On Degradable Quantum Channels by Y ingkai Ouyang.. Main Reference : - PDF document

On Degradable Quantum Channels by Y ingkai Ouyang.. Main Reference : quant-ph/0802.1360v2, The struc- ture of degradable quantum channels, Cubitt, Ruskai, Smith 1 Complementary Channels Φ : M d A → M d B . A k ρA † A † � � Φ( ρ ) = k , k A k = I d A . k k Now define � | k � ⊗ A k . W = k Then WρW † = | j �� k | ⊗ A j ρA † � k . j,k Tr E ( WρW † ) = Φ( ρ ) For convention, define Φ C ( ρ ) = Tr A ( WρW † ) . ‘A’ labels the system, ‘E’ labels the environment. Choi rank   d A − 1 � | i �� j | ⊗ Φ( | i �� j | ) d E := rk ( J (Φ)) = rk   i,j =0 is the minimal number of Kraus operators needed to represent Φ. One can check that � Φ C ( ρ ) = R µ ρR † µ µ 1

where � j | R µ | k � = � µ | A j | k � , µ ∈ { 0 , ..., d B − 1 } , k ∈ { 0 , ..., d A − 1 } . The j -th row of R µ is the µ -th row of A j . 2 Degradable Channels Definition: A channel is degradable if there exists a CPT Ψ such that Ψ ◦ Φ = Φ C , that is Ψ(Φ( ρ )) = Φ C ( ρ ) ∀ ρ ∈ M d A ⇒ ker Φ ⊆ ker Φ C . Fact: Φ degradable = Easy to show Facts: ⇒ Φ , Φ C degradable, anti-degradable • d A = 1 = • d B = 1 = ⇒ Φ = Tr = ⇒ Φ antidegradable Φ C = Φ( ρ ) = UρU † , U † U = I d A • d E = 1 = ⇒ = ⇒ ⇒ Φ degradable, Φ = Tr Tr = Thm 1: Suppose Φ : M d A → M d B maps every pure state to a pure state. Then either 2

1. d A ≤ d B and Φ( ρ ) = UρU † , U † U = I d A , Φ degradable, d E = 1. 2. Φ( ρ ) = Tr ( ρ ) | φ �� φ | , Φ antidegradable. holds Thm3 Let Φ be CPT such that there exists | ψ � ∈ C d A with rank (Φ( | ψ �� ψ | )) = d B . Then if Φ is degradable, then d E = d B . Thm4: Let Φ : M d A → M 2 be a CPT map with qubit output . If Φ is degradable, then (i) d E ≤ 2, (ii) and d A ≤ 3. Proof of Thm4: (i) If maxrank ρ (Φ( ρ )) = 1, then by Thm1 , d E = 1. If maxrank ρ (Φ( ρ )) = 2, then by Thm3 , d E = d B = 2. ⇒ Φ( ρ ) = AρA † + BρB † . For a 1 , a 2 ∈ [0 , 1], (ii) d E ≤ 2 = � √ a 1 � 0 0 ... 0 √ a 2 0 ... 0 A = 0 and B † B = I d A − A † A = diag (1 − a 1 , 1 − a 2 , 1 , ..., 1) rk ( B † B ) ≤ But B is 2 × d A matrix ⇒ rk ( B ) ≤ 2 ⇒ = = 2 = ⇒ d A ≤ 4. If d A = 4, then a 1 = a 2 = 1. But ker Φ is not contained in ker Φ C which is a contradiction. Hence d A ≤ 3. end of Thm4 � Thm by Wolf, Perez-Garcia Let Φ : M 2 → M 2 have Choi rank 2. If Φ is degradable or antidegradable, its Kraus operators 3

are � 0 � cos α � � 0 sin α A 0 = , A 1 = . (2.1) 0 cos β sin β 0 Significance of Thm4 Thm10: Let Φ have qutrit output. If Φ degradable, d E ≤ 3 Question : What about results with d B = 4 , 5 , 6 , ... ? d E ≤ d B ? 4

Answer: No. If d B = 2 d A then we can have d E > d B (Construction in reference). CRS also construct channels with d A = d B = 6 d , and d E = 3( d 2 + 1) > 6 d = d B . But what about d B = 4 , 5. 3 What other channels are degradable? Thm11 Every channel with rank 1 Kraus operators is anti- degradable. Proof is constructive. Many more examples of antidegradable channels given. LOTS of such channels. (although it is remarked that most channels are neither degradable/ antidegradable) 4 Applications The qubit amplitude damping channel (degradable) has been used by SSW to improve the upper-bound for the depolarization chan- nel. If N , M are degradable, then Q ( λ N + (1 − λ ) M ) ≤ λQ ( N ) + (1 − λ ) Q ( M ) . Quantum capacity of degradable channels can be efficiently evaluated because of several results. • I coh is additive for Φ degradable. • I coh (Φ , ρ ) is concave function of ρ for Φ degradable, implies that we only need to consider diagonal ρ . 5

• I coh (Φ , ρ ) = S (Φ( ρ )) − S (Φ C ( ρ )) 2 A γ ( ρ ) + 1 1 2 XA γ ( ρ ) X = N γ ( ρ ) where N γ has Kraus operators √ p x X, √ p y Y, √ p z Z, � 1 − p x − p y − p z I and 2 − √ 1 − γ 4 , p z = 1 − γ p x = p y = γ 2 Now define H = X + Z 2 , H yz = Y + Z 2 , H xy = X + Y 2 . Conjugation of a nontrivial Pauli by H takes X → Z, Y → Y, Z → X . Conjugation of a nontrivial Pauli by H yz takes X → X, Y → Z, Z → Y . Conjugation of a nontrivial Pauli by H xy takes X → Y, Y → X, Z → Z . Now let Φ p be a quantum channel such that Φ p ( ρ ) = N γ ( ρ ) + H N γ ( H † ρH ) H † + H yz N γ ( H † yz ρH yz ) H † yz . 3 Then Φ p is a depolarization channel of noise parameter p = ( p x + p y + p z ) / 3. 5 Further questions Can we use a dimension 2 m dimension amplitude damping chan- nel to also obtain an upper bound for the quantum capacity of the depolarization channel? Tensor product of m qubit ampli- tude damping channels is not equal to a 2 m dimension amplitude damping channel in general. 6

Let � γ = ( γ 1 , ..., γ d − 1 ) It turns out that the following channel A ( d ) : M d → M d , with Kraus operators � γ d − 1 � � A 0 = | 0 �� 0 | + 1 − γ i | i �� i | (5.1) i =1 A i = √ γ i | 0 �� i | , i ∈ [ d − 1] (5.2) for real γ i ∈ [0 , 1]. It follows that the complementary channel for A ( d ) have the Kraus operators d − 1 √ γ i | i �� i | � R 0 = | 0 �� 0 | + (5.3) i =1 � 1 − γ i | 0 �� i | , i ∈ [ d − 1] R i = (5.4) Now let � λ = ( λ 1 , ..., λ d − 1 ) such that λ i = 1 − 2 γ i 1 − γ i . If 0 ≤ γ i ≤ 1 2 for all i ∈ [ d − 1], then A ( d ) is a well defined CPT channel and � λ A ( d ) γ ◦ A ( d ) λ = A ( d ) γ . Thus A ( d ) is degradable if 0 ≤ γ i ≤ 1 2 for all � � 1 − � � γ i ∈ [ d − 1]. References [1] M. M. Wolf and D. Perez-Garcia, “Quantum capacities of channels with small environment,” Phys. Rev. A , vol. 75, no. 012303, 2007. [2] T. S. Cubitt, M. B. Ruskai, and G. Smith, “The structure of degradable quantum channels,” Journal of Mathematical Physics , vol. 49, no. 10, 2008. 7