Multiaccess quantum communication and product higher rank numerical - PowerPoint PPT Presentation

Multiaccess quantum communication and product higher rank numerical ranges Maciej Demianowicz ∗ ( joint work with P. Horodecki and K. Życzkowski ) support: ERC Advanced Grant QOLAPS (leader R. Horodecki) ∗ Department of Atomic Physics and Luminescence Faculty of Applied Physics and Mathematics Gdańsk University of Technology Sep 19th, 2012 2012/09/19 Multiaccess quantum communication and product higher rank ...

Outline Outline introduction: quantum channels + quantum error correction (QEC) higher rank numerical range approach to QEC product higher rank numerical range: properties zero entropy codes (decoherence free subspaces) higher entropy codes conclusions + open problems 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC 1 Quantum channels Quantum channel = a model of noise a trace preserving (TP) completely positive (CP) map i A i ̺ A † i A † Choi–Kraus representation L ( ̺ ) = � i A i = i , � bipartite, multiple access: ̺ = ̺ 1 ⊗ ̺ 2 ⊗ · · · , broadcast, km –user i p i U i ̺ U † Random unitary channel: L ( ̺ ) = � i p i = 1. i , � Biunitary channel (BUC): L ( ̺ ) = pU 1 ̺ U 1 † +( 1 − p ) U 2 ̺ U 2 † → p ̺ +( 1 − p ) U ̺ U † , U = U † 1 U 2 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC Error correction Error correction = a way to combat noise Definition A quantum error correction code (QECC) is a subspace C ⊆ H . Equivalently, it is the projection P C onto this subspace. | ψ � = a | 0 � + b | 1 � , | 0 � − → | 000 � , | 1 � − → | 111 � | ψ � → | φ � = a | 000 � + b | 111 � C ⊂ ( C 2 ) ⊗ 3 , P C = | 000 �� 000 | + | 111 �� 111 | C is correctable: D ◦ Λ( ̺ ) = ̺ for P C ̺ P C = ̺ Theorem (Knill–Laflamme conditions) [Knill&Laflamme 1997] C is correctable iff R C A † i A j R C = β ij R C for some hermitian matrix [ B ] ij = β ij . 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC Higher rank numerical range and QEC Λ k ( A ) = { λ ∈ C : PAP = λ P for P ∈ P k } . � λ 1 BUC: L ( ̺ ) = p ̺ + ( 1 − p ) U ̺ U † − → KL condition: RUR = λ R Definition [Choi et al. 2006] Higher rank (or rank– k ) numerical range of an operator A is defined to be the following set: � ∗ ∗ ∗ Example.  0 0 0 0  0 1 0 0 A 4 =   ,  Λ 2 ( A 4 ) = � 1 ; 2 � ,   0 0 2 0  0 0 0 3 P 2 = | φ 1 �� φ 1 | + | φ 2 �� φ 2 | , 1 2 ( | 0 � + | 3 � ) , | φ 2 � = 1 2 ( | 1 � + | 2 � ) − → P 2 A 4 P 2 = 3 2 P 2 | φ 1 � = √ √ 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC Entropy of a code code entropy=an effort to perform recovery= ancillary qubits for D Definition [Kribs et al. 2008] The von Neumann entropy S ( C ) := S ( B ) is called the entropy of a code C . � � p p ( 1 − p ) � ± λ For BUC: B = � p ( 1 − p ) 1 − p ± λ S ( C ) = 0 iff λ = ± 1 — zero–entropy codes → decoherence free subspaces (DFS; trivial recovery) S ( C ) > 0 iff λ � = ± 1 — higher entropy codes 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC Multiple access channels and QEC Observation Local codes C i are correctable for a MAC with Kraus operators { A i } with k inputs if and only if R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k A † i A j R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k = β ij R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k for some hermitian matrix [ B ] ij = β ij . 2012/09/19 Multiaccess quantum communication and product higher rank ...

Channels and error correction Quantum channels and quantum error correction (QEC) Higher rank numerical range, code entropy MACs and QEC Multiple access channels and QEC Observation Local codes C i are correctable for a MAC with Kraus operators { A i } with k inputs if and only if R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k A † i A j R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k = β ij R C 1 ⊗ R C 2 ⊗· · ·⊗ R C k for some hermitian matrix [ B ] ij = β ij . 2012/09/19 Multiaccess quantum communication and product higher rank ...

Definitions Product higher rank numerical range Properties Product higher rank numerical ranges: definitions Definition [MD,PH&KŻ 2012] The k 1 ⊗ k 2 ⊗· · · product higher rank numerical range of an operator A is defined to be Λ k 1 ⊗ k 2 ⊗··· ( A ) = { λ ∈ C : R ⊗ R ′ ⊗· · · AR ⊗ R ′ ⊗· · · = λ R ⊗ R ′ ⊗· · ·} for some R ∈ P k 1 , R ′ ∈ P k 2 , . . . k 1 ⊗ k 2 — bipartite , k 1 ⊗ k 2 ⊗ k 3 ⊗ · · · — multipartite symmetric product higher rank numerical range: R = R ′ = . . . locally symmetric : R = R ′ , R ′′ � = R ′′′ � = . . . joint: Λ k 1 ⊗ k 2 ( A 1 , A 2 ) = { ( λ 1 , λ 2 ) ∈ C 2 : R ⊗ R ′ A i R ⊗ R ′ = λ i R ⊗ R ′ } common : k 1 ⊗ k 2 ( A 1 , A 2 ) = { λ ∈ C : R ⊗ R ′ A i R ⊗ R ′ = λ R ⊗ R ′ } Λ comm . 2012/09/19 Multiaccess quantum communication and product higher rank ...

Definitions Product higher rank numerical range Properties Product higher rank numerical ranges: definitions Definition [MD,PH&KŻ 2012] The k 1 ⊗ k 2 ⊗· · · product higher rank numerical range of an operator A is defined to be Λ k 1 ⊗ k 2 ⊗··· ( A ) = { λ ∈ C : R ⊗ R ′ ⊗· · · AR ⊗ R ′ ⊗· · · = λ R ⊗ R ′ ⊗· · ·} for some R ∈ P k 1 , R ′ ∈ P k 2 , . . . k 1 ⊗ k 2 — bipartite , k 1 ⊗ k 2 ⊗ k 3 ⊗ · · · — multipartite symmetric product higher rank numerical range: R = R ′ = . . . locally symmetric : R = R ′ , R ′′ � = R ′′′ � = . . . joint: Λ k 1 ⊗ k 2 ( A 1 , A 2 ) = { ( λ 1 , λ 2 ) ∈ C 2 : R ⊗ R ′ A i R ⊗ R ′ = λ i R ⊗ R ′ } common : k 1 ⊗ k 2 ( A 1 , A 2 ) = { λ ∈ C : R ⊗ R ′ A i R ⊗ R ′ = λ R ⊗ R ′ } Λ comm . 2012/09/19 Multiaccess quantum communication and product higher rank ...

Definitions Product higher rank numerical range Properties Properties of the product higher rank numerical range 1 ⊗ UA 1 ⊗ U † � Λ m ⊗ n ( A ) = Λ m ⊗ n ( U ⊗ VAU † ⊗ V † ) Λ m ⊗ n ( A ) is a compact set Λ m ⊗ n ( A ) ⊆ Λ mn ( A ) Λ m ⊗ n ( A ) ⊆ Λ loc ( A ) , Λ loc ( A ) = { λ : λ = � φ ⊗ ψ | A | φ ⊗ ψ �} m ⊗ n ( A , B ) ⊆ Λ joint Λ comm . m ⊗ n ( A , B ) Λ symm . m ⊗ n ⊗ p ( A ) ⊆ Λ loc . symm . m ⊗ n ⊗ p ( A ) ⊆ Λ m ⊗ n ⊗ p ( A ) U Λ symm . Λ m ⊗ n ( A ) = � � m ⊗ n Λ comm . ( A 1 , A 2 ) ⊆ Λ k ( α A 1 + ( 1 − α ) A 2 ) k � 1 � Λ k 1 ⊗ k 2 ( A ) ⊆ W loc . A , ’optimization’ bound R ⊗ R ′ k 1 k 2 W loc . ( A ) = { tr C † ( U ⊗ V ) † A ( U ⊗ V ) , U , V ∈ U} C 2012/09/19 Multiaccess quantum communication and product higher rank ...

Definitions Product higher rank numerical range Properties Properties of the product higher rank numerical range 1 ⊗ UA 1 ⊗ U † � Λ m ⊗ n ( A ) = Λ m ⊗ n ( U ⊗ VAU † ⊗ V † ) Λ m ⊗ n ( A ) is a compact set Λ m ⊗ n ( A ) ⊆ Λ mn ( A ) Λ m ⊗ n ( A ) ⊆ Λ loc ( A ) , Λ loc ( A ) = { λ : λ = � φ ⊗ ψ | A | φ ⊗ ψ �} m ⊗ n ( A , B ) ⊆ Λ joint Λ comm . m ⊗ n ( A , B ) Λ symm . m ⊗ n ⊗ p ( A ) ⊆ Λ loc . symm . m ⊗ n ⊗ p ( A ) ⊆ Λ m ⊗ n ⊗ p ( A ) U Λ symm . Λ m ⊗ n ( A ) = � � m ⊗ n Λ comm . ( A 1 , A 2 ) ⊆ Λ k ( α A 1 + ( 1 − α ) A 2 ) k � 1 � Λ k 1 ⊗ k 2 ( A ) ⊆ W loc . A , ’optimization’ bound R ⊗ R ′ k 1 k 2 W loc . ( A ) = { tr C † ( U ⊗ V ) † A ( U ⊗ V ) , U , V ∈ U} C 2012/09/19 Multiaccess quantum communication and product higher rank ...

Model Codes Zero entropy codes Higher entropy codes Noise model L : C d ⊗ C d → C d 2 BUC: L ( ̺ ) = p ̺ + ( 1 − p ) U ̺ U † , ̺ = ̺ 1 ⊗ ̺ 2 U — hermitian ⇒ U = P − Q R ⊗ R ′ UR ⊗ R ′ = λ R ⊗ R ′ γ = 1 − λ 2 R ⊗ R ′ PR ⊗ R ′ = ( 1 − γ ) R ⊗ R ′ R ⊗ R ′ QR ⊗ R ′ = γ R ⊗ R ′ or γ = 0 , 1: decoherence free subspaces 0 < γ < 1: higher entropy codes 2012/09/19 Multiaccess quantum communication and product higher rank ...