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A sandwich theorem and a capacity bound for non-commutative graphs

Gareth Boreland (Queen’s University Belfast) Joint work with Ivan Todorov (Queen’s University Belfast) and Andreas Winter (Universitat Autonoma de Barcelona). (arXiv: 1907.11504)

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The confusability graph of a classical channel

Example Consider classical channel N : X = {1, 2, 3, 4} → Y = {a, b, c, d}. (An arrow from x ∈ X to y ∈ Y denotes that p(y|x) > 0.)

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• a
• 2
• b
• 3
• c
• 4
• d

Channel N has confusability graph GN:

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• 4
• 3

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One-shot zero-error capacity

The one-shot zero-error capacity of channel N : X → Y is the maximum cardinality of a subset X0 ⊆ X such that when the sender chooses letters only from X0, there is no potential confusion. Given by α(GN), the independence number of GN. Example

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{1, 3, 4} is the largest independent set in GN and α(GN) = 3.

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Shannon capacity

If G, H are graphs, the strong product G ⊠ H has vertex set V (G) × V (H) with (i, p) ≃ (j, q) when i ≃ j in G and p ≃ q in H. Suppose channel N is used k times. Regard as a single use of channel Nk : X k → Yk. Two message strings in X k are confusable iff they are confusable or equal at every co-ordinate. So GNk = G ⊠k

N

where G ⊠k is the kth strong power of G. Definition (Shannon) The zero-error capacity of GN is given by c(GN) = lim

n→∞

n

• α(G ⊠n

N ).

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The Lov´ asz number

Definition (Lov´ asz) The Lov´ asz number θ(G) of graph G is given by θ(G) = max

• I +T : T = (tij), i ≃ j in G ⇒ tij = 0, I +T ≥ 0
• .

The Lov´ asz number satisfies i θ(G) ≥ α(G), and ii θ(G ⊠ H) ≤ θ(G)θ(H). This gives α(G ⊠n) ≤ θ(G ⊠n) ≤ θ(G)n. Immediately we have the following. Theorem (Lov´ asz) For a graph G, c(G) ≤ θ(G).

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Convex corners in Rn

A set A ⊆ Rn is an Rn-convex corner if A is (i) closed, (ii) convex, (iii) non-empty, (iv) non-negative, that is a ≥ 0 for all a ∈ A, in the sense that ai ≥ 0, i = 1, . . . , n, (v) hereditary, in the sense that if a ∈ A and b ≤ a, then b ∈ A.

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Two definitions

Definition If A ⊆ Rn

+, then A♭, the antiblocker of A, is defined by

A♭ =

• v ∈ Rn

+ : v, u ≤ 1 ∀u ∈ A

• .

Definition For a set A ⊆ Rn

+, we define

γ(A) = max n

• i=1

ai : a ∈ A

• .

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The vertex packing polytope

Consider graph G. Form the characteristic vectors of each independent set. (For example, in

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{1, 3, 4} has characteristic vector

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1 1 t.) Let the convex hull of these characteristic vectors be VP(G), the vertex packing polytope of G. VP(G) is a convex corner. γ(VP(G)) = α(G).

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The theta convex body

An orthonormal labelling (o.l.) of G is a set (a(i))i∈V (G) of unit vectors in Rk satisfying

• a(i), a(j)

= 0 when i ≃ j in G. Definition (Gr¨

• tschel, Lov´

asz and Schrijver) The convex corner TH(G), known as the theta convex body of graph G, is given by TH(G) =

• |a(i), c|2

i∈V (G) : (a(i))i∈V (G) an o.l., c ≤ 1

♭ . Proposition (Gr¨

• tschel, Lov´

asz and Schrijver) If G is a graph, then γ(TH(G)) = θ(G).

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The fractional vertex packing polytope

Function f : V (G) → R+ is a fractional clique of G if

• i∈S f (i) ≤ 1 for every independent set S, or equivalently if

(f (i))i∈V (G) ∈ VP(G)♭. Definition The fractional clique number of graph G is given by ωf(G) = γ(VP(G)♭). Definition The fractional vertex packing polytope of G is the convex corner VP(G)♭. It then holds that γ(VP(G)♭) = ωf(G).

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The Sandwich Theorem

Theorem (Gr¨

• tschel, Lov´

asz and Schrijver) For a graph G, VP(G) ⊆ TH(G) ⊆ VP(G)♭. Corollary For a graph G, α(G) ≤ θ(G) ≤ ωf(G).

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Quantum channels and non-commutative graphs

Consider quantum channel Φ : Md → Mk. We define α(Φ), the one-shot zero-error capacity of Φ, as the largest n st

1 v1, . . . , vn ∈ Cd are orthonormal, 2 Φ(v1v∗

1 ), . . . , Φ(vnv∗ n) are perfectly distinguishable

(that is, Φ(viv∗

i )Φ(vjv∗ j ) = 0 ∀i = j.)

Definition A subspace S ⊆ Md is a non-commutative graph (n.c.g.) if i I ∈ S and ii A ∈ S ⇒ A∗ ∈ S.

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Non-commutative graphs

Example (1) With confusability graph G on d vertices we associate the n.c.g. SG = span{Eij : i ≃ j in G} ⊆ Md. Distinct i, j ∈ V (G) are distinguishable iff Eij = eie∗

j ∈ S⊥ G .

For example, G =

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gives SG =            a11 a12 a21 a22 a23 a32 a33 a44     : aij ∈ C        .

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How do n.c.g.s generalise graphs?

Example (2) Let quantum channel Φ : Md → Mk have Kraus representation Φ(ρ) =

m

• i=1

EiρE ∗

i .

The subspace SΦ ⊆ Md is a n.c.g. where SΦ = span{E ∗

i Ej : i, j ∈ [m]}.

For orthonormal u, v ∈ Cd, Φ(uu∗), Φ(vv∗) are distinguishable iff uv∗ ∈ S⊥

Φ .

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Independence number and zero-error capacity of a n.c.g.

For n.c.g. S ⊆ Md, the orthonormal set {v1, . . . , vn} is an independent set when viv∗

j ∈ S⊥ for all i = j.

The size of the largest independent set in S is called α(S), the independence number of S. We have α(Φ) = α(SΦ). Definition (Duan, Severini, Winter) The zero-error capacity of n.c.g. S is given by c(S) = lim

n→∞

n

• α(S⊗n).

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Generalising the Lov´ asz number

For graph G, the Lov´ asz number satisfies θ(G) = max{I + T : T ∈ S⊥

G , I + T ≥ 0}.

Definition (Duan, Severini, Winter) For n.c.g. S, let ϑ(S) = max{I + T : T ∈ S⊥, I + T ≥ 0}, with ‘complete version’ ˜ ϑ(S) = sup

m∈N

ϑ(Mm(S)).

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Generalising the Lov´ asz number

Theorem (Duan, Severini, Winter) The parameter ˜ ϑ satisfies i ˜ ϑ(SG) = θ(G), ii ˜ ϑ(S) ≥ α(S), iii ˜ ϑ(S ⊗ T ) ≤ ˜ ϑ(S)˜ ϑ(T ). Then ˜ ϑ(S) is a ‘quantum generalisation’ of the Lov´ asz number and c(S) ≤ ˜ ϑ(S).

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Convex corners in Md

A set A ⊆ Md is an Md-convex corner if A is (i) closed, (ii) convex, (iii) non-empty, (iv) non-negative, that is A ≥ 0 for all A ∈ A, (v) hereditary, in the sense that if A ∈ A and B ≤ A, then B ∈ A.

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Two definitions

Definition If A ⊆ M+

d , then A♯, the antiblocker of A, is defined by

A♯ =

• B ∈ M+

d : B, A ≤ 1 ∀A ∈ A

• .

Definition For a set A ⊆ M+

d , we define

γ(A) = max

• Tr A : A ∈ A
• .

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The abelian projection convex corner

Recall that orthonormal set {v1, . . . , vn} is S-independent when viv∗

j ∈ S⊥ for all i = j.

If {v1, . . . , vn} is S-independent, we call n

i=1 viv∗ i an S-abelian

projection. Definition We define ap(S), the abelian projection convex corner to be the convex corner generated by the abelian projections. Lemma If S is a n.c.g., then γ(ap(S)) = α(S). Definition Recalling that ωf(G) = γ(VP(G)♭), for any n.c.g. S we define ωf(S) = γ(ap(S)♯).

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Relating convex corners in Rd and Md

Let Dd denote the diagonal d × d matrices. Let ∆ : Md → Dd be given by ∆(M) = d

i=1 ei, Mei eie∗ i .

Regard a subset of Dd as a subset of Rd in the canonical way. Lemma VP(G) = ∆(ap(SG)) = Dd ∩ ap(SG) Corollary For graph G we have α(SG) = α(G) and ωf(SG) = ωf(G).

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The full projection convex corner

Orthonormal set {v1, . . . , vn} is S-full when viv∗

j ∈ S for all i, j.

If {v1, . . . , vn} is S-full, n

i=1 viv∗ i is called an S-full projection.

Definition We define fp(S), the full projection convex corner, to be the convex corner generated by the full projections. Lemma VP(G) = ∆(fp(SG)) = Dd ∩ fp(SG). Definition We define ϕ(S) = γ(fp(S)♯). Corollary For graph G we have ϕ(SG) = ωf(G).

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Towards a quantum sandwich theorem

Recall the Sandwich Theorem for graph G: VP(G) ⊆ TH(G) ⊆ VP(G)♭. ap(S) is the ‘quantum version’ of VP(G). fp(S)♯ is the ‘quantum version’ of VP(G)♭. It is not hard to show that ap(S) ⊆ fp(S)♯. Question Can we find a quantum version of TH(G) to complete the quantum sandwich?

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Generalising TH(G): the theta corner

Definition If S ⊆ Md is a n.c.g., we define th(S), the theta corner of S, by th(S) = {T ∈ M+

d : Φ(T) ≤ I ∀ c.p.t.p. Φ satisfying SΦ ⊆ S}.

We define the Lov´ asz number of S by θ(S) = γ(th(S)). The next results show th(S) and θ(S) are quantum versions of TH(G) and θ(G). Theorem For graph G, TH(G) = ∆(th(SG)) = Dd ∩ th(SG). Corollary θ(SG) = θ(G).

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Filling the sandwich: the theta corner

Theorem (Sandwich Theorem) For n.c.g. S, it holds that ap(S) ⊆ th(S) ⊆ fp(S)♯. Corollary α(S) ≤ θ(S) ≤ ϕ(S). (i) We note θ(S) = ˜ ϑ(S) in general. (ii) It is not known if θ(S) is sub-multiplicative and hence an upper bound on c(S).

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Another quantum version of θ(G)

The Lov´ asz number of graph G can be expressed as θ(G) = min

• max

i∈V (G)

1

• a(i), c
• 2 : c = 1, (a(i))i∈V (G) an o.l. of G
• .

This motivates the following definition. Definition For n.c.g. S ⊆ Md, we define ˆ θ(S) = inf{Φ∗(σ)−1 : σ a state, SΦ ⊆ S}. Proposition If G is a graph, ˆ θ(SG) = θ(G).

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Properties of ˆ θ(S)

If S and T are n.c.g.s then: α(S) ≤ ˆ θ(S); ˆ θ(S ⊗ T ) ≤ ˆ θ(S)ˆ θ(T ); Thus c(S) ≤ ˆ θ(S). Remark We note that ˆ θ(S) = ˜ ϑ(S) in general. Indeed, the ratio

ˆ θ(S) ˜ ϑ(S) can be arbitrarily small.

Questions Is θ(S) = ˆ θ(S) in general? Does ˆ θ(S) arise form a convex corner?

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Summary: the classical case embeds in the quantum

Classical

• Quantum

graph G

• non-commutative graph S

classical channel N

• Quantum channel Φ

confusability graph GN

• n.c.g. SΦ

Rd-convex corner A

• Md-convex corner A

γ(A) = max{d

i=1 ai : a ∈ A}

• γ(A) = max{Tr A : A ∈ A}

Antiblocker A♭

• Antiblocker A♯

VP(G) ⊆ TH(G) ⊆ VP(G)♭

• ap(S) ⊆ th(S) ⊆ fp(S)♯

α(G) ≤ θ(G) ≤ ωf(G)

• α(S) ≤ θ(S) ≤ φ(S)

Lov´ asz number, θ(G)

• θ(S), ˆ

θ(S), ϑ(S), ˜ ϑ(S) c(G) ≤ θ(G)

• c(S) ≤ ˆ

θ(S)

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Thanks for your time.

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