Zero-error quantum information theory

withGareth Boreland, QUBRupert Levene, Dublin

Vern Paulsen, IQC WaterlooAndreas Winter, Barcelona

April 2019, Shanghai Jiao Tong

Ivan Todorov QUB

Outline

(1) The zero-error scenario in information transmission

(2) Zero-error capacity of classical channels

(3) The Sandwich Theorem

(4) Convex corners

(5) Quantum channels and zero-error transmission

(6) Non-commutative confusability graphs

(7) Non-commutative graph parameters

(8) The Quantum Sandwich Theorem

Ivan Todorov QUB

The Shannon model

Source → Encoder → Channel → Decoder → Target

A channel N transmits symbols from an alphabet X into analphabet Y :

Ivan Todorov QUB

The Shannon model

Source → Encoder → Channel → Decoder → Target

A channel N transmits symbols from an alphabet X into analphabet Y :

Ivan Todorov QUB

Formalism

A channel N : X → Y is a family {(p(y |x))y∈Y : x ∈ X} ofprobability distributions over Y , one for each input symbol x .

A zero-error code for N : a subset A ⊆ X such that each symbolfrom A can be identified unambiguously after its receipt, despitethe noise.

Equivalently: A ⊆ X such that

support(p(·|x))∩ support(p(·|x ′)) = ∅ whenever x , x ′ ∈ A distinct.

The confusability graph GN of N :

vertex set: X

x ∼ x ′ if support(p(·|x)) ∩ support(p(·|x ′)) 6= ∅.

(x ∼ x ′ iff ∃ y ∈ Y s.t. p(y |x) > 0 and p(y |x ′) > 0)

Ivan Todorov QUB

Formalism

A channel N : X → Y is a family {(p(y |x))y∈Y : x ∈ X} ofprobability distributions over Y , one for each input symbol x .

A zero-error code for N : a subset A ⊆ X such that each symbolfrom A can be identified unambiguously after its receipt, despitethe noise.

Equivalently: A ⊆ X such that

support(p(·|x))∩ support(p(·|x ′)) = ∅ whenever x , x ′ ∈ A distinct.

The confusability graph GN of N :

vertex set: X

x ∼ x ′ if support(p(·|x)) ∩ support(p(·|x ′)) 6= ∅.

(x ∼ x ′ iff ∃ y ∈ Y s.t. p(y |x) > 0 and p(y |x ′) > 0)

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An example

A channel (i) and its confusability graph C5 (ii)

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One shot zero-error capacity

One shot zero-error capacity: the size of a largest zero-error code.

Equivalently: the independence number α(GN ) of the graph GN .

By definition, α(G ) is the largest independent set in G , i.e. thelargest set A ⊆ X such that

x , x ′ ∈ A, x 6= x ′ implies x 6∼ x ′.

For C5, we have α(C5) = 2.

Ivan Todorov QUB

One shot zero-error capacity

One shot zero-error capacity: the size of a largest zero-error code.

Equivalently: the independence number α(GN ) of the graph GN .

By definition, α(G ) is the largest independent set in G , i.e. thelargest set A ⊆ X such that

x , x ′ ∈ A, x 6= x ′ implies x 6∼ x ′.

For C5, we have α(C5) = 2.

Ivan Todorov QUB

Product channels

N1 : X1 → Y1, N2 : X2 → Y2 channels. The product channel

N1 ×N2 : X1 × X2 → Y1 × Y2

is given byp(y1, y2|x1, x2) = p(y1|x1)p(y2|x2).

GN1×N2 = GN1 � GN2

G1 � G2: the strong graph product:

vertex set: X1 × X2

(x1, x2) ' (x ′1, x′2) if x1 ' x ′1 and x2 ' x ′2.

Ivan Todorov QUB

Product channels

N1 : X1 → Y1, N2 : X2 → Y2 channels. The product channel

N1 ×N2 : X1 × X2 → Y1 × Y2

is given byp(y1, y2|x1, x2) = p(y1|x1)p(y2|x2).

GN1×N2 = GN1 � GN2

G1 � G2: the strong graph product:

vertex set: X1 × X2

(x1, x2) ' (x ′1, x′2) if x1 ' x ′1 and x2 ' x ′2.

Ivan Todorov QUB

Parallel repetition and zero-error capacity

Parallel repetition of N : forming products N×n, n = 1, 2, 3, . . . .

The zero-error capacity

c0(N ) = limn→∞

n

√α(G�nN

).

Note: The limit exists due to the fact that

α(G1 � G2) ≥ α(G1)α(G2).

Strict inequality may occur you can do better on the average byusing N repeatedly.

Question: What is c0(C5)?

Ivan Todorov QUB

Parallel repetition and zero-error capacity

Parallel repetition of N : forming products N×n, n = 1, 2, 3, . . . .

The zero-error capacity

c0(N ) = limn→∞

n

√α(G�nN

).

Note: The limit exists due to the fact that

α(G1 � G2) ≥ α(G1)α(G2).

Strict inequality may occur you can do better on the average byusing N repeatedly.

Question: What is c0(C5)?

Ivan Todorov QUB

The Lovasz number

Answer (Lovasz, 1979): c0(C5) =√

5.

Method: Introduced a parameter θ(G ) such that

α(G ) ≤ θ(G ) and

θ(G1 � G2) = θ(G1)θ(G2).

The inequality θ(G1 � G2) ≤ θ(G1)θ(G2) alone will then give

c0(G ) ≤ θ(G ).

Note: θ(G ) remains the best general computable bound for c0(G ).

Ivan Todorov QUB

The Lovasz number

Answer (Lovasz, 1979): c0(C5) =√

5.

Method: Introduced a parameter θ(G ) such that

α(G ) ≤ θ(G ) and

θ(G1 � G2) = θ(G1)θ(G2).

The inequality θ(G1 � G2) ≤ θ(G1)θ(G2) alone will then give

c0(G ) ≤ θ(G ).

Note: θ(G ) remains the best general computable bound for c0(G ).

Ivan Todorov QUB

The Lovasz number

G a graph with vertex set X , |X | = d . For A ⊆ Rd+, let

A[ = {b ∈ Rd+ : 〈b, a〉 ≤ 1, ∀ a ∈ A}.

Orthogonal labelling (o.l.): a family (ax)x∈X of unit vectors s.t.

x 6' y ⇒ ax ⊥ ay .

P0(G ) ={(|〈ax , c〉|2

)x∈X : (ax)x∈X o.l. and ‖c‖ ≤ 1

}.

thab(G ) = P0(G )[

The Lovasz number

θ(G ) = max{∑

x∈X λx : (λx)x∈X ∈ thab(G )}.

Equivalently: θ(G ) = minc maxx∈X1

|〈ax ,c〉|2

Ivan Todorov QUB

The Lovasz number

G a graph with vertex set X , |X | = d . For A ⊆ Rd+, let

A[ = {b ∈ Rd+ : 〈b, a〉 ≤ 1, ∀ a ∈ A}.

Orthogonal labelling (o.l.): a family (ax)x∈X of unit vectors s.t.

x 6' y ⇒ ax ⊥ ay .

P0(G ) ={(|〈ax , c〉|2

)x∈X : (ax)x∈X o.l. and ‖c‖ ≤ 1

}.

thab(G ) = P0(G )[

The Lovasz number

θ(G ) = max{∑

x∈X λx : (λx)x∈X ∈ thab(G )}.

Equivalently: θ(G ) = minc maxx∈X1

|〈ax ,c〉|2

Ivan Todorov QUB

The Lovasz number

G a graph with vertex set X , |X | = d . For A ⊆ Rd+, let

A[ = {b ∈ Rd+ : 〈b, a〉 ≤ 1, ∀ a ∈ A}.

Orthogonal labelling (o.l.): a family (ax)x∈X of unit vectors s.t.

x 6' y ⇒ ax ⊥ ay .

P0(G ) ={(|〈ax , c〉|2

)x∈X : (ax)x∈X o.l. and ‖c‖ ≤ 1

}.

thab(G ) = P0(G )[

The Lovasz number

θ(G ) = max{∑

x∈X λx : (λx)x∈X ∈ thab(G )}.

Equivalently: θ(G ) = minc maxx∈X1

|〈ax ,c〉|2

Ivan Todorov QUB

The Lovasz Sandwich Theorem

α(G ) ≤ c0(G ) ≤ θ(G ) ≤ χf(G ).

χf(G ): the fractional chromatic number of the complement of G .

χf(G ) = max{∑

x∈X λx :∑

x∈K λx ≤ 1, ∀ clique K}

The Strong Sandwich Theorem

vp(G ) ⊆ thab(G ) ⊆ fvp(G ).

vp(G ) = conv{χS : S ⊆ X independent set}fvp(G ) = conv{χK : K ⊆ X clique}[

vp(G ), thab(G ) and fvp(G ) are convex corners.

Ivan Todorov QUB

The Lovasz Sandwich Theorem

α(G ) ≤ c0(G ) ≤ θ(G ) ≤ χf(G ).

χf(G ): the fractional chromatic number of the complement of G .

χf(G ) = max{∑

x∈X λx :∑

x∈K λx ≤ 1, ∀ clique K}

The Strong Sandwich Theorem

vp(G ) ⊆ thab(G ) ⊆ fvp(G ).

vp(G ) = conv{χS : S ⊆ X independent set}fvp(G ) = conv{χK : K ⊆ X clique}[

vp(G ), thab(G ) and fvp(G ) are convex corners.

Ivan Todorov QUB

Convex corners and dualities

Convex corner

A ⊆ Rd+ : convex, closed, hereditary

(Hereditary: a ∈ A, 0 ≤ b ≤ a ⇒ b ∈ A.)

vp and fvp are dual to each other

vp(G )[ = fvp(G )

thab is self-dual

thab(G )[ = thab(G )

Second anti-blocker theorem

If A ⊆ Rd+ is a convex corner then

A[[ = A.

Ivan Todorov QUB

Convex corners and dualities

Convex corner

A ⊆ Rd+ : convex, closed, hereditary

(Hereditary: a ∈ A, 0 ≤ b ≤ a ⇒ b ∈ A.)

vp and fvp are dual to each other

vp(G )[ = fvp(G )

thab is self-dual

thab(G )[ = thab(G )

Second anti-blocker theorem

If A ⊆ Rd+ is a convex corner then

A[[ = A.Ivan Todorov QUB

From classical to quantum

Replace Cd by Md .

Md – the algebra of all d × d complex matrices; (Ex ,x ′)x ,x ′∈Xmatrix units.

Trace Tr((λx ,y )) =∑

x∈X λx ;

Inner product (A,B) = Tr(B∗A);

Duality 〈A,B〉 = Tr(AB), making it into a self-dual space;

Positivity A ≥ 0 if (Aξ, ξ) ≥ 0, ∀ ξ.

Let DX ⊆ Md be the subalgebra of all diagonal matrices.

Going from classical to quantum, we move from DX to Md ,and from sets to projections.

Ivan Todorov QUB

From classical to quantum

Replace Cd by Md .

Md – the algebra of all d × d complex matrices; (Ex ,x ′)x ,x ′∈Xmatrix units.

Trace Tr((λx ,y )) =∑

x∈X λx ;

Inner product (A,B) = Tr(B∗A);

Duality 〈A,B〉 = Tr(AB), making it into a self-dual space;

Positivity A ≥ 0 if (Aξ, ξ) ≥ 0, ∀ ξ.

Let DX ⊆ Md be the subalgebra of all diagonal matrices.

Going from classical to quantum, we move from DX to Md ,and from sets to projections.

Ivan Todorov QUB

Quantum channels

Classical channels revisited: N = {(p(y |x))y∈Y : x ∈ X}.Here, p(y |x) ≥ 0 and

∑y∈Y p(y |x) = 1, for all x ∈ X .

Let ΦN : DX → DY be the linear map

ΦN (Ex ,x) =∑y∈Y

p(y |x)Ey ,y .

ΦN is positive: A ≥ 0 ⇒ ΦN (A) ≥ 0.

ΦN is trace preserving: Tr(ΦN (A)) = Tr(A).

Quantum channel

Φ : Md → Mk linear, completely positive, trace preserving.

Completely positive: Φ⊗ idd : Md ⊗Md → Mk ⊗Md is positive.

Ivan Todorov QUB

Quantum channels

Classical channels revisited: N = {(p(y |x))y∈Y : x ∈ X}.Here, p(y |x) ≥ 0 and

∑y∈Y p(y |x) = 1, for all x ∈ X .

Let ΦN : DX → DY be the linear map

ΦN (Ex ,x) =∑y∈Y

p(y |x)Ey ,y .

ΦN is positive: A ≥ 0 ⇒ ΦN (A) ≥ 0.

ΦN is trace preserving: Tr(ΦN (A)) = Tr(A).

Quantum channel

Φ : Md → Mk linear, completely positive, trace preserving.

Completely positive: Φ⊗ idd : Md ⊗Md → Mk ⊗Md is positive.

Ivan Todorov QUB

Kraus representation

The representation theorem

Let Φ : Md → Mk be a linear map. The following are equivalent:

Φ is a quantum channel;

there exist Ap : Cd → Ck , p = 1, . . . , r , such that

Φ(T ) =r∑

p=1

ApTA∗p, T ∈ Md ,

andr∑

p=1

A∗pAp = I .

Ivan Todorov QUB

Quantum communication

Quantum channels transmit states in Md to states in Mk .

A state in Md is a matrix ρ ∈ Md with ρ ≥ 0 and Tr(ρ) = 1.

The set of all states is convex; its extreme points are known aspure states.

Pure states: for a unit vector ξ, consider ξξ∗: (ξξ∗)(η) = (η, ξ)ξ.

Two states ρ, σ are perfectly distinguishable if Tr(ρσ) = 0.

Equivalently: there are orthogonal projections P ⊥ Q with

ρ = PρP and σ = QσQ.

Effect of noise: Pure states are not necessarily mapped to purestates.

Ivan Todorov QUB

Quantum zero-error communication

Let Φ : Md → Mk be a quantum channel.

A zero-error code for Φ is a set {ξi}mi=1 of unit vectors in Cd suchthat the states

Φ(ξ1ξ∗1),Φ(ξ2ξ

∗2), . . . ,Φ(ξmξ

∗m)

are perfectly distinguishable.

An abelian projection for Φ is a projection P in Md whose range isthe span of a zero-error code.

Write Φ(T ) =∑r

p=1 ApTA∗p. This means

ξiξ∗j ⊥ ApA

∗q, for all i , j , p, q.

One shot zero-error capacity α(Φ) of Φ: the maximum rank of anabelian projection.

Ivan Todorov QUB

Quantum zero-error communication

Let Φ : Md → Mk be a quantum channel.

A zero-error code for Φ is a set {ξi}mi=1 of unit vectors in Cd suchthat the states

Φ(ξ1ξ∗1),Φ(ξ2ξ

∗2), . . . ,Φ(ξmξ

∗m)

are perfectly distinguishable.

An abelian projection for Φ is a projection P in Md whose range isthe span of a zero-error code.

Write Φ(T ) =∑r

p=1 ApTA∗p. This means

ξiξ∗j ⊥ ApA

∗q, for all i , j , p, q.

One shot zero-error capacity α(Φ) of Φ: the maximum rank of anabelian projection.

Ivan Todorov QUB

Non-commutative confusability graphs

For a quantum channel Φ(T ) =∑r

p=1 ApTA∗p, let

SΦ = span{ApA∗q : p, q = 1, . . . , r}.

SΦ ⊆ Md ;

SΦ is an operator system: A ∈ SΦ ⇒ A∗ ∈ SΦ and I ∈ SΦ;

SΦ is independent of the Kraus representation of Φ;

P is an abelian projection for Φ if and only if PSΦP is acommutative.

Definition (Duan-Severini-Winter, 2013)

A non-commutative graph in Md is an operator system in Md ;

SΦ is called the non-commutative confusability graph of Φ.

Ivan Todorov QUB

Non-commutative confusability graphs

For a quantum channel Φ(T ) =∑r

p=1 ApTA∗p, let

SΦ = span{ApA∗q : p, q = 1, . . . , r}.

SΦ ⊆ Md ;

SΦ is an operator system: A ∈ SΦ ⇒ A∗ ∈ SΦ and I ∈ SΦ;

SΦ is independent of the Kraus representation of Φ;

P is an abelian projection for Φ if and only if PSΦP is acommutative.

Definition (Duan-Severini-Winter, 2013)

A non-commutative graph in Md is an operator system in Md ;

SΦ is called the non-commutative confusability graph of Φ.

Ivan Todorov QUB

Operator systems historically

Originated in the 1960’s in the work of Arveson;

Two viewpoints: concrete and abstract. Choi-Effros Theoremshows they are equivalent.

Lied at the base of Quantised Functional Analyisis, developedsince the 1980’s (Arveson, Christensen, Blecher, Effros,Haagerup, Pisier, Ruan, Sinclair and many others);

The natural domains of completely positive maps due torichness of positivity structure.

Applications to Quantum Information Theory: quantumcorrelations, Bell’s Theorem, non-local games, zero-errorquantum information.

Ivan Todorov QUB

Classical graphs as non-commutative graphs

Let G be a graph with vertex set X of cardinality d .

SG = span{Ex ,x ′ ,Ey ,y : y ∈ X , x ∼ x ′},

a graph operator system.

SG1∼= SG2 iff G1

∼= G2;

Let N : X → Y a classical channel. Then SGN = SΦN

justification for calling SΦ a confusability graph.

Consistency: α(G ) = α(SG )

Non-commutative graph theory: Combinatorial properties ofoperator systems.

Some successes: Non-commutative graph parameters, Ramseytheory.

Ivan Todorov QUB

Product quantum channels

If Φ1 : Md1 → Mk1 and Φ2 : Md2 → Mk2 are quantum channelsthen

Φ1 ⊗ Φ2 : Md1 ⊗Md2 → Mk1 ⊗Mk2

is a quantum channel.

For classical channels N1 and N2 we haveΦN1×N2 = ΦN1 ⊗ ΦN2 .

SΦ1⊗Φ2 = SΦ1 ⊗ SΦ2

Ivan Todorov QUB

Zero-error capacity: the quantum case

Let Φ : Md → Mk be a quantum channel.Parallel repetition of Φ : forming products Φ⊗n, n = 1, 2, 3, . . . .

Set S = SΦ.

The zero-error capacity

c0(Φ) = c0(S) = limn→∞

n√α (S⊗n).

Note: The limit exists due to the fact that

α(S1 ⊗ S2) ≥ α(S1)α(S2).

Strict inequality may occur in an extreme way:

Superactivation (Duan, 2008): ∃ Φ: α(Φ) = 1 and α(Φ⊗ Φ) > 1.

Ivan Todorov QUB

Zero-error capacity: the quantum case

Let Φ : Md → Mk be a quantum channel.Parallel repetition of Φ : forming products Φ⊗n, n = 1, 2, 3, . . . .

Set S = SΦ.

The zero-error capacity

c0(Φ) = c0(S) = limn→∞

n√α (S⊗n).

Note: The limit exists due to the fact that

α(S1 ⊗ S2) ≥ α(S1)α(S2).

Strict inequality may occur in an extreme way:

Superactivation (Duan, 2008): ∃ Φ: α(Φ) = 1 and α(Φ⊗ Φ) > 1.

Ivan Todorov QUB

The non-commutative graphs Sk

Let Sk = span{Ei ,j ,El ,l : i 6= j} ⊆ Mk , k ∈ N.

S2 ={(

λ ab λ

): λ, a, b ∈ C

},

the smallest non-trivial genuinely non-commutative graph.

α (Sk1 ⊗ · · · Skm) = 1;

c0 (Sk1 ⊗ · · · Skm) = 1;

If α(T ) = 1 then α(S2 ⊗ T ) = 1.

Ivan Todorov QUB

A quantum Lovasz number

Let S ⊆ Md be a non-commutative graph.

Duan-Severini-Winter, 2013:

θDSW(S) = max{‖I + T‖ : T ∈ S⊥, I + T ≥ 0}.

α(S) ≤ θDSW(A);

Supermultiplicativity: θDSW(S1 ⊗ S2) ≥ θDSW(S1)θDSW(S2),not useful.

θDSW(S) = maxk∈N

θDSW(S ⊗Mk).

θDSW(S1 ⊗ S2) = θDSW(S1)θDSW(S2) and so

c0(S) ≤ θDSW(S).

Ivan Todorov QUB

A quantum Lovasz number

Let S ⊆ Md be a non-commutative graph.

Duan-Severini-Winter, 2013:

θDSW(S) = max{‖I + T‖ : T ∈ S⊥, I + T ≥ 0}.

α(S) ≤ θDSW(A);

Supermultiplicativity: θDSW(S1 ⊗ S2) ≥ θDSW(S1)θDSW(S2),not useful.

θDSW(S) = maxk∈N

θDSW(S ⊗Mk).

θDSW(S1 ⊗ S2) = θDSW(S1)θDSW(S2) and so

c0(S) ≤ θDSW(S).

Ivan Todorov QUB

Advantages, disadvantages and questions

θDSW(S) is computable via semi-definite program. . .

. . . but is not always a useful bound:

θDSW(Sk) = k, θDSW(Sk) = k2.

Questions

Can we find better bounds on c0(S)?

Is there a Strong Sandwich Theorem, involving convexcorners?

Answers: YES

Ivan Todorov QUB

Advantages, disadvantages and questions

θDSW(S) is computable via semi-definite program. . .

. . . but is not always a useful bound:

θDSW(Sk) = k, θDSW(Sk) = k2.

Questions

Can we find better bounds on c0(S)?

Is there a Strong Sandwich Theorem, involving convexcorners?

Answers: YES

Ivan Todorov QUB

Orthogonal rank – classical and quantum

β(G ) = min{k : ∃ o.l. of G in Ck}

β(S) = min{k : ∃ Φ : Md → Mk quantum channel with SΦ ⊆ S}(Levene-Paulsen-T, 2018). Relation to min. semi-definite rank.

β(SG ) = β(G )

α(S) ≤ β(S)

Submultiplicativity: β(S1 ⊗ S2) ≤ β(S1)β(S2).

Can be genuinely better:

β(Sk ⊗ Sk2) ≤ k2 < k3 ≤ θDSW(Sk ⊗ Sk2).

Ivan Todorov QUB

Non-commutative convex corners

Convex corners in Md (Boreland - T - Winter)

A ⊆ M+d : convex, closed, hereditary

(A ∈ A, 0 ≤ B ≤ A =⇒ B ∈ A.)

Examples of “trivial” convex corners:

{T ∈ M+d : Tr(T ) ≤ 1} and {T ∈ M+

d : ‖T‖ ≤ 1}.

Anti-blocker: A] = {T ∈ M+d : Tr(ST ) ≤ 1, ∀ S ∈ A}.

Second anti-blocker theorem (Boreland - T - Winter)

If A is a convex corner in M+d then A]] = A.

Note: For any non-empty A ⊆ M+d , the anti-blocker A] is a

convex corner.Ivan Todorov QUB

Abelian and full projections

Let S ⊆ Md be a non-commutative graph.

Recall: A projection P ∈ Md is called abelian if it spans azero-error code for S.

Equivalently: PSP is a commutative family of matrices.

A projection P ∈ Md is called full if L(PH)⊕ 0P⊥H ⊆ S.

If G is a graph with vertex set X , a subset K is called a cliqueif x , x ′ ∈ K ⇒ x ' x ′.

If K ⊆ X is a clique for G then the projection PK ontospan{ex : x ∈ K} is full for SG . Every full projection for SG isof this form.

Ivan Todorov QUB

Convex corners from non-commutative graphs

Let S ⊆ Md be a non-commutative graph.

ap(S) = her(conv{P : an abelian projection})

(her(A) = {B ≥ 0 : ∃A ∈ A such that B ≤ A})

fp(S) = her(conv{P : a full projection})

Consistency (Boreland - T - Winter)

The convex corners ap(S) and fp(S)] are quantisations of vp(G )and fvp(G ):

ap(SG ) ∩ DX = ∆(ap(SG )) = vp(G );

fp(SG )] ∩ DX = ∆(fp(SG )]) = fvp(G ).

ap(S) ⊆ fp(S)]

Ivan Todorov QUB

Convex corners from non-commutative graphs

Let S ⊆ Md be a non-commutative graph.

ap(S) = her(conv{P : an abelian projection})

(her(A) = {B ≥ 0 : ∃A ∈ A such that B ≤ A})

fp(S) = her(conv{P : a full projection})

Consistency (Boreland - T - Winter)

The convex corners ap(S) and fp(S)] are quantisations of vp(G )and fvp(G ):

ap(SG ) ∩ DX = ∆(ap(SG )) = vp(G );

fp(SG )] ∩ DX = ∆(fp(SG )]) = fvp(G ).

ap(S) ⊆ fp(S)]

Ivan Todorov QUB

The Lovasz non-commutative corner

Let S ⊆ L(H) be an operator system.

C(S) = {Φ : Md → Mk : quantum channel with SΦ ⊆ S, k ∈ N} .

th(S) ={T ∈ L(H)+ : Φ(T ) ≤ I for every Φ ∈ C(S)

}th(S) = {Φ∗(σ) : Φ ∈ C(S), σ ≥ 0,Tr(σ) ≤ 1} .

th(S) is a convex corner and th(S) = th(S)].

Consistency (Boreland - T - Winter)

If G is a graph with a vertex set X then

th(SG ) ∩ DX = ∆(th(SG )) = thab(G ).

Ivan Todorov QUB

The Lovasz non-commutative corner

Let S ⊆ L(H) be an operator system.

C(S) = {Φ : Md → Mk : quantum channel with SΦ ⊆ S, k ∈ N} .

th(S) ={T ∈ L(H)+ : Φ(T ) ≤ I for every Φ ∈ C(S)

}th(S) = {Φ∗(σ) : Φ ∈ C(S), σ ≥ 0,Tr(σ) ≤ 1} .

th(S) is a convex corner and th(S) = th(S)].

Consistency (Boreland - T - Winter)

If G is a graph with a vertex set X then

th(SG ) ∩ DX = ∆(th(SG )) = thab(G ).

Ivan Todorov QUB

The strong Lovasz Sandwich Theorem

A quantum sandwich (Boreland - T - Winter)

Let S ⊆ Md be a non-commutative graph. Then

ap(S) ⊆ th(S) ⊆ fp(S)].

Ivan Todorov QUB

Non-commutative parameters

Maximising the trace functional yields the parameters:

max {Tr(T ) : T ∈ ap(S)} = α(S);

max {Tr(T ) : T ∈ th(S)} =: θ(S);

max{Tr(T ) : T ∈ fp(S)]

}=: ωf(S).

By the consistency results, for a graph G we have:

θ(SG ) = θ(G ) and ωf(SG ) = ωf(G )

ωf(G ) = χf(G )

Ivan Todorov QUB

The Lovasz Sandwich Theorem

Let S be a non-commutative graph in Md .

α(S) ≤ θ(S) ≤ ωf(S)

Question: Is θ a bound on the zero-error capacity?

– open

Ivan Todorov QUB

The Lovasz Sandwich Theorem

Let S be a non-commutative graph in Md .

α(S) ≤ θ(S) ≤ ωf(S)

Question: Is θ a bound on the zero-error capacity? – open

Ivan Todorov QUB

The parameter θ

Let S be a non-commutative graph.

θ(S) = inf{∥∥Φ∗(σ)−1

∥∥ : σ ≥ 0,Tr(σ) ≤ 1,Φ ∈ C(S),Φ∗(σ) invertible}

θ vs. θ

(i) θ(S)−1 = sup {inf {‖Φ(ρ)‖ : ρ a state on H} : Φ ∈ C(S)};

(ii) θ(S)−1 = inf {sup {‖Φ(ρ)‖ : Φ ∈ C(S)} : ρ a state on H}.

d inf{‖Φ(Id)‖−1 : Φ ∈ C(S)

}≤ θ(S) ≤ θ(S) ≤ β(S) ≤ d .

Ivan Todorov QUB

The parameter θ

Let S be a non-commutative graph.

θ(S) = inf{∥∥Φ∗(σ)−1

∥∥ : σ ≥ 0,Tr(σ) ≤ 1,Φ ∈ C(S),Φ∗(σ) invertible}

θ vs. θ

(i) θ(S)−1 = sup {inf {‖Φ(ρ)‖ : ρ a state on H} : Φ ∈ C(S)};

(ii) θ(S)−1 = inf {sup {‖Φ(ρ)‖ : Φ ∈ C(S)} : ρ a state on H}.

d inf{‖Φ(Id)‖−1 : Φ ∈ C(S)

}≤ θ(S) ≤ θ(S) ≤ β(S) ≤ d .

Ivan Todorov QUB

The parameter θ

Let S be a non-commutative graph.

θ(S) = inf{∥∥Φ∗(σ)−1

∥∥ : σ ≥ 0,Tr(σ) ≤ 1,Φ ∈ C(S),Φ∗(σ) invertible}

θ vs. θ

(i) θ(S)−1 = sup {inf {‖Φ(ρ)‖ : ρ a state on H} : Φ ∈ C(S)};

(ii) θ(S)−1 = inf {sup {‖Φ(ρ)‖ : Φ ∈ C(S)} : ρ a state on H}.

d inf{‖Φ(Id)‖−1 : Φ ∈ C(S)

}≤ θ(S) ≤ θ(S) ≤ β(S) ≤ d .

Ivan Todorov QUB

Consistency

Let G be a graph. Then θ(SG ) = θ(G ).

θ(G ) = θ(SG ) ≤ θ(SG )

Let (ax)x∈X ⊆ Ck be an orthogonal labelling.

Φ(S) =∑x∈X

(axe∗x )S(exa

∗x), S ∈ Md .

If x 6' y then (exa∗x)(aye

∗y ) = 〈ay , ax〉exe∗y = 0 ⇒ SΦ ⊆ SG .

Let c ∈ Ck s.t. 〈ax , c〉 6= 0, x ∈ X . Then∥∥Φ∗(cc∗)−1∥∥ = max

x∈X

1

|〈ax , c〉|2.

⇒ θ(SG ) ≤ θ(G ).

Ivan Todorov QUB

Consistency

Let G be a graph. Then θ(SG ) = θ(G ).

θ(G ) = θ(SG ) ≤ θ(SG )

Let (ax)x∈X ⊆ Ck be an orthogonal labelling.

Φ(S) =∑x∈X

(axe∗x )S(exa

∗x), S ∈ Md .

If x 6' y then (exa∗x)(aye

∗y ) = 〈ay , ax〉exe∗y = 0 ⇒ SΦ ⊆ SG .

Let c ∈ Ck s.t. 〈ax , c〉 6= 0, x ∈ X . Then∥∥Φ∗(cc∗)−1∥∥ = max

x∈X

1

|〈ax , c〉|2.

⇒ θ(SG ) ≤ θ(G ).

Ivan Todorov QUB

Submultiplicativity

θ(S1 ⊗ S2) ≤ θ(S1)θ(S2)

Let σi ∈ M+di

s.t. Tr(σi ) ≤ 1 and Φ∗i (σi ) invertible, andΦi : Mdi → Mki be q. c. with SΦi

⊆ Si , s. t.∥∥Φ∗i (σi )−1∥∥ ≤ θ1(Si ) + ε, i = 1, 2.

Φ1 ⊗ Φ2 : Md1d2 → Mk1k2 is a q. c. with SΦ1⊗Φ2 ⊆ S1 ⊗ S2.

θ(S1 ⊗ S2) ≤∥∥(Φ1 ⊗ Φ2)∗(σ1 ⊗ σ2)−1

∥∥=

∥∥Φ∗1(σ1)−1∥∥∥∥Φ∗2(σ2)−1

∥∥≤ (θ(S1) + ε)(θ(S2) + ε).

Ivan Todorov QUB

Submultiplicativity

θ(S1 ⊗ S2) ≤ θ(S1)θ(S2)

Let σi ∈ M+di

s.t. Tr(σi ) ≤ 1 and Φ∗i (σi ) invertible, andΦi : Mdi → Mki be q. c. with SΦi

⊆ Si , s. t.∥∥Φ∗i (σi )−1∥∥ ≤ θ1(Si ) + ε, i = 1, 2.

Φ1 ⊗ Φ2 : Md1d2 → Mk1k2 is a q. c. with SΦ1⊗Φ2 ⊆ S1 ⊗ S2.

θ(S1 ⊗ S2) ≤∥∥(Φ1 ⊗ Φ2)∗(σ1 ⊗ σ2)−1

∥∥=

∥∥Φ∗1(σ1)−1∥∥∥∥Φ∗2(σ2)−1

∥∥≤ (θ(S1) + ε)(θ(S2) + ε).

Ivan Todorov QUB

θ is a bound on the zero-error capacity

Since α(S) ≤ θ(S), the submultiplicativity of θ immediately yields:

c0(S) ≤ θ(S)

Note: θ can be a genuinely better bound on the zero-error capacitythan θDSW .

Ivan Todorov QUB

Some more properties

θ can be efficiently computed: it suffices to consider channelsfrom Md into Md2 .

No need for higher dimension of the output system.

the map S → θ(S) is continuous.

θ is monotone: S → T implies θ(S) ≤ θ(T ) andθ(S) ≤ θ(T ).

Consequence: θ(Mn(S)) = θ(S) and θ(Mn(S)) = θ(S).

θ(S) = 1 iff θ(S) = 1 iff S = Md .

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = θ(S)?

Perhaps not – counterexample?

Is the map S → θ(S) continuous?

Difficulty: Unboundedness of dimensions no compactnessarguments applicable

Is there a duality theorem for the non-commutative thetacorner, th(S)] = th(S)?

Difficulty: The non-commutative graph complement.

Does θ(S) arise from a convex corner?

What are the values of θ(Sk) and θ(Sk)?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = θ(S)?

Perhaps not – counterexample?

Is the map S → θ(S) continuous?

Difficulty: Unboundedness of dimensions no compactnessarguments applicable

Is there a duality theorem for the non-commutative thetacorner, th(S)] = th(S)?

Difficulty: The non-commutative graph complement.

Does θ(S) arise from a convex corner?

What are the values of θ(Sk) and θ(Sk)?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = θ(S)?

Perhaps not – counterexample?

Is the map S → θ(S) continuous?

Difficulty: Unboundedness of dimensions no compactnessarguments applicable

Is there a duality theorem for the non-commutative thetacorner, th(S)] = th(S)?

Difficulty: The non-commutative graph complement.

Does θ(S) arise from a convex corner?

What are the values of θ(Sk) and θ(Sk)?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = θ(S)?

Perhaps not – counterexample?

Is the map S → θ(S) continuous?

Difficulty: Unboundedness of dimensions no compactnessarguments applicable

Is there a duality theorem for the non-commutative thetacorner, th(S)] = th(S)?

Difficulty: The non-commutative graph complement.

Does θ(S) arise from a convex corner?

What are the values of θ(Sk) and θ(Sk)?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = θ(S)?

Perhaps not – counterexample?

Is the map S → θ(S) continuous?

Difficulty: Unboundedness of dimensions no compactnessarguments applicable

Is there a duality theorem for the non-commutative thetacorner, th(S)] = th(S)?

Difficulty: The non-commutative graph complement.

Does θ(S) arise from a convex corner?

What are the values of θ(Sk) and θ(Sk)?

Ivan Todorov QUB

THANK YOU VERY MUCH

Ivan Todorov QUB