Quantum Graphs! Priyanga Ganesan November 11, 2020 ASU - PowerPoint PPT Presentation

Quantum Graphs! — Priyanga Ganesan November 11, 2020 ASU C*-Seminar Priyanga Ganesan Quantum Graphs

Classical Graphs G = ( V , E , A G ) Vertex set: V Edge set: E ⊆ V × V Adjacency matrix: A G = [ a ij ], where a ij = 1 if ( i , j ) ∈ E , else 0. 2 Example Essential structure: V = { 1 , 2 , 3 } E = { (1 , 2) , (1 , 3) }   0 ∗ ∗ 3   0 1 1 ∗ 0 0   1 A G = 1 0 0 ∗ 0 0   1 0 0 Figure: G = ( V , E , A G )     0 ∗ ∗    where ∗ ∈ C S G := ∗ 0 0  ⊆ M 3 ( C )  ∗ 0 0  Priyanga Ganesan Quantum Graphs

When the graph is reflexive....    ∗ ∗ ∗     where ∗ ∈ C S G := ∗ ∗ 0  ⊆ M 3 ( C )  ∗ 0 ∗  Properties of S G : 2 Linear subspace ⇒ A ∗ ∈ S G ) Self-adjointness ( A ∈ S G ⇐ 3 1 Contains identity S G is an operator system! Operator System A subspace S ⊆ B ( H ) is called an operator system if I ∈ S . ⇒ A ∗ ∈ S . A ∈ S = Priyanga Ganesan Quantum Graphs

Matrix Quantum Graphs Let G = ( V , E ) be a graph on n -vertices and let V = { 1 , 2 . . . n } . Non-commutative Graph [DSW, 2013] The non-commutative graph associated with the classical graph G is the operator system S G defined as S G = span { e ij : ( i , j ) ∈ E or i = j , ∀ i , j ∈ V } ⊆ M n .   0 0 0 . . . . . ... . .  . . 0   with 1 in the i th − row and Here e ij are the matrix units     0 0 1 0  0 0 0 . . . j th − column. Definition (DSW, 2013) An operator system in M n is called a Matrix quantum graph. Priyanga Ganesan Quantum Graphs

Motivation from Information theory Matrix quantum graphs generalize the confusability graph of classical channels. Confusability graphs − > zero-error classical communication. Quantum graphs − > analogous role in zero-error quantum communication. Classical Channel Φ ← → Probability transition function [ P ( y | x )]. Φ (Input messages) X − → Y (Output messages) Φ { x 1 , x 2 . . . x m } − → { y 1 , y 2 . . . y n } Priyanga Ganesan Quantum Graphs

Confusability graph of classical channel (Φ : X → Y ) ← → Probability transition function [ P ( y | x )]. Confusability graph of Φ Vertex set: X = { x 1 , x 2 . . . x m } . Edges: x i ∼ x j if there exists y ∈ Y such that P ( y | x i ) P ( y | x j ) > 0 . Φ Input messages ( X ) Output messages( Y ) X 1 X 1 Y 1 X 2 Y 2 X 2 X 3 X 3 Y 3 X 4 X 5 X 4 Y 4 X 5 Y 5 Priyanga Ganesan Quantum Graphs

Quantum Channels Quantum communication channel take quantum states to quantum states. Φ : B ( H A ) linear − → B ( H B ) TP : Trace preserving: Tr( ρ ) = Tr(Φ( ρ )). CP : Completely positive: Φ is positive and all extensions Φ ⊗ I E are also positive. CPTP maps have several representations : Kraus form Φ( ρ ) = � r i , where K i ∈ B ( H A , H B ) satisfying � r i =1 K i ρ K ∗ i =1 K ∗ i K i = I A . The Kraus operators are not unique. Priyanga Ganesan Quantum Graphs

Classical embedded in Quantum Input A = { 1 , 2 . . . m } − → B = { 1 , 2 . . . n } Output CLASSICAL QUANTUM Input: | i � = e i ∈ C m Input: matrix units e ii ∈ M m ( e ii = | i � � i | ) Output: | j � = e j ∈ C n Output: matrix units e jj ∈ M n ( e jj = | j � � j | ) Φ Φ C m → C n − M m − → M n � K ab ( X ) K ∗ Φ( v ) = Pv , where Φ( X ) = ab , where P = [ P ( b | a )] a ∈ A , b ∈ B a ∈ A , b ∈ B � Kraus operators K ab = P ( b | a ) e ba ∈ M n × m K ∗ � Confusability graph G ab K cd = P ( b | a ) P ( d | c ) δ bd e ac K ∗ a ∼ c ⇐ ⇒ ∃ b with ab K cd � = 0 ⇐ ⇒ P ( b | a ) P ( b | c ) � = 0 b = d and P ( b | a ) P ( d | c ) � = 0 S Φ = span { K ∗ S Φ = span { e ac : a ∼ c } ab K cd : a , c ∈ A and b , d ∈ B } Priyanga Ganesan Quantum Graphs

Quantum Graphs Non-commutative confusability graph [DSW, 2013] Given a quantum channel Φ : M m → M n with Φ( x ) = � r i =1 K i xK ∗ i , the confusability graph of Φ is the operator system: S Φ = span { K ∗ i K j : 1 ≤ i , j ≤ r } ⊆ M m . This is independent of the Choi-Kraus representation of Φ. Every operator system arises from a quantum channel! Proposition Let S ⊆ M m be an operator system. Then there is n ∈ N and a quantum channel Ψ : M m → M n such that S = S Ψ . Priyanga Ganesan Quantum Graphs

Applications in zero-error communication Goal Send messages through a channel without confusion. Classical: x i and x j are not confusable ⇐ ⇒ x i �∼ x j in the confusability graph. Φ Input messages ( X ) Output messages( Y ) X 1 Y 1 X 1 X 2 Y 2 X 2 X 3 X 3 Y 3 X 4 X 5 X 4 Y 4 X 5 Y 5 One-shot zero error capacity of φ = Independence number of G = maximum number of messages transmitted without confusion. Priyanga Ganesan Quantum Graphs

Zero-error quantum communication Quantum states: ρ, σ ∈ B ( H ) are distinguishable ⇐ ⇒ � ρ, σ � = 0. r � K i ρ K ∗ i , S Φ := span { K ∗ Φ( ρ ) = j K i : 1 ≤ i , j ≤ r } . i =1 Encode input message x �→ ρ x = | x � � x | ∈ B ( H ). ρ x , ρ y are not confusable ⇐ ⇒ Φ( ρ x ) , Φ( ρ y ) are distinguishable. � Φ( ρ x ) , Φ( ρ y ) � = 0, with respect to Hilbert-Schmidt inner product. r i K j x �| 2 = 0 � Tr(Φ( ρ y ) ∗ Φ( ρ x )) = 0 ⇐ |� y , K ∗ ⇒ i , j =1 ⇒ Tr( | x � � y | K ∗ ⇒ ( | x � � y | ) ⊥ K ∗ ⇐ i K j ) = 0 ⇐ j K i , ∀ i , j . Result Input messages x , y are not confusable ⇐ ⇒ | x � � y | ⊥ S Φ .

Other approaches to quantum graphs Classical graph G = ( V , E , A G ) Quantize confusability graph of classical channels [DSW, 2010] Matrix quantum graphs and Operator systems Projection P S onto the operator system S Quantize edge set E ⊆ V × V [Weaver, 2010, 2015] Quantum relations Projection P E from χ E Quantize adjacency matrix [MRV, 2018] Categorical theory of quantum sets and quantum functions Projection P G using A G Unification Under appropriate identifications, range of these projections is the same operator system! Priyanga Ganesan Quantum Graphs

Quantum Relations Quantum set: von-Neumann algebra M ⊆ B ( H ) M ′ := { A ∈ B ( H ) such that AM = MA , ∀ M ∈ M} . Quantum relation [Weaver, 2010] A quantum relation on M is a weak*-closed subspace S ⊆ B ( H ) that is a bi-module over its commutant M ′ , i.e. M ′ S M ′ ⊆ S . Independent of the representation M ⊆ B ( H ). Quantum relations on l ∞ ( V ) ← → subsets of V × V ← → relations on V . S contains operators that ”connect adjacent vertices”. Priyanga Ganesan Quantum Graphs

Quantum graphs as quantum relations Classical graph: E ⊆ V × V - reflexive, symmetric relation on V . Quantum Graph [Weaver, 2015] A quantum graph on M is a reflexive and symmetric quantum relation on M . Quantum relation S ⊆ B ( H ) on M is: ⇒ M ′ ⊆ S ( = Reflexive ⇐ ⇒ 1 ∈ S ). ⇒ S ∗ = S . Symmetric ⇐ Connection to operator system Quantum graph S is a weak*-closed operator system that is a bimodule over M ′ . Priyanga Ganesan Quantum Graphs

Projection picture Motivation from commutative setting: Classical graph G = ( V , E ) with vertex set V and edge set E ⊆ V × V : ∼ χ E ∈ C ( V × V ) C ( V ) ⊗ C ( V ) = � ← → δ xy ( χ x ⊗ χ y ) χ E x , y ∈ V where δ xy = 1 if ( x , y ) ∈ E and 0 otherwise. Properties Idempotent: χ E = χ ∗ E = χ 2 E Reflexive: m ( χ E ) = 1 V Symmetric: σ ( χ E ) = χ E multiply swap where m : C ( V ) ⊗ C ( V ) − → C ( V ) and σ : C ( V ) ⊗ C ( V ) − → C ( V ) ⊗ C ( V ). Priyanga Ganesan Quantum Graphs

Quantum graph as projections Quantum set: finite dimensional C*-algebra M with fixed tracial state. Definition A quantum graph is a quantum set M ⊆ B ( H ) with a projection p ∈ M ⊗ M op satisfying p = p ∗ = p 2 m ( p ) = 1 M σ ( p ) = p p ∈ M ⊗ M op ∼ π M ′ CB M ′ � � B ( H ) = Connection to operator system S := Range( π ( p )) ⊆ B ( H ) is a weak*-closed operator system in B ( H ) that is a bimodule over M ′ . Priyanga Ganesan Quantum Graphs

Quantizing adjacency matrix... Definition A quantum graph is a pair ( M , A G ) containing Quantum set M Quantum adjacency matrix A G : M linear − → M with Idempotency: m ( A G ⊗ A G ) m ∗ = A G Reflexivity: m ( A G ⊗ I ) m ∗ = I Symmetry: ( η ∗ m ⊗ I )( I ⊗ A G ⊗ I )( I ⊗ m ∗ η ) = A G Back to projections: Get p ∈ M ⊗ M op as p := ( I ⊗ A G ) m ∗ η. Advantage of quantum adjacency matrix Allows us to define the spectrum of a quantum graph! Priyanga Ganesan Quantum Graphs

Comparing different notions of quantum graphs Quantum set M : finite dimensional C*-algebra with fixed tracial state ψ . CLASSICAL MATRIX QUANTUM PROJECTIONS ADJACENCY GRAPH Q.GRAPH RELATIONS MATRIX G = ( V , E , A G ) S ⊆ M n is ( M , M ′ S M ′ ) ( M , p ) ( M , A G ) an operator p ∈ M ⊗ M op A G ∈ M n { 0 , 1 } system. weak*-closed A G : M → M operator sys in B ( H ), bimodule over M ′ . M ′ SM ′ ⊆ S p = p ∗ = p 2 Idempotency: A G ⊙ ( M n ) m ( A G ⊗ A G ) m ∗ = A G ⊙ A G = A G = S A G M ′ ⊆ S m ( A G ⊗ I ) m ∗ = I Reflexivity: 1s 1 ∈ S m ( p ) = 1 M on the diagonal Undirected: S = S ∗ S = S ∗ σ ( p ) = p ( η ∗ m ⊗ I )( I ⊗ A G ⊗ A G = A T I )( I ⊗ m ∗ η ) = A G G Priyanga Ganesan Quantum Graphs