Output entropy of tensor products of random quantum channels 1

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1 Output entropy of tensor products of random quantum channels 1 Motohisa Fukuda 1 [Collins, Fukuda and Nechita], [Collins, Fukuda and Nechita] 1 / 13

2 1 Introduction Aim and background Existence of one large eigenvalue 2 2 / 13

3 Aim and background Existence of one large eigenvalue Aim 1 Let Φ be a random quantum channel: Φ : M Nn (C) M n (C) with its environment space being C k. 2 Then, we ask: what is the asymptotic behavior of output eigenvalues of Φ Φ( a n a n ) as n and N n grow? 2 Here, Φ is the complex conjugate of Φ. 3 To make this question tractable, we need to define a good measure on quantum channels and impose some conditions on the sequence of a n C Nn. 2 The case where a n is a Bell state is done in [Collins and Nechita] 3 / 13

4 Aim and background Existence of one large eigenvalue Background 1 It is shown in [Hastings] that for some channels Φ Here, S min (Φ Φ) < S min (Φ) + S min ( Φ) S min (Φ) = min S(Φ(ρ)) ρ where S( ) be the von Neumann entropy. 2 This phenomena, called additivity violation, solved other important problems in quantum information theory via the equivalence proven in [Shor]. 3 An important factor in the proof on additivity violation is that Φ Φ( b b ) has some eigenvalue more than N/kn when b is a Bell state, proven in [Hayden and Winter]. 4 / 13

5 Aim and background Existence of one large eigenvalue Hayden - Winter trick 1 Let s prove the property for random unitary channels: Ψ(ρ) = k p i U i ρui i=1 where U i U(n) are unitary matrices and {p i } is a probability distribution. 2 Remember that, for a Bell state b, we have U Ū b = b 3 Now, with the fact k i=1 p2 i 1 k Ψ Ψ( b b ) = i=1 p 2 i b b + i j p i p j (U i Ū j ) b b (U i Ū T j ) prove the statement because N = n. 5 / 13

6 Conditions on input sequences 3 1 Assumption 0: Of course, in order to satisfy unit trace condition, we have 2 Assumption 1: Tr [A n ] Nn Tr[A n A n] = 1 = m + O where m is an important parameter. 3 Assumption 2: A n = O ( ) 1 n 2 ( 1 n ) where A n must be stable for the distribution to have the limit. 3 We have the identification a n = A n where A n M Nn (C) 6 / 13

7 Examples of input sequences 1 Example 1: Bell state a n = 1 Nn N n j=1 j j A n = 1 Nn diag{1,..., 1} m = 1 2 Example 2: Bell state with phase a n = 1 Nn N n j=1 A n = 1 Nn diag m = 0 { 2πi exp j N n { exp { 2πi N n } j j }, exp { 4πi N n } },..., 1 7 / 13

8 Setup for random quantum channels 1 A quantum channel Φ : M Nn M n with its environment space being of dimension k can be written as Φ(ρ) = Tr C k [V ρv ] for some isometry V : C Nn C kn. 2 Then, we define a measure on quantum channels with respect to the Haar measure on U(kn), i.e., we can get the random isometry V by truncating U U(kn) 3 Its complex conjugate is defined as [ Φ(ρ) ] = Tr C k V ρv T 8 / 13

9 Our random matrix 1 Our random matrix Z n M n 2(C) to be analyzed is : Z n = Φ Φ( a n a n ) 2 We set N = tkn with 0 < t < 1 where k is fixed. 3 We investigate asymptotic behavior of the empirical eigenvalue distribution of a matrix, which is the probability measure: 1 k 2 δ λi, k 2 i=1 where λ 1,..., λ k 2 are (asymptotically) the non-zero eigenvalues of Z n. 9 / 13

10 Limiting measure 1 Under the assumptions on input sequences, the empirical eigenvalue distribution of the matrix Z n converges almost surely, as n, to the probability measure 1 [ k 2 δλ1 + (k 2 ] 1)δ λ2 where the Dirac masses are located at λ 1 = t m t m 2 k 2 and λ 2 = 1 t m 2 k 2. 2 In other words, the output state has asymptotically the following eigenvalues: t m t m 2 k, with multiplicity one; 2 1 t m 2 k, with multiplicity k / 13

11 Analysis 1 Among inputs under the assumptions a Bell input gives the least entropy for random conjugate pair Φ Φ. 2 It is because the entropy is monotone in m but on the other hand, Tr[AA ] = 1 implies that Tr [A n ] 1 Nn i.e. m 1 The inequality is attained when A n = I / N n up to a global complex phase. 3 This is a positive evidence supporting the conjecture that Bell states give the least entropy for generic conjugate pairs Φ Φ. 4 4 I.e., we ultimately want to fix Φ Φ and then know the optimizers. 11 / 13

12 Setup for random random unitary channels 1 A random unitary channel Φ : M n M n is defined by Ψ(ρ) = k w i U i ρui i=1 where U i U(n) and (w i ) i is a probability distribution. 2 If we take (U i ) i to be i.i.d. with respect to the Haar measure on U(n), we can define random random unitary channels. 3 Its complex conjugate is defined by Ψ(ρ) = k w i Ū i ρui T i=1 12 / 13

13 Result for random unitary channels 1 Similarly as before, we are interested in the following random matrix Z n M n 2(C) : Z n = Ψ Ψ( a n a n ) 2 For simplicity, we set w i = 1/k for all 1 i k in this slide. 5 3 Then, assuming the same conditions on the sequence (a n ) n, the limit eigenvalue distribution turns out to be m 2 k + 1 m 2 k 2, 1 m 2 k m 2 }{{ k 2 } k 1 times, 1 k 2,..., 1 }{{ k 2 } k 2 k times 5 In our paper, (w i ) i is another parameters. 13 / 13