A relation between completely bounded norms and conjugate channels

arXiv:quant-ph/0601071v1 11 Jan 2006

A relation between completely bounded norms and conjugate channels. Anna Jenˇcov´a Mathematical Institute, Slovak Academy of Sciences, Bratislava,Slovakia e-mail:[email protected]

Abstract.We show a relation between a quantum channel Φ and its conjugate ΦC , which implies that the p → p Schatten norm of the channel is the same as the 1 → p completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.

1. Introduction. A quantum channel is a completely positive trace preserving (CPT) map Φ : Md → Md′ , Md is the set of d × d complex matrices. Any channel can be viewed as a map Lq (Md ) → Lp (Md′ ), where Lq (Md ) denotes the space Md with the Schatten norm kAkq = Tr(|A|q )1/q , 1 ≤ q ≤ ∞. Let kΦkq→p be the corresponding norm of Φ, kΦ(A)kp kΦ(A)kp = sup , kΦkq→p = sup kAkq A∈Md ,A≥0 kAkq A∈Md the second equality was proved in [1, 8]. Multiplicativity of this type of norms is an important conjecture in quantum information theory. The spaces Lq (Md ) and Lp (Md′ ) can be endowed with an operator space structure as in [7], then Φ is a completely bounded map. Multiplicativity of the corresponding completely bounded norms kΦkCB,q→p for all 1 ≤ p, q ≤ ∞ was proved in [3]. In particular, this implies multiplicativity of kΦkq→p for q ≥ p, since this is equal to kΦkCB,q→p for CPT maps. It was shown that the norm kΦkCB,1→p is equal to the quantity ωp (Φ) =

sup ψ∈Cd ⊗Cd

k(I ⊗ Φ)(|ψihψ|)kp kTr2 (|ψihψ|)kp

Multiplicativity of ωp then yields the additivity for the CB minimal conditional entropy, defined as SCB,min =

inf

ψ∈Cd ⊗Cd

(S [(I ⊗ Φ)(|ψihψ|)] − S [Tr2 (|ψihψ|)])

In the present note, we show that there is a relation between ωp (Φ) and the norm kΦC kp→p of the conjugate map ΦC . This relation is then used for an alternative proof of multiplicativity of both quantities, avoiding the use of the deep results of the theory of operator spaces and CB norms, involved in the proofs in [3]. 1

2

2. Representations of CPT maps and conjugate channels. P Let ed1 , . . . , edd be the standard basis in Cd and let β0 = d1 i edi ⊗ edi be a maximally entangled vector. Let Φ : Md → Md′ be a CPT map. Then Φ is uniquely represented by its Choi-Jamiolkowski matrix Xφ ∈ Md ⊗ Md′ , defined by (1)

XΦ = d2 (I ⊗ Φ)(|β0 ihβ0 |) =

X

|edi ihedj | ⊗ Φ(|edi ihedj |).

i,j

Other representations of Φ can be obtained from the Stinespring representation, which in the case of matrices has the form [6] (2)

Φ(ρ) = V † (ρ ⊗ Iκ )V,

′

V : Cd → Cd ⊗ H,

Tr2 V V † = Id

where H is an auxiliary Hilbert space, κ = dim H ≤ dd′ . The Lindblad-Stinespring representation of Φ is Φ(ρ) = Tr2 U (ρ ⊗ |φihφ|)U †

(3)

′

where φ is a unit vector in H, and U : Cd ⊗ H → Cd ⊗ H is a partial isometry. ˆ The This can be obtained from the Stinespring representation of the dual map Φ. Kraus representation (4)

Φ(ρ) =

κ X

Fk ρFk† ,

′

Fk : Cd → Cd ,

k=1

X

Fk† Fk = Id

k

is related to (2) and (3) by V

=

Fk

=

κ X

Fk† ⊗ |eκk i

k=1

Tr2 U (I ⊗ |φiheκk |), k = 1, . . . , κ,

where eκ1 , . . . , eκκ is an orthonormal basis in H. Let Φ be given by (3). The conjugate channel to Φ is the map ΦC : Md → B(H), defined as [4, 5]   X (5) ΦC (ρ) = Tr1 U (ρ ⊗ |φihφ|)U † = Tr Fj ρFk† |eκj iheκk | j,k

The next Lemma shows a relation between the Stinespring representation (2) of Φ and the Choi-Jamiolkowski matrix (1) of its conjugate. Lemma 1. Let Φ be a CPT map, such that Φ(ρ) = V † (ρ ⊗ Iκ )V is the Stinespring representation. Then XΦC = (V V † )T T where B T is the transpose of the matrix B, Bij = Bji .

3

Proof: Let V = VV†

=

Pκ

k=1

κ X

Fk† ⊗ |eκk i, then using (5), we get

Fi† Fj ⊗ |eκi iheκj | =

=

|edk ihedl | ⊗

=

k,l=1

κ X

i,j=1

k,l=1 d X

hedk |Fi† Fj |edl i |edk ihedl | ⊗ |eκi iheκj |

i,j=1 k,l=1

i,j=1 d X

κ d X X

  Tr Fj |edl ihedk |Fi† |eκi iheκj |

 T |edk ihedl | ⊗ ΦC (|edl ihedk |) = XΦTC



Theorem 1. For a CPT map Φ and 1 ≤ p ≤ ∞, kΦkp→p = ωp (ΦC ) Proof. Note first that for any CPT map, we have ([3]) (6)

ωp (Φ) =

k(A ⊗ Id′ )XΦ (A ⊗ Id′ )kp ,

sup A≥0,kAk2p ≤1

Let the Stinespring representation (2) of Φ be Φ(ρ) = V † (ρ ⊗ Iκ )V . Then by Lemma 1, kΦkp→p

=

sup

kΦ(A)kp =

A≥0,kAkp ≤1

=

sup

kV † (B 2 ⊗ Iκ )V kp =

B≥0,kBk2p ≤1

sup B≥0,kBk2p ≤1

k(B ⊗ Iκ )XΦTC (B ⊗ Iκ )kp = ωp (ΦC ),

the last equality follows from the fact that B T ≥ 0 if B ≥ 0 and kB T kp = kBkp .  Remark. Let q > p. Exactly as in the above proof, we get that kΦkq→p =

sup A≥0,kAk2q ≤1

k(A ⊗ Id′ )XΦC (A ⊗ Id′ )kp ,

The last expression is equal to the Lr (Md , Lp (Md′ )) norm kXΦC k(r,p) for 1q + r1 = p1 , see Eq. (3.18) in [3]. This is an operator space type of norm, but not a CB norm, in general. 3. Multiplicativity. To prove multiplicativity, we need the following observation (7)

ωp (Φ ⊗ Tr) = ωp (Tr ⊗ Φ) = ωp (Φ)

This follows from Lemma 2, proved in the Appendix. We remark that this equality implies that the supremum in the definition of ωp can be taken over all Md ⊗ Md , that is, (8)

ωp (Φ) =

sup X∈Md ⊗Md

k(I ⊗ Φ)(X)kp kTr2 (X)kp

4

To show this, we first note that the supremum in (8) may be restricted to positive 2 X. Let X ≥ 0 and let |ψ123 i ∈ Cd ⊗ Cd ⊗ Cd be a purification of X, X = Tr3 (|ψ123 ihψ123 |). Then k(I ⊗ Φ)(X)kp kTr2 (X)kp

=

k(I1 ⊗ Φ)(Tr3 (|ψ123 ihψ123 |))kp = kTr23 (|ψ123 ihψ123 |)kp

=

k(I1 ⊗ Φ ⊗ Tr)(|ψ123 ihψ123 |))kp kTr23 (|ψ123 ihψ123 |)kp

Consequently, ωp (Φ)

≤

sup X∈Md ⊗Md

≤

k(I ⊗ Φ)(X)kp ≤ kTr2 (X)kp

sup ψ∈(Cd ⊗Cd2 )⊗2

k(I12 ⊗ Φ ⊗ Tr)(|ψihψ|)kp = ωp (Φ ⊗ Tr) = ωp (Φ), kTr34 (|ψihψ|)kp

hence the assertion. We now obtain an alternative proof of multiplicativity of k · kp→p and ωp . Theorem 2. For CPT maps Φ1 : Md1 → Md′1 and Φ2 : Md2 → Md′2 and for 1 ≤ p ≤ ∞, kΦ1 ⊗ Φ2 kp→p ωp (Φ1 ⊗ Φ2 )

= kΦ1 kp→p kΦ2 kp→p = ωp (Φ1 )ωp (Φ2 )

Proof. We first show that the p → p norm of a channel Φ is not changed by tensoring with identity. Indeed, by Theorem 1 and (7), kΦ ⊗ Ikp→p = ωp ((Φ ⊗ I)C ) = ωp (ΦC ⊗ Tr) = ωp (ΦC ) = kΦkp Similarly, kI ⊗ Φkp→p = kΦkp→p . Let now A ∈ Md1 ⊗ Md2 , B = (I ⊗ Φ2 )(A) and compute sup A

k(Φ1 ⊗ Φ2 )(A)kp k(Φ1 ⊗ I)(B)kp k(I ⊗ Φ2 )(A)kp = sup ≤ kΦ1 kp→p kΦ2 kp→p kAkp kBkp kAkp A

Since the opposite inequality is easy, we get kΦ1 ⊗ Φ2 kp→p = kΦ1 kp→p kΦ2 kp→p , which in turn implies the multiplicativity of ωp .  Acknowledgements. This work was done during a visit to Tufts University and thereby partially supported by NSF grant DMS-0314228. The author wishes to thank Mary Beth Ruskai and Christopher King for discussions and valuable comments. The research was supported by Center of Excellence SAS Physics of Information I/2/2005 and Science and Technology Assistance Agency under the contract No. APVT-51-032002. Appendix. The following Lemma is due to C. King.

5

Lemma 2. Let Ω : Mn → Mm be a channel with the covariance property Ω(U ρU † ) = U ′ Ω(ρ)¯(U ′ )† where U ′ is a unitary in Mm , for any unitary U ∈ Mn . Then for any CPT map Φ, we have ωp (Ω ⊗ Φ) = ωp (Φ ⊗ Ω) = ωp (Φ)ωp (Ω) Proof. The proof uses the fact that there are n2 unitary operators in Mn , such Pn2 −1 that k=0 Uk AUk† = n(TrA)In for any n × n matrix A, and therefore X (Uk ⊗ Id )A12 (Uk† ⊗ Id ) = nIn ⊗ A2 k

for A12 ∈ Mn ⊗ Md , A2 = Tr1 A12 . Let us define ip h ip h 1/2 1/2 gp (ρ, Φ) = Tr (ρ1/2p ⊗ Id′ )XΦ (ρ1/2p ⊗ Id′ ) = Tr XΦ (ρ1/p ⊗ Id′ )XΦ

so that ωp (Φ)p = supρ≥0,Trρ≤1 gp (ρ, Φ). Then by [2], ρ 7→ gp (ρ, Φ) is concave. It is easy to see that gp (ρ, Ω) = gp (U ρU † , Ω) for any unitary operator U on Cn . It follows that for any ρ ≥ 0, Trρ = 1 ! 1 X 1 X † † gp (Uk ρUk , Ω) ≤ gp Uk ρUk , Ω gp (ρ, Ω) = n2 n2 k

=

k

1 gp ( In , Ω) = n−1/p kXΩ kp = ωp (Ω) n

Similarly, we have gp (ρ12 , Ω ⊗ Φ) =

1 X gp ((Uk ⊗ Id )ρ12 (Uk† ⊗ Id ), Ω ⊗ Φ) n2 k

≤ =

! 1 X gp (Uk ⊗ Id )ρ12 (Uk† ⊗ Id ), Ω ⊗ Φ n2 k   1 1 In ⊗ ρ2 , Ω ⊗ Φ = gp ( In , Ω)gp (ρ2 , Φ) gp n n

The easy inequality ωp (Ω)ωp (Φ) ≤ ωp (Ω ⊗ Φ) now finishes the proof. The equality ωp (Φ ⊗ Ω) = ωp (Φ)ωp (Ω) is proved similarly.  References [1] K.M.R. Audenaert, A note on the p → q norms of completely positive maps, math-ph/0505085 [2] H. Epstein, Remarks on two theorems of E. Lieb, Commun. Math. Phys. 31, (1973), 317–325 [3] I. Devetak, M. Junge, C. King, M.B. Ruskai, Multiplicativity of completely bounded p-norms implies a new additivity result, quant-ph/0506196 [4] A. S. Holevo, On complementary channels and the additivity problem, quant-ph/0509101 [5] C. King, K. Matsumoto, M. Nathanson, M.B. Ruskai, Properties of conjugate channels with applications to additivity and multiplicativity, quant-ph/0509126 [6] V. Paulsen, Completely bounded maps and operator algebras, Cambridge University Press, (2002) [7] G. Pisier, Non-commutative vector valued Lp -spaces and completely p-summing maps, Ast´ erisque (Soc. Math. France) 247 (1998), 1–131 [8] J. Watrous, Notes on super-operator norms induced by Schatten norms, Quantum. Inf. Comput. 5, 57–67 (2005), quant-ph/0411077