Structure of Irreducibly Covariant Quantum Channels Marek Mozrzymas - PowerPoint PPT Presentation

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Structure of Irreducibly Covariant Quantum Channels Marek Mozrzymas 1 , joint work with: Michał Studziński 2 , Nilanjana Datta 3 1 Institute for Theoretical Physics, University of Wrocław, Wrocław, Poland 2 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK 3 Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Outline Preliminary Informations 1 Towards Irreducibly Covariant Quantum Channels 2 Characterisation of the Irreducibly Covariant Quamntum 3 Channels Applications 4

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Contragradient and Adjoint Representation Let G be a finite group and let: U : G → M ( n , C ) be a unitary irreducible representation (irrep, in short) of G . The contragradient representation U c : G → M ( n , C ) is given by U c ( g ) = U ( g − 1 ) T ≡ U ( g ) ∀ g ∈ G . The map Ad G U : G − → End [ M ( n , C )] is called the adjoint representation of the group G with respect to the unitary irrep U , and is defined through its action on any X ∈ M ( n , C ) as follows: Ad U ( g ) ( X ) ≡ U ( g ) XU † ( g ) ∀ g ∈ G .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Irreducibly Covariant Maps and Quantum Channels Definition (Irreducibly covariant- linear maps (ICLM) and quantum channels (ICQC)) A linear map Φ ∈ End [ M ( n , C )] is said to be irreducibly covariant with respect to the unitary irrep U : G → M ( n , C ) of a finite group G , if ∀ g ∈ G , ∀ X ∈ M ( n , C ) Ad U ( g ) [Φ( X )] = Φ[ Ad U ( g ) ( X )] , i.e. Φ ∈ Int G ( Ad U ) . Further, if the linear map Φ is completely positive and trace-preserving, then it is referred to as an irreducibly covariant quantum channel. We denote an irreducibly covariant linear map by the acronym ICLM , and an irreducibly covariant quantum channel by the acronym ICQC .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Commutant of the Adjoint Representation Definition (Commutant of the adjoint representation) Let Int G ( Ad U ) denote the set of intertwiners of Ad U , i.e. the set of maps in End [ M ( n , C )] whose action commutes with that of Ad U : Int G ( Ad U ) = { Ψ ∈ End [ M ( n , C )] : Ψ ◦ Ad U = Ad U ◦ Ψ } . We have: mat ( Ad U ( g ) ) = U ( g ) ⊗ U ( g ) . Thus the operator Ad U ( g ) ∈ End [ M ( n , C )] may be represented as a matrix U ( g ) ⊗ U ( g ) ∈ M ( n 2 , C ) .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Lemma A linear map Φ ∈ End [ M ( n , C )] is irreducibly covariant with respect to the irrep U : G → M ( n , C ) of a finite group G (i.e. Φ ∈ Int G ( Ad U )) if and only if � � mat (Φ) ∈ Int G ( U ⊗ U c ) = Int G U ⊗ U . From this it follows that the commutant of the representation U ⊗ U c : Int G ( U ⊗ U c ) = � � � � � � � � n 2 , C = A ∈ M : ∀ g ∈ G A U ( g ) ⊗ U ( g ) = U ( g ) ⊗ U ( g ) A .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Characteristics of the Adjoint Representation Lemma The character of the adjoint representation U : G → End [ M ( n , C )] , denoted by χ Ad G U ≡ χ Ad : G → C , is Ad G given by � � � � χ Ad G U ( g ) := Tr mat ( Ad U ( g ) ) = Tr U ( g ) ⊗ U ( g ) , ∀ g ∈ G . In particular, we have χ Ad ( g ) = | χ U ( g ) | 2 , ∀ g ∈ G , where χ U : G → C is the character of the representation U : G → M ( n , C ) , i.e. χ U ( g ) = Tr ( U ( g )) , ∀ g ∈ G .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Characteristics of the Adjoint Representation Lemma Let U : G → M ( n , C ) be a unitary irreducible representation of a given finite group G . Then we have U ⊗ U c = ϕ id ⊕ α � = id m α ϕ α , i.e. the identity irrep, ϕ id , is always included in the representation U ⊗ U c with multiplicity one. Moreover, � � � 1 4 , � � dim [ Int G ( U ⊗ U c )] = � χ U ( g ) � | G | g ∈ G where χ U : G → C is the character of the representation U : G → M ( n , C ) .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Adjoint Representation - Examples Example For G = S ( 3 ) and its two-dimensional, unitary irrep U = ϕ ( 2 , 1 ) characterised by the partition λ = ( 2 , 1 ) we have � � U ⊗ U c = ϕ id ⊕ ϕ sgn ⊕ ϕ ( 2 , 1 ) , Int S ( 3 ) ( U ⊗ U c ) dim = 3 . Example For G = S ( 4 ) and its two-dimensional, unitary irrep U = ϕ ( 2 , 2 ) characterised by the partition λ = ( 2 , 2 ) we have � � U ⊗ U c = ϕ id ⊕ ϕ sgn ⊕ ϕ ( 2 , 2 ) , Int S ( 4 ) ( U ⊗ U c ) dim = 3 .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum The Structure of the Commutant U ⊗ U c Proposition Suppose that an unitary irrep U : G → M ( n , C ) , of a finite group G is such that U ⊗ U c is multiplicity-free , i.e. � U ⊗ U c = ϕ α . α ∈ Θ Then, � � Π α : α ∈ Θ � Int G ( U ⊗ U c ) = span C , where χ α � g − 1 � Π α = | ϕ α | � � U ( g ) ⊗ U ( g ) ∈ M ( n 2 , C ) . | G | g ∈ G

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum General Irreducibly Covariant Linear Map Corollary A linear map Φ ∈ End [ M ( n , C )] , which is irreducibly covariant with respect to a unitary irrep U : G → M ( n , C ) of a finite group G , can be expressed in the form � Φ = l id Π id + l α Π α : l α ∈ C , α ∈ Θ ,α � = id where χ α � g − 1 � Π α = | ϕ α | � Ad U ( g ) ∈ End [ M ( n , C )] , α ∈ Θ , | G | g ∈ G and the operators Π α have the same properties as their matrix Π α ≡ mat (Π α ) . representants �

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Outline Preliminary Informations 1 Towards Irreducibly Covariant Quantum Channels 2 Characterisation of the Irreducibly Covariant Quamntum 3 Channels Applications 4

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Irreducibly Covariant Linear Maps - Trace Preservation Proposition An ICLM Φ = l id Π id + � α ∈ Θ ,α � = id l α Π α ∈ Int G ( Ad U ) is trace preserving if and only if l id = 1 , so that it is of the form: � Φ = Π id + l α Π α , α ∈ Θ ,α � = id where the coefficient l α for α ∈ Θ , with α � = id , can be arbitrary.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Choi-Jamiołkowski Isomorphism Consider a linear map Φ ∈ End [ M ( n , C )] , i.e. Φ : M ( n , C ) → M ( n , C ) . Its Choi-Jamiołkowski image J (Φ) is given by: n � E ij ⊗ Φ( E ij ) ∈ M ( n 2 , C ) . J (Φ) := i , j = 1 It is well-known that a linear map Φ ∈ End [ M ( n , C )] is completely positive map ( CP ) if and only if its Choi-Jamiołkowski image J (Φ) is a positive semidefinite matrix, i.e. J (Φ) � 0.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Irreducibly Covariant Linear Maps - Complete Positivity Goal Find the spectrum of the Choi-Jamiołkowski image for a given irreducibly covariant trace preserving map Φ ∈ End [ M ( n , C )] . Proposition The Choi-Jamiołkowski image of a trace-preserving ICLM Φ ∈ End [ M ( n , C )] is given by 1 J (Φ) = | U | ✶ n ⊗ ✶ n +   � � � l α | ϕ α | χ α � g − 1 � � � † + 1   U C ( i ) ( g ) E ij ⊗ U C ( j ) ( g ) | G | ij g ∈ G α ∈ Θ ,α � = id

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum Eigenvectors of the Choi-Jamiołkowski Image Proposition (Part I) Let V α i ∈ M ( n , C ) be the normalised eigenvectors of the operators Π α i ∈ End [ M ( n , C )] , which form an orthonormal basis of M ( n , C ) . Let us define the set of n 2 vectors n � � � kl vec ( E lk ) ∈ C n 2 , | v β V β i = 1 , . . . , | ϕ β | . i � ≡ β ∈ Θ , i k , l = 1 Then ∀ α ∈ Θ , the Choi-Jamiołkowski images of the operators Π α satisfy J (Π α ) | v β i � = µ i ( α, β ) | v β i � .