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

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) QUB

� � � � 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 . ) • 1 • a Channel N has confusability graph G N : • 2 • b • 1 • 2 � • c • 3 • 4 • 3 � • d • 4 QUB

One-shot zero-error capacity The one-shot zero-error capacity of channel N : X → Y is the maximum cardinality of a subset X 0 ⊆ X such that when the sender chooses letters only from X 0 , there is no potential confusion. Given by α ( G N ), the independence number of G N . Example • 1 ◦ 2 • 4 • 3 { 1 , 3 , 4 } is the largest independent set in G N and α ( G N ) = 3. QUB

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 N k : X k → Y k . Two message strings in X k are confusable iff they are confusable or equal at every co-ordinate. where G ⊠ k is the k th strong power of G . So G N k = G ⊠ k N Definition (Shannon) The zero-error capacity of G N is given by � n α ( G ⊠ n c ( G N ) = lim N ) . n →∞ QUB

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 = ( t ij ) , i ≃ j in G ⇒ t ij = 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 ) . QUB

Convex corners in R n A set A ⊆ R n is an R n -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 a i ≥ 0, i = 1 , . . . , n , (v) hereditary, in the sense that if a ∈ A and b ≤ a , then b ∈ A . QUB

Two definitions Definition + , then A ♭ , the antiblocker of A , is defined by If A ⊆ R n A ♭ = v ∈ R n � � + : � v , u � ≤ 1 ∀ u ∈ A . Definition For a set A ⊆ R n + , we define � n � � γ ( A ) = max a i : a ∈ A . i =1 QUB

The vertex packing polytope Consider graph G . Form the characteristic vectors of each independent set. (For example, in • 1 ◦ 2 • 4 • 3 � t .) � { 1 , 3 , 4 } has characteristic vector 1 0 1 1 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 ) . QUB

The theta convex body An orthonormal labelling (o.l.) of G is a set ( a ( i ) ) i ∈ V ( G ) of unit vectors in R k satisfying a ( i ) , a ( j ) � � = 0 when i �≃ j in G . Definition (Gr¨ otschel, Lov´ asz and Schrijver) The convex corner TH ( G ), known as the theta convex body of graph G , is given by � ♭ �� |� a ( i ) , c �| 2 � i ∈ V ( G ) : ( a ( i ) ) i ∈ V ( G ) an o.l. , � c � ≤ 1 TH ( G ) = . Proposition (Gr¨ otschel, Lov´ asz and Schrijver) If G is a graph, then γ ( TH ( G )) = θ ( G ) . QUB

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 ω f ( 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 ) ♭ ) = ω f ( G ) . QUB

The Sandwich Theorem Theorem (Gr¨ otschel, Lov´ asz and Schrijver) For a graph G, VP ( G ) ⊆ TH ( G ) ⊆ VP ( G ) ♭ . Corollary For a graph G, α ( G ) ≤ θ ( G ) ≤ ω f ( G ) . QUB

Quantum channels and non-commutative graphs Consider quantum channel Φ : M d → M k . We define α (Φ), the one-shot zero-error capacity of Φ, as the largest n st 1 v 1 , . . . , v n ∈ C d are orthonormal, 2 Φ( v 1 v ∗ 1 ) , . . . , Φ( v n v ∗ n ) are perfectly distinguishable (that is, Φ( v i v ∗ i )Φ( v j v ∗ j ) = 0 ∀ i � = j . ) Definition A subspace S ⊆ M d is a non-commutative graph (n.c.g.) if i I ∈ S and ii A ∈ S ⇒ A ∗ ∈ S . QUB

Non-commutative graphs Example (1) With confusability graph G on d vertices we associate the n.c.g. S G = span { E ij : i ≃ j in G } ⊆ M d . Distinct i , j ∈ V ( G ) are distinguishable iff E ij = e i e ∗ j ∈ S ⊥ G . For example, • 1 • 2 G = • 4 • 3    0 0  a 11 a 12     a 21 a 22 a 23 0     gives S G =  : a ij ∈ C .   0 0 a 32 a 33      0 0 0 a 44   QUB

How do n.c.g.s generalise graphs? Example (2) Let quantum channel Φ : M d → M k have Kraus representation m � E i ρ E ∗ Φ( ρ ) = i . i =1 The subspace S Φ ⊆ M d is a n.c.g. where S Φ = span { E ∗ i E j : i , j ∈ [ m ] } . For orthonormal u , v ∈ C d , Φ( uu ∗ ) , Φ( vv ∗ ) are distinguishable iff uv ∗ ∈ S ⊥ Φ . QUB

Independence number and zero-error capacity of a n.c.g. For n.c.g. S ⊆ M d , the orthonormal set { v 1 , . . . , v n } is an j ∈ S ⊥ for all i � = j . independent set when v i v ∗ The size of the largest independent set in S is called α ( S ), the independence number of S . We have α (Φ) = α ( S Φ ) . Definition (Duan, Severini, Winter) The zero-error capacity of n.c.g. S is given by � n α ( S ⊗ n ) . c ( S ) = lim n →∞ QUB

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 ϑ ( M m ( S )) . m ∈ N QUB

Generalising the Lov´ asz number Theorem (Duan, Severini, Winter) The parameter ˜ ϑ satisfies i ˜ ϑ ( S G ) = 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 ) . QUB

Convex corners in M d A set A ⊆ M d is an M d -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 . QUB

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 . QUB

The abelian projection convex corner Recall that orthonormal set { v 1 , . . . , v n } is S -independent when j ∈ S ⊥ for all i � = j . v i v ∗ If { v 1 , . . . , v n } is S -independent, we call � n i =1 v i v ∗ 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 ω f ( G ) = γ ( VP ( G ) ♭ ), for any n.c.g. S we define ω f ( S ) = γ ( ap ( S ) ♯ ) . QUB

Relating convex corners in R d and M d Let D d denote the diagonal d × d matrices. Let ∆ : M d → D d be given by ∆( M ) = � d i =1 � e i , Me i � e i e ∗ i . Regard a subset of D d as a subset of R d in the canonical way. Lemma VP ( G ) = ∆( ap ( S G )) = D d ∩ ap ( S G ) Corollary For graph G we have α ( S G ) = α ( G ) and ω f ( S G ) = ω f ( G ) . QUB

The full projection convex corner Orthonormal set { v 1 , . . . , v n } is S -full when v i v ∗ j ∈ S for all i , j . If { v 1 , . . . , v n } is S -full, � n i =1 v i v ∗ 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 ( S G )) = D d ∩ fp ( S G ) . Definition ϕ ( S ) = γ ( fp ( S ) ♯ ) . We define Corollary ϕ ( S G ) = ω f ( G ) . For graph G we have QUB

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? QUB

Generalising TH ( G ): the theta corner Definition If S ⊆ M d 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 S Φ ⊆ 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 ( S G )) = D d ∩ th ( S G ) . Corollary θ ( S G ) = θ ( G ) . QUB