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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)
Ivan Todorov QUB
An example
A channel (i) and its confusability graph C5 (ii)
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
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
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