Zero-error quantum information theory with Gareth Boreland, QUB - - PowerPoint PPT Presentation

Zero-error quantum information theory

with Gareth Boreland, QUB Rupert Levene, Dublin Vern Paulsen, IQC Waterloo Andreas 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

Ivan Todorov QUB

The Shannon model

Source → Encoder → Channel → Decoder → Target A channel N transmits symbols from an alphabet X into an alphabet Y :

Ivan Todorov QUB

Formalism

A channel N : X → Y is a family {(p(y|x))y∈Y : x ∈ X} of probability distributions over Y , one for each input symbol x. A zero-error code for N: a subset A ⊆ X such that each symbol from A can be identified unambiguously after its receipt, despite the noise. Equivalently: A ⊆ X such that support(p(·|x)) ∩ support(p(·|x′)) = ∅ whenever x, x′ ∈ A distinct.

Ivan Todorov QUB

Formalism

A channel N : X → Y is a family {(p(y|x))y∈Y : x ∈ X} of probability distributions over Y , one for each input symbol x. A zero-error code for N: a subset A ⊆ X such that each symbol from A can be identified unambiguously after its receipt, despite the 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′)) = ∅. (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 .

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. the largest set A ⊆ X such that x, x′ ∈ A, x = x′ implies x ∼ 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 by p(y1, y2|x1, x2) = p(y1|x1)p(y2|x2).

Ivan Todorov QUB

Product channels

N1 : X1 → Y1, N2 : X2 → Y2 channels. The product channel N1 × N2 : X1 × X2 → Y1 × Y2 is given by p(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) = lim

n→∞

n

• α
• G ⊠n

N

• .

Note: The limit exists due to the fact that α(G1 ⊠ G2) ≥ α(G1)α(G2).

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) = lim

n→∞

n

• α
• G ⊠n

N

• .

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 by using N repeatedly. Question: What is c0(C5)?

Ivan Todorov QUB

The Lov´ asz number

Answer (Lov´ asz, 1979): c0(C5) = √ 5. Method: Introduced a parameter θ(G) such that α(G) ≤ θ(G) and θ(G1 ⊠ G2) = θ(G1)θ(G2).

Ivan Todorov QUB

The Lov´ asz number

Answer (Lov´ asz, 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 Lov´ asz number

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

+, let

A♭ = {b ∈ Rd

+ : b, a ≤ 1, ∀ a ∈ A}.

Ivan Todorov QUB

The Lov´ asz 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 ≃ y ⇒ ax ⊥ ay. P0(G) =

• |ax, c|2

x∈X : (ax)x∈X o.l. and c ≤ 1

• .

thab(G) = P0(G)♭ The Lov´ asz number θ(G) = max

• x∈X λx : (λx)x∈X ∈ thab(G)
• .

Ivan Todorov QUB

The Lov´ asz 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 ≃ y ⇒ ax ⊥ ay. P0(G) =

• |ax, c|2

x∈X : (ax)x∈X o.l. and c ≤ 1

• .

thab(G) = P0(G)♭ The Lov´ asz number θ(G) = max

• x∈X λx : (λx)x∈X ∈ thab(G)
• .

Equivalently: θ(G) = minc maxx∈X

1 |ax,c|2

Ivan Todorov QUB

The Lov´ asz 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

• Ivan Todorov

QUB

The Lov´ asz 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.)

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′∈X matrix 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.

Ivan Todorov QUB

From classical to quantum

Replace Cd by Md. Md – the algebra of all d × d complex matrices; (Ex,x′)x,x′∈X matrix 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).

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, and

r

• 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 as pure 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 pure states.

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}m

i=1 of unit vectors in Cd such

that 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 is the span of a zero-error code.

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}m

i=1 of unit vectors in Cd such

that 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 is the 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 an abelian 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 a commutative.

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 a commutative. 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 Theorem shows they are equivalent. Lied at the base of Quantised Functional Analyisis, developed since the 1980’s (Arveson, Christensen, Blecher, Effros, Haagerup, Pisier, Ruan, Sinclair and many others); The natural domains of completely positive maps due to richness of positivity structure. Applications to Quantum Information Theory: quantum correlations, Bell’s Theorem, non-local games, zero-error quantum 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 of

• perator systems.

Some successes: Non-commutative graph parameters, Ramsey theory.

Ivan Todorov QUB

Product quantum channels

If Φ1 : Md1 → Mk1 and Φ2 : Md2 → Mk2 are quantum channels then Φ1 ⊗ Φ2 : Md1 ⊗ Md2 → Mk1 ⊗ Mk2 is a quantum channel. For classical channels N1 and N2 we have ΦN1×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) = lim

n→∞

n

• α (S⊗n).

Note: The limit exists due to the fact that α(S1 ⊗ S2) ≥ α(S1)α(S2).

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) = lim

n→∞

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 = j} ⊆ Mk, k ∈ N. S2 = λ a

b 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 Lov´ asz 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.

Ivan Todorov QUB

A quantum Lov´ asz 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) = max

k∈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. . .

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 convex corners? 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 a zero-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 clique if x, x′ ∈ K ⇒ x ≃ x′. If K ⊆ X is a clique for G then the projection PK onto span{ex : x ∈ K} is full for SG. Every full projection for SG is

• f 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).

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 Lov´ asz 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)♯.

Ivan Todorov QUB

The Lov´ asz 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 Lov´ asz 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 Lov´ asz 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?

Ivan Todorov QUB

The Lov´ asz 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 ˆ θ

Let S be a non-commutative graph. ˆ θ(S) = inf

• Φ∗(σ)−1

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

• Ivan Todorov

QUB

The parameter ˆ θ

Let S be a non-commutative graph. ˆ θ(S) = inf

• Φ∗(σ)−1

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

• θ vs. ˆ

θ (i) ˆ 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}.

Ivan Todorov QUB

The parameter ˆ θ

Let S be a non-commutative graph. ˆ θ(S) = inf

• Φ∗(σ)−1

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

• θ vs. ˆ

θ (i) ˆ 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) ≤ ˆ

θ(S) ≤ β(S) ≤ d.

Ivan Todorov QUB

Consistency

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

Ivan Todorov QUB

Consistency

Let G be a graph. Then ˆ θ(SG) = θ(G). θ(G) = θ(SG) ≤ ˆ θ(SG) Let (ax)x∈X ⊆ Ck be an orthogonal labelling. Φ(S) =

• x∈X

(axe∗

x)S(exa∗ x),

S ∈ Md. If x ≃ y then (exa∗

x)(aye∗ y) = ay, axexe∗ y = 0 ⇒ SΦ ⊆ SG.

Let c ∈ Ck s.t. ax, c = 0, x ∈ X. Then

• Φ∗(cc∗)−1

= max

x∈X

1 |ax, c|2 . ⇒ ˆ θ(SG) ≤ θ(G).

Ivan Todorov QUB

Submultiplicativity

ˆ θ(S1 ⊗ S2) ≤ ˆ θ(S1)ˆ θ(S2) Let σi ∈ M+

di s.t. Tr(σi) ≤ 1 and Φ∗ i (σi) invertible, and

Φi : Mdi → Mki be q. c. with SΦi ⊆ Si, s. t.

• Φ∗

i (σi)−1

≤ ˆ θ1(Si) + ǫ, i = 1, 2.

Ivan Todorov QUB

Submultiplicativity

ˆ θ(S1 ⊗ S2) ≤ ˆ θ(S1)ˆ θ(S2) Let σi ∈ M+

di s.t. Tr(σi) ≤ 1 and Φ∗ i (σi) invertible, and

Φi : Mdi → Mki be q. c. with SΦi ⊆ Si, s. t.

• Φ∗

i (σi)−1

≤ ˆ θ1(Si) + ǫ, i = 1, 2. Φ1 ⊗ Φ2 : Md1d2 → Mk1k2 is a q. c. with SΦ1⊗Φ2 ⊆ S1 ⊗ S2. ˆ θ(S1 ⊗ S2) ≤

• (Φ1 ⊗ Φ2)∗(σ1 ⊗ σ2)−1
• =
• Φ∗

1(σ1)−1

Φ∗

2(σ2)−1

• ≤

(ˆ θ(S1) + ǫ)(ˆ θ(S2) + ǫ).

Ivan Todorov QUB

ˆ θ is a bound on the zero-error capacity

Since α(S) ≤ ˆ θ(S), the submultiplicativity of ˆ θ immediately yields: c0(S) ≤ ˆ θ(S) Note: ˆ θ can be a genuinely better bound on the zero-error capacity than θDSW .

Ivan Todorov QUB

Some more properties

θ can be efficiently computed: it suffices to consider channels from 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 ) and ˆ θ(S) ≤ ˆ θ(T ). Consequence: θ(Mn(S)) = θ(S) and ˆ θ(Mn(S)) = ˆ θ(S). θ(S) = 1 iff ˆ θ(S) = 1 iff S = Md.

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = ˆ θ(S)? Perhaps not – counterexample?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = ˆ θ(S)? Perhaps not – counterexample? Is the map S → ˆ θ(S) continuous? Difficulty: Unboundedness of dimensions no compactness arguments applicable

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = ˆ θ(S)? Perhaps not – counterexample? Is the map S → ˆ θ(S) continuous? Difficulty: Unboundedness of dimensions no compactness arguments applicable Is there a duality theorem for the non-commutative theta corner, th(S)♯ = th( ¯ S)? Difficulty: The non-commutative graph complement.

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = ˆ θ(S)? Perhaps not – counterexample? Is the map S → ˆ θ(S) continuous? Difficulty: Unboundedness of dimensions no compactness arguments applicable Is there a duality theorem for the non-commutative theta corner, th(S)♯ = th( ¯ S)? Difficulty: The non-commutative graph complement. Does ˆ θ(S) arise from a convex corner?

Ivan Todorov QUB

Some open questions

Is it true that θ(S) = ˆ θ(S)? Perhaps not – counterexample? Is the map S → ˆ θ(S) continuous? Difficulty: Unboundedness of dimensions no compactness arguments applicable Is there a duality theorem for the non-commutative theta corner, th(S)♯ = th( ¯ S)? Difficulty: The non-commutative graph complement. Does ˆ θ(S) arise from a convex corner? What are the values of θ(Sk) and ˆ θ(Sk)?

Ivan Todorov QUB

THANK YOU VERY MUCH

Ivan Todorov QUB