji

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1 Intermission – graphs as operator systems

Classical graphs

Classical graphs also fit into this framework. To a graph we associate the following space V := span{eij :ij or i=j}.

Quantum invariants

It turns out that two graphs are isomorphic if and only if the corre- sponding operator systems are isomorphic (Ortiz-Paulsen). This allows to associate “quantum” invariants to graphs, using quantities appearing in the theory of operator systems.

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1 Intermission – graphs as operator systems

Classical graphs

Classical graphs also fit into this framework. To a graph we associate the following space V := span{eij :ij or i=j}.

Quantum invariants

It turns out that two graphs are isomorphic if and only if the corre- sponding operator systems are isomorphic (Ortiz-Paulsen). This allows to associate “quantum” invariants to graphs, using quantities appearing in the theory of operator systems.

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1 Automorphisms of quantum graphs

The easiest way to define automorphisms is in the operator system framework. We say that a unitary matrixUMn is an automorphism of an operator system VMn ifU V U =V.

Translated to the adjacency matrix, it means that A(U xU) = U A(x)U.

The degree matrix

Note that if U is an automorphism of our quantum graph then it com- mutes with the degree matrix D:=A1.

If the spectrum of D is simple then the automorphism group is auto- matically abelian.

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1 Automorphisms of quantum graphs

The easiest way to define automorphisms is in the operator system framework. We say that a unitary matrixUMn is an automorphism of an operator system VMn ifU V U =V.

Translated to the adjacency matrix, it means that A(U xU) = U A(x)U.

The degree matrix

Note that if U is an automorphism of our quantum graph then it com- mutes with the degree matrix D:=A1.

If the spectrum of D is simple then the automorphism group is auto- matically abelian.

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1 Automorphisms of quantum graphs

The easiest way to define automorphisms is in the operator system framework. We say that a unitary matrixUMn is an automorphism of an operator system VMn ifU V U =V.

Translated to the adjacency matrix, it means that A(U xU) = U A(x)U.

The degree matrix

Note that if U is an automorphism of our quantum graph then it com- mutes with the degree matrix D:=A1.

If the spectrum of D is simple then the automorphism group is auto- matically abelian.

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1 Automorphisms of quantum graphs

The easiest way to define automorphisms is in the operator system framework. We say that a unitary matrixUMn is an automorphism of an operator system VMn ifU V U =V.

Translated to the adjacency matrix, it means that A(U xU) = U A(x)U.

The degree matrix

Note that if U is an automorphism of our quantum graph then it com- mutes with the degree matrix D:=A1.

If the spectrum of D is simple then the automorphism group is auto- matically abelian.

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2 Outline

1 Quantum graphs

2 Random quantum graphs

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2 Random models

The easiest thing to do is the following. We fix 0 6 d 6 n2 −1.

Then we take dindependent random Hermitian matricesX1,...,Xdand consider Vd:= span{1, X1, . . . , Xd}.

GUE

The most natural choice is to take Xi from the GUE ensemble. G(n, M)

The model above corresponds to the Erd˝os-R´enyi random graph G(n, M) with a fixed number of edges. We can also build a version of the G(n, p), where we fix the number of vertices and the probability that a given pair of vertices is connected by an edge.

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2 Random models

The easiest thing to do is the following. We fix 0 6 d 6 n2 −1.

Then we take dindependent random Hermitian matricesX1,...,Xdand consider Vd:= span{1, X1, . . . , Xd}.

GUE

The most natural choice is to take Xi from the GUE ensemble.

G(n, M)

The model above corresponds to the Erd˝os-R´enyi random graph G(n, M) with a fixed number of edges. We can also build a version of the G(n, p), where we fix the number of vertices and the probability that a given pair of vertices is connected by an edge.

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2 Random models

The easiest thing to do is the following. We fix 0 6 d 6 n2 −1.

Then we take dindependent random Hermitian matricesX1,...,Xdand consider Vd:= span{1, X1, . . . , Xd}.

GUE

The most natural choice is to take Xi from the GUE ensemble.

G(n, M)

The model above corresponds to the Erd˝os-R´enyi random graph G(n, M) with a fixed number of edges. We can also build a version of the G(n, p), where we fix the number of vertices and the probability that a given pair of vertices is connected by an edge.

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2 The results

Theorem (Chirvasitu-W.)

If 16d6n2−2 then the degree matrix Dhas almost surely simple spectrum.

Corollary (Chirvasitu-W.)

If16d6n2−2then the automorphism group is almost surely abelian. Theorem (Chirvasitu-W.)

If26d6n2−3then the automorphism group is almost surely trivial.

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2 The results

Theorem (Chirvasitu-W.)

If 16d6n2−2 then the degree matrix Dhas almost surely simple spectrum.

Corollary (Chirvasitu-W.)

If16d6n2−2then the automorphism group is almost surely abelian.

Theorem (Chirvasitu-W.)

If26d6n2−3then the automorphism group is almost surely trivial.

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2 The results

Theorem (Chirvasitu-W.)

If 16d6n2−2 then the degree matrix Dhas almost surely simple spectrum.

Corollary (Chirvasitu-W.)

If16d6n2−2then the automorphism group is almost surely abelian.

Theorem (Chirvasitu-W.)

If26d6n2−3then the automorphism group is almost surely trivial.

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2 Diagonal degree matrices

The degree matrix

Let VMn be an operator system with an orthonormal basis (A1, . . . , Ad)consisting of Hermitian matrices. Then the degree matrix is equal to D=Pdi=1A2i.

It is clear that the set of tuples (X1, . . . , Xd) such that the degree matrixDhas no repeated eigenvalues is Zariski open. It suffices to find one example to prove that it is of full measure.

Definefij := eij+eji

2 for i < j andfij := i(eij−eji)

2 for i > j. Then the set (fij)is orthonormal and fij2 is adiagonal matrix.

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2 Diagonal degree matrices

The degree matrix

Let VMn be an operator system with an orthonormal basis (A1, . . . , Ad)consisting of Hermitian matrices. Then the degree matrix is equal to D=Pdi=1A2i.

It is clear that the set of tuples (X1, . . . , Xd) such that the degree matrixDhas no repeated eigenvalues is Zariski open. It suffices to find one example to prove that it is of full measure.

Definefij := eij+eji

2 for i < j andfij := i(eij−eji)

2 for i > j. Then the set (fij)is orthonormal and fij2 is adiagonal matrix.

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2 Diagonal degree matrices

The degree matrix

Let VMn be an operator system with an orthonormal basis (A1, . . . , Ad)consisting of Hermitian matrices. Then the degree matrix is equal to D=Pdi=1A2i.

It is clear that the set of tuples (X1, . . . , Xd) such that the degree matrixDhas no repeated eigenvalues is Zariski open. It suffices to find one example to prove that it is of full measure.

Definefij := eij+eji

2 for i < j andfij := i(eij−eji)

2 for i > j. Then the set (fij)is orthonormal and fij2 is a diagonal matrix.

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2 Degree matrices with a simple spectrum

Let d∈ {1, . . . , n2−2}. If d6n−2 then we take d random mutu- ally orthogonal diagonal matrices (rows of a random Haar orthogonal matrix).

Ifd > n−2then we taked−n+ 2matrices from the set(fij)andn−2 random mutually orthogonal diagonal matrices. The degree matrix is diagonal and for almost all choices it has distinct entries.

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2 Degree matrices with a simple spectrum

Let d∈ {1, . . . , n2−2}. If d6n−2 then we take d random mutu- ally orthogonal diagonal matrices (rows of a random Haar orthogonal matrix).

Ifd > n−2then we taked−n+ 2matrices from the set(fij)andn−2 random mutually orthogonal diagonal matrices. The degree matrix is diagonal and for almost all choices it has distinct entries.

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2 Trivial automorphism group

The set of tuples (X1, . . . , Xd) for which the automorphism group is trivial is Zariski open (more difficult than before). It therefore suffices again to construct a single example.

The idea is to construct an operator system with a diagonal degree ma- trix such that the onlydiagonalunitaries preserving it upon conjugation are trivial.

Then we add some random diagonal matrices so that the degree matrix has distinct entries and conclude, because the only possible automor- phisms had to be diagonal to begin with.

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2 Trivial automorphism group

The set of tuples (X1, . . . , Xd) for which the automorphism group is trivial is Zariski open (more difficult than before). It therefore suffices again to construct a single example.

The idea is to construct an operator system with a diagonal degree ma- trix such that the onlydiagonalunitaries preserving it upon conjugation are trivial.

Then we add some random diagonal matrices so that the degree matrix has distinct entries and conclude, because the only possible automor- phisms had to be diagonal to begin with.

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2 Trivial automorphism group

The set of tuples (X1, . . . , Xd) for which the automorphism group is trivial is Zariski open (more difficult than before). It therefore suffices again to construct a single example.

The idea is to construct an operator system with a diagonal degree ma- trix such that the onlydiagonalunitaries preserving it upon conjugation are trivial.

Then we add some random diagonal matrices so that the degree matrix has distinct entries and conclude, because the only possible automor- phisms had to be diagonal to begin with.

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2 The construction

This simple approach works only forn>6andd∈ {4, . . . , n2−5}, as opposed to n>3and d∈ {2, . . . , n2−3}.

We start with two matrices X1 := P[(n−1)/2]i=1 f2i,2i+1 and X2 := P[n/2]

i=1 f2i−1,2i. Their span is preserved by a diagonal unitary matrixU iff all its entries are ±1. Moreover these entries have to be4-periodic. We have u1 = 1 and u4 = u2u3, so only u2 and u3 have to be de- termined. To this end one has to add another matrix, for example Y :=f14+f25+f37 (and something slightly different for n= 6).

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2 The construction

This simple approach works only forn>6andd∈ {4, . . . , n2−5}, as opposed to n>3and d∈ {2, . . . , n2−3}.

We start with two matrices X1 := P[(n−1)/2]i=1 f2i,2i+1 and X2 :=

P[n/2]

i=1 f2i−1,2i. Their span is preserved by a diagonal unitary matrixU iff all its entries are ±1. Moreover these entries have to be4-periodic.

We have u1 = 1 and u4 = u2u3, so only u2 and u3 have to be de- termined. To this end one has to add another matrix, for example Y :=f14+f25+f37 (and something slightly different for n= 6).

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