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 G N 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 C 5 (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 ( G N ) of the graph G N . 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 ( G N ) of the graph G N . 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 C 5 , we have α ( C 5 ) = 2. Ivan Todorov QUB

Product channels N 1 : X 1 → Y 1 , N 2 : X 2 → Y 2 channels. The product channel N 1 × N 2 : X 1 × X 2 → Y 1 × Y 2 is given by p ( y 1 , y 2 | x 1 , x 2 ) = p ( y 1 | x 1 ) p ( y 2 | x 2 ) . Ivan Todorov QUB

Product channels N 1 : X 1 → Y 1 , N 2 : X 2 → Y 2 channels. The product channel N 1 × N 2 : X 1 × X 2 → Y 1 × Y 2 is given by p ( y 1 , y 2 | x 1 , x 2 ) = p ( y 1 | x 1 ) p ( y 2 | x 2 ) . G N 1 ×N 2 = G N 1 ⊠ G N 2 G 1 ⊠ G 2 : the strong graph product: vertex set: X 1 × X 2 ( x 1 , x 2 ) ≃ ( x ′ 1 , x ′ 2 ) if x 1 ≃ x ′ 1 and x 2 ≃ 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 � � � G ⊠ n c 0 ( N ) = lim n α . N n →∞ Note: The limit exists due to the fact that α ( G 1 ⊠ G 2 ) ≥ α ( G 1 ) α ( G 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 � � � G ⊠ n c 0 ( N ) = lim n α . N n →∞ Note: The limit exists due to the fact that α ( G 1 ⊠ G 2 ) ≥ α ( G 1 ) α ( G 2 ). Strict inequality may occur � you can do better on the average by using N repeatedly. Question: What is c 0 ( C 5 )? Ivan Todorov QUB

The Lov´ asz number √ Answer (Lov´ asz, 1979): c 0 ( C 5 ) = 5. Method: Introduced a parameter θ ( G ) such that α ( G ) ≤ θ ( G ) and θ ( G 1 ⊠ G 2 ) = θ ( G 1 ) θ ( G 2 ). Ivan Todorov QUB

The Lov´ asz number √ Answer (Lov´ asz, 1979): c 0 ( C 5 ) = 5. Method: Introduced a parameter θ ( G ) such that α ( G ) ≤ θ ( G ) and θ ( G 1 ⊠ G 2 ) = θ ( G 1 ) θ ( G 2 ). The inequality θ ( G 1 ⊠ G 2 ) ≤ θ ( G 1 ) θ ( G 2 ) alone will then give c 0 ( G ) ≤ θ ( G ). Note: θ ( G ) remains the best general computable bound for c 0 ( G ). Ivan Todorov QUB

The Lov´ asz number G a graph with vertex set X , | X | = d . For A ⊆ R d + , let A ♭ = { b ∈ R d + : � b , a � ≤ 1 , ∀ a ∈ A} . Ivan Todorov QUB

The Lov´ asz number G a graph with vertex set X , | X | = d . For A ⊆ R d + , let A ♭ = { b ∈ R d + : � b , a � ≤ 1 , ∀ a ∈ A} . Orthogonal labelling (o.l.): a family ( a x ) x ∈ X of unit vectors s.t. x �≃ y ⇒ a x ⊥ a y . �� � |� a x , c �| 2 � P 0 ( G ) = x ∈ X : ( a x ) x ∈ X o.l. and � c � ≤ 1 . thab ( G ) = P 0 ( G ) ♭ The Lov´ asz number �� � x ∈ X λ x : ( λ x ) x ∈ X ∈ thab ( G ) θ ( G ) = max . Ivan Todorov QUB

The Lov´ asz number G a graph with vertex set X , | X | = d . For A ⊆ R d + , let A ♭ = { b ∈ R d + : � b , a � ≤ 1 , ∀ a ∈ A} . Orthogonal labelling (o.l.): a family ( a x ) x ∈ X of unit vectors s.t. x �≃ y ⇒ a x ⊥ a y . �� � |� a x , c �| 2 � P 0 ( G ) = x ∈ X : ( a x ) x ∈ X o.l. and � c � ≤ 1 . thab ( G ) = P 0 ( G ) ♭ The Lov´ asz number �� � x ∈ X λ x : ( λ x ) x ∈ X ∈ thab ( G ) θ ( G ) = max . 1 Equivalently: θ ( G ) = min c max x ∈ X |� a x , c �| 2 Ivan Todorov QUB

The Lov´ asz Sandwich Theorem α ( G ) ≤ c 0 ( G ) ≤ θ ( G ) ≤ χ f ( ¯ G ). χ f ( ¯ G ): the fractional chromatic number of the complement of G . �� � x ∈ X λ x : � χ f ( G ) = max x ∈ K λ x ≤ 1 , ∀ clique K Ivan Todorov QUB

The Lov´ asz Sandwich Theorem α ( G ) ≤ c 0 ( G ) ≤ θ ( G ) ≤ χ f ( ¯ G ). χ f ( ¯ G ): the fractional chromatic number of the complement of G . �� � x ∈ X λ x : � χ f ( G ) = max x ∈ K λ x ≤ 1 , ∀ clique K The Strong Sandwich Theorem vp ( G ) ⊆ thab ( G ) ⊆ fvp ( G ) . vp ( G ) = conv { χ S : S ⊆ X independent set } fvp ( G ) = conv { χ K : K ⊆ X clique } ♭ vp ( G ), thab ( G ) and fvp ( G ) are convex corners. Ivan Todorov QUB

Convex corners and dualities Convex corner A ⊆ R d + : convex, closed, hereditary (Hereditary: a ∈ A , 0 ≤ b ≤ a ⇒ b ∈ A .) Ivan Todorov QUB

Convex corners and dualities Convex corner A ⊆ R d + : 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 ⊆ R d + is a convex corner then A ♭♭ = A . Ivan Todorov QUB

From classical to quantum Replace C d by M d . M d – the algebra of all d × d complex matrices; ( E x , x ′ ) x , x ′ ∈ X matrix units. Trace Tr (( λ x , y )) = � x ∈ X λ x ; Inner product ( A , B ) = Tr ( B ∗ A ); Duality � A , B � = Tr ( AB ), making it into a self-dual space; Positivity A ≥ 0 if ( A ξ, ξ ) ≥ 0 , ∀ ξ . Ivan Todorov QUB

From classical to quantum Replace C d by M d . M d – the algebra of all d × d complex matrices; ( E x , x ′ ) x , x ′ ∈ X matrix units. Trace Tr (( λ x , y )) = � x ∈ X λ x ; Inner product ( A , B ) = Tr ( B ∗ A ); Duality � A , B � = Tr ( AB ), making it into a self-dual space; Positivity A ≥ 0 if ( A ξ, ξ ) ≥ 0 , ∀ ξ . Let D X ⊆ M d be the subalgebra of all diagonal matrices. Going from classical to quantum, we move from D X to M d , 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 Φ N : D X → D Y be the linear map � Φ N ( E x , x ) = p ( y | x ) E y , y . y ∈ Y Φ N is positive: A ≥ 0 ⇒ Φ N ( A ) ≥ 0. Φ N is trace preserving: Tr (Φ N ( 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 Φ N : D X → D Y be the linear map � Φ N ( E x , x ) = p ( y | x ) E y , y . y ∈ Y Φ N is positive: A ≥ 0 ⇒ Φ N ( A ) ≥ 0. Φ N is trace preserving: Tr (Φ N ( A )) = Tr ( A ). Quantum channel Φ : M d → M k linear, completely positive, trace preserving. Completely positive: Φ ⊗ id d : M d ⊗ M d → M k ⊗ M d is positive. Ivan Todorov QUB

Kraus representation The representation theorem Let Φ : M d → M k be a linear map. The following are equivalent: Φ is a quantum channel; there exist A p : C d → C k , p = 1 , . . . , r , such that r � A p TA ∗ Φ( T ) = T ∈ M d , p , p =1 and � r A ∗ p A p = I . p =1 Ivan Todorov QUB