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 Mozrzymas1, joint work with: Michał Studziński2, Nilanjana Datta3

1Institute for Theoretical Physics, University of Wrocław, Wrocław, Poland 2DAMTP, Centre for Mathematical Sciences, University of Cambridge,

Cambridge, UK

3Statistical Laboratory, Centre for Mathematical Sciences, University of

Cambridge, Cambridge, UK

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Outline

1

Preliminary Informations

2

Towards Irreducibly Covariant Quantum Channels

3

Characterisation of the Irreducibly Covariant Quamntum Channels

4

Applications

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 Uc : G → M(n, C) is given by Uc(g) = U(g−1)T ≡ U(g) ∀ g ∈ G. The map AdG

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: AdU(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) AdU(g)[Φ(X)] = Φ[AdU(g)(X)], i.e. Φ ∈ IntG(AdU). 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 IntG(AdU) denote the set of intertwiners of AdU, i.e. the set

• f maps in End [M(n, C)] whose action commutes with that of

AdU: IntG(AdU) = {Ψ ∈ End [M(n, C)] : Ψ ◦ AdU = AdU ◦Ψ}. We have: mat(AdU(g)) = U(g) ⊗ U(g). Thus the operator AdU(g) ∈ End[M(n, C)] may be represented as a matrix U(g) ⊗ U(g) ∈ M(n2, 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 ∈ IntG(AdU)) if and only if mat(Φ) ∈ IntG (U ⊗ Uc) = IntG

• U ⊗ U
• .

From this it follows that the commutant of the representation U ⊗ Uc:

IntG (U ⊗ Uc) = =

• A ∈ M
• n2, C
• : ∀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 AdG

U : G → End [M(n, C)], denoted by χAdG

U ≡ χAd : G → C, is

given by χAdG

U(g) := Tr

• mat(AdU(g))
• = Tr
• U(g) ⊗ U(g)
• , ∀g ∈ G.

In particular, we have χAd(g) = |χU(g)|2, ∀g ∈ G, where χU : G → C is the character of the representation U : G → M(n, C), i.e. χU(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 ⊗ Uc = ϕid ⊕α=id mαϕα, i.e. the identity irrep, ϕid, is always included in the representation U ⊗ Uc with multiplicity one. Moreover, dim [IntG (U ⊗ Uc)] = 1 |G|

• g∈G
• χU(g)
• 4 ,

where χU : 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 ⊗ Uc = ϕid ⊕ ϕsgn ⊕ ϕ(2,1), dim

• IntS(3) (U ⊗ Uc)
• = 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 ⊗ Uc = ϕid ⊕ ϕsgn ⊕ ϕ(2,2), dim

• IntS(4) (U ⊗ Uc)
• = 3.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

The Structure of the Commutant U ⊗ Uc

Proposition Suppose that an unitary irrep U : G → M(n, C), of a finite group G is such that U ⊗ Uc is multiplicity-free, i.e. U ⊗ Uc =

• α∈Θ

ϕα. Then, IntG (U ⊗ Uc) = spanC

• Πα : α ∈ Θ
• ,

where

• Πα = |ϕα|

|G|

• g∈G

χα g−1 U(g) ⊗ U(g) ∈ M(n2, C).

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 = lidΠid +

• α∈Θ,α=id

lαΠα : lα ∈ C, where Πα = |ϕα| |G|

• g∈G

χα g−1 AdU(g) ∈ End [M (n, C)] , α ∈ Θ, and the operators Πα have the same properties as their matrix representants Πα ≡ mat(Πα).

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Outline

1

Preliminary Informations

2

Towards Irreducibly Covariant Quantum Channels

3

Characterisation of the Irreducibly Covariant Quamntum Channels

4

Applications

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Irreducibly Covariant Linear Maps - Trace Preservation

Proposition An ICLM Φ = lidΠid +

α∈Θ,α=id lαΠα ∈ IntG (AdU) is trace

preserving if and only if lid = 1, so that it is of the form: Φ = Πid +

• α∈Θ,α=id

lαΠα, where the coefficient lα 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: J(Φ) :=

n

• i,j=1

Eij ⊗ Φ(Eij) ∈ M(n2, C). 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

J(Φ) = 1 |U|✶n ⊗ ✶n+ + 1 |G|

• ij

Eij⊗

• g∈G

 

• α∈Θ,α=id

lα |ϕα| χα g −1   UC(i)(g)

• UC(j)(g)

†

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

Πα

i ∈ End [M(n, C)], which form an orthonormal basis of M(n, C).

Let us define the set of n2 vectors |vβ

i ≡ n

• k,l=1
• V β

i

• kl vec(Elk) ∈ Cn2,

β ∈ Θ, i = 1, . . . , |ϕβ|. Then ∀α ∈ Θ, the Choi-Jamiołkowski images of the operators Πα satisfy J(Πα)|vβ

i = µi(α, β)|vβ i .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Spectrum of the Choi-Jamiołkowski Image

Proposition (Part II) The vectors |vβ

i are common eigenvectors of all J(Πα) with

eigenvalues µi(α, β) given by µi(α, β) = |ϕα| |G|

• g∈G

χα g−1

• Tr
• V β

i U†(g)

• 2 .

Since the vectors defined in are simultaneous eigenvectors of J(Πα), ∀ α ∈ Θ, then ∀ α, β ∈ Θ, [J(Πα), J(Πβ)] = 0.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Spectrum of the Choi-Jamiołkowski Image

Corollary The Choi-Jamiołkowski image J(Φ) given by is positive semi-definite if and only if its eigenvalues, which we denote by ǫβ

i ,

are non-negative, i.e. for any β ∈ Θ, and i = 1, . . . , |ϕβ| ǫβ

i ≡

• α∈Θ

lαµi(α, β) = = 1 |G|

• g∈G

 

α∈Θ

lα |ϕα| χα g−1

 

• Tr
• V β

i U†(g)

• 2 0,

where V β

i ∈ M(n, C) is the normalized eigenvector of the projector

Πβ

i .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Outline

1

Preliminary Informations

2

Towards Irreducibly Covariant Quantum Channels

3

Characterisation of the Irreducibly Covariant Quamntum Channels

4

Applications

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Main Results - General Characterisation

Theorem (Part I) Suppose that a unitary irrep U : G → M(n, C) (of a finite group G) is such that U ⊗ Uc is simply reducible, i.e., U ⊗ Uc =

• α∈Θ

ϕα. Then a linear map Φ ∈ End [M(n, C)], is an ICQC with respect to the irrep U if and only if it has a decomposition of the following form: Φ = lidΠid +

• α∈Θ,α=id

lαΠα, with lid = 1, lα ∈ C; Πid, Πα ∈ End [M(n, C)] .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Main Results - General Characterisation

Theorem (Part II) The coefficients lα are eigenvalues of Φ and satisfy ∀β ∈ Θ, i ∈ {1, . . . , |ϕβ|} the following inequalities:

• g∈G

 

α∈Θ

lα |ϕα| χα g−1

 

• Tr
• V β

i U†(g)

• 2 0.

In the above, V β

i ∈ M(n, C) denote the normalized eigenvectors of

rank-one projectors Πβ

i ∈ End [M(n, C)] such that Πβ = i Πβ i .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Main Results - Kraus Representation

Theorem The Kraus operators of any ICQC Φ ∈ End [M(n, C)], which satisfy the assumptions of the previous theorem, have the following form: Ki(β) =

• ǫβ

i

• V β

i

T ,

β ∈ Θ, i = 1, . . . , |Θ|, where ǫβ

i are eigenvalues of the Choi-Jamiołkowski image J(Φ) and

Vi(β) =

• vec−1

|vβ

i

T ∈ M(n, C). It follows that the Kraus

representation of ICQC Φ is of the form: Φ(X) =

• β∈Θ

|ϕβ|

• i=1

Ki(β)XK †

i (β),

X ∈ M (n, C) .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Outline

1

Preliminary Informations

2

Towards Irreducibly Covariant Quantum Channels

3

Characterisation of the Irreducibly Covariant Quamntum Channels

4

Applications

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

A Wide Class of ICQC

Theorem Let K(g) ⊂ G be the conjugacy class of g ∈ G, and let f : G → C be a function on the group G such that,

• g∈G

f (g) = |G|, ∀K(g)

• h∈K(g)

f (h) 0. Then a family of ICQC are those for which the coefficients lα are given by: lα = 1 |G| 1 |ϕα|

• g∈G

χα(g)f (g), α ∈ Θ.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Geometric Properties of ICQCs

Let us consider he system of linear equations: ǫβ

i =

• α∈Θ

lαµi(α, β), β ∈ Θ, i = 1, . . . , |ϕβ|, without the assumption lid = 1. This system of linear equations may also be written in the matrix form: E = ML, M = (mβi,α) ≡ (µi(α, β)) ∈ M

• n2 × |Θ|, C
• .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Geometric Properties of ICQCs

Proposition The matrix M = (mβi,α) ≡ (µi(α, β)) has maximal possible rank, equal to |Θ|, which means that the columns MC(α) ∈ Cn2 of the matrix M are linearly independent. This implies that the matrix M is invertible from the left, and denoting the left inverse as Minv, we have: MinvM = ✶|Θ| : Minv = G −1M† ∈ M

• |Θ| × n2, C
• ,

where G =

• (MC(α), MC(α′)
• ∈ M (|Θ|, C) is the Gram matrix of

the columns MC(α) ∈ Cn2 of the matrix M.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Geometric Properties of ICQCs

Proposition The eigenvalues E = (ǫβ

i ) of the Choi-Jamiołkowski image J(Φ) of

an ICQC Φ =

α∈Θ lαΠα lie in the intersection of the scaled

simplex Σ(U) ⊂ Cn2 and the subspace E ⊂ Cn2, i.e., E = (ǫβ

i ) ∈ Σ(U) ∩ E.

Where E = M(C|Θ|), and Σ(U) =

  (x1, x2, . . . , xn2) ∈ Rn2 : xi 0,

n2

• i=1

xi = |U|

   ⊂ Cn2.

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Geometric Properties of ICQCs

Corollary An ICLM Φ =

α∈Θ lαΠα is an ICQC if and only if its vector of

eigenvalues L = (lα) ∈ C|Θ| is an inverse image, in the mapping M : C|Θ| → E = M

• C|Θ|

, of some vector E = (ǫβ

i ) ∈ Σ(U) ∩ E,

i.e. L = (lα) = Minv(E).

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Quaternion Group Q

The quaternion group Q = {±Qe, ±Q1, ±Q2, ±Q3} is a non-abelian group of order eight. Q Qe −Qe Q1 Q2 Q3 −Q1 −Q2 −Q3 χid 1 1 1 1 1 1 1 1 χt1 1 1

• 1

1

• 1
• 1

1

• 1

χt2 1 1 1

• 1
• 1

1

• 1
• 1

χt3 1 1

• 1
• 1

1

• 1
• 1

1 χt4 2

• 2

Table of characters for the quaternion group Q.

The group Q can be represented as a subgroup of GL(2, C). The matrix representation R : Q → GL(2, C) is given by Qe =

• 1

1

• , Q1 =
• i

− i

• , Q2 =
• 1

−1

• , Q3 =
• i

i

• .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Quaternion Group Q

We construct an ICQC w.r.t the two-dimensional irrep t4. We have: U⊗Uc = Uid⊕Ut1⊕Ut2⊕Ut3, dim [IntQ (U ⊗ Uc)] = 4. The matrix representation of the trace preserving ICLM is the following:

• Φt4 =

Πid+lt1 Πt1+lt2 Πt2+lt3 Πt3 = 1 2     1 + lt2 1 − lt2 lt1 + lt3 lt3 − lt1 lt3 − lt1 lt1 + lt3 1 − lt2 1 + lt2     .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Quaternion Group Q

We have E = ML →

    

ǫid ǫt1 ǫt2 ǫt3

     = 1

2

    

1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1

         

lid lt1 lt2 lt3

    

so L = M−1E →

    

lid lt1 lt2 lt3

     = 1

2

    

1 1 1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 1

         

ǫid ǫt1 ǫt2 ǫt3

    

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Quaternion Group Q

The conditions ǫβ 0 reduce to: |lα| 1 2

• β∈Θ

|ǫβ| = 1 2

• β∈Θ

ǫβ = 1, ∀α ∈ Θ \ {id}, so all lα where α ∈ Θ \ {id} are included in a three dimensional cube. Kraus operators are given as: K(t1) =

• ǫt4

2

• 1

−1

• ,

K(t2) =

• ǫt4

2

• −1

1

• ,

K(t3) =

• ǫt4

2

• 1

1

• ,

K(id) =

• ǫid

2

• 1

1

• .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Permutation Group S(3)

In the case of G = S(3) we have three inequivalent irreducible representations. In the case of the two-dimensional irrep we have: U⊗Uc = Uid⊕Usgn⊕U, and dim

• IntS(3) (U ⊗ Uc)
• = 3.

The matrix representation Φ of the trace-preserving ICLM Φ is given by the following expression:

• Φ =

Πid+lsgn Πsgn+lλ Πλ = 1 2

    

1 + lsgn 1 − lsgn 2lλ 2lλ 1 − lsgn 1 + lsgn

     .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Permutation Group S(3)

We use the set of linear constraints E = ML:

    

ǫid ǫsgn ǫλ

1

ǫλ

2

     = 1

2

    

1 1 2 1 1 −2 1 −1 1 −1

       

1 lsgn lλ

   .

Conditions ǫβ

i 0 and 1 2(ǫid + ǫsgn + ǫλ 1 + ǫλ 2) = 1 yields to

the following statement: J (Φ) 0 ⇔ 1 lsgn −1, 1 2(1 + lsgn) |lλ| .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

ICQC based on the Permutation Group S(3)

The Kraus operators of the ICQC Φ are of the form:

K1(λ) =

• ǫλ

1

1

• ,

K2(λ) =

• ǫλ

2

1

• ,

K(sgn) =

• ǫsgn

2

• −1

1

• ,

K(id) =

• ǫid

2

• 1

1

• .

Preliminary Informations Towards Irreducibly Covariant Quantum Channels Characterisation of the Irreducibly Covariant Quamntum

Conclusions and Open Questions

Conclusions: Explicit characterisation of linear maps (in particular quantum channels) which are covariant with respect to irrep U of the finite group G. We give spectral decomposition of any such map only in terms of representation characteristics of G. We present if and only if conditions under which ICLM is ICQC. Open questions: Extensions of the presented results on the non-multiplicity free case. Construction of the irreducibly positive maps, but not completely positive. New classes of the entanglement witnesses.