Gaussian Classical capacity of Gaussian Quantum channels

Transcription

1 Gaussian Classical capacity of Gaussian Quantum channels E. KARPOV, J. SCHÄFER, O. PILYAVETS and N. J. CERF Centre for Quantum InforamIon and CommunicaIon Université Libre de Bruxelles MathemaIcal Challenges of Quantum Transport St. Petersburg March 2013

2 Outline Quantum channels for informaion transmission Classical capacity of quantum channels Gaussian channels AddiIve (classical) noisuantum channel Gaussian capacity MathemaIcal challenges Conclusion (Gaussian)

3 Classical capacity of quantum channel Quantum channel completely posiive map InformaIon encoding into quantum states modulaion { in P i, ρ } i Average input: input ensemble ρ in = in P i ρ i, average output InformaIon content Von Neumann entropy i ρ out = T! " ρ in InformaIon capacity [Holevo-Schumacher-Westmoreland] C 1 # [ T ] = max % in { P i,ρ i } $ S ρ ( ) P i S ρ i out i S ρ ( ) ρ = out P i ρ i & ( ' i # $ ( ) = Tr [ ρ Log 2 ρ]

4 AddiIvity problem When using entangled inputs is classical capacity superaddiive? C 1 (T 1 T 2 ) > C 1 (T 1 )+ C 1 (T 2 ) Capacity : C T ( ) = lim n someimes yes [HasIngs] Memory channels / correlated channel uses : Parallel n-mode channel C T ( ) = lim n 1 n C 1 1 n C 1 ( T n ) ( T ( n ))

5 Bosonic channels coninuous variables Quadratures of electromagneic field mode!â, â " # $ =1 ( ) ˆp = 1 ˆq = 1 2 â + â i 2 â â Gaussian states are fully determined by ρ G (d,v in ) displacement vector first moments covariance matrix (CM) second moments Gaussian channels transform first and second moments d out = Xd in + d V out = XV in X +Y d = q, p V X ( ) T, - Real matrix Y, - CM = ω =1 ( )

6 AddiIve noise channel Classical noise displacement with a Gaussian probability distribuion Input energy constraint X = I V out = V in +Y Non-trivial opimizaion problem Tr V in +V m [ ] E Challenge 1: Classical capacity of Gaussian channels is addiive True for entanglement breaking channels [Holevo 2008] RestricIon to Gaussian states: Gaussian modulaion displacement with a Gaussian probability distribuion Modulated output CM V = V in +V m +Y

7 Gaussian capacity C 1 C 1 Von Neumann entropy is a funcion of CM T G [ ] = max All signal states at the output have the same entropy T G [ ] = max $ S V { Tr(V in +V m ) E}% # S V { Tr(V in +V m ) E} $ ( ) dq dpp ( ) S V out ( ) Challenge 2: Gaussian capacity is the capacity Proven for lossy channel with vacuum noise [V. Giovanneg et al., Phys. Rev. Leh. 92, (2004)] % & q, p ( )S V out ( ) & '

8 One modroblem 2x2 CM: SymplecIc eigenvalue Entropy: S V ( ) = g ν 1/ 2 ( ) V diag v, v ( ) g( x) = ( x +1)Log 2 ( x +1) ( x)log 2 ( x) OpImizaIon in terms of CM C 1 [ T G ] = max #$ S Tr(V in +V m ( V in +V m +Y ) S( V in +Y ) % & { ) E} If noise CM is diagonalized by a passive symplecic transformaion then opimal input and modulaion are diagonal In the eigenbasis of the CM of noise V ν = ( + m q + e )( q + + e ) p

9 Waterfilling soluion Uniform distribuion of the output energy is opimal OpImal input is a minimum output entropy Gaussian state C 1 [ T ] = g E ( ) (( ) / 2) g V q = V p Challenge 3: minimum output entropy is achieved on a Gaussian state (vacuum state) E > E th = + = [Holevo, et al. PRA 59, 1820 (1999)] m q

10 OpImal states for mulimode channel Non-trivial input energy distribuion between the modes common Lagrange muliplier waterfilling E > E th = m q m q e λ =1 q + i q i q i q m q = =1

11 SoluIon of 2 d type 1< E < E th Only onuadrature modulated Lagrange mulipliers no explicit soluion An implicit soluion given by a transcendent equaion Input energy also spent on the non-modulated quadrature OpImal input state is not the minimum output entropy state [Schäfer, et al. PRA 80, (2009)] 1< <

12 SoluIon of 2 d type g' ν 1/ 2 ( ) ν ( E + 2i ) q = g' ν 1/ 2 out g' x ( ) = Log 2 x +1 ( ) Log 2 x ( ) ν out " $ # % ' & ν = ( + )(E + ) ν out = ( + )( + ) + + = E =1/ 4 g' ν 1/ 2 ( ) ν ( + e ) q = λ - Lagrange muliplier

13 OpImal states for mulimode channel Non-trivial energy distribuion among the modes common Lagrange muliplier generalized waterfilling 1< E < E th E =1 E > E th i q m q m q m q i q i p e i q q = 1< < =1

14 ApplicaIon - memory channel Correlated noise correlated opimal input mulimodroblem 1 Parallel channel C( T ) = lim n n C 1 Gauss Markov noise { Μ( φ) } i, j = Nφ i j # Y = % % $ Μ q ( φ) 0 ( ) 0 Μ p φ ( ( T n )) SymplecIc and orthogonal diagonalizaion is possible as Μ q ( φ) and Μ q ( φ ) commute in the limit n & ( ( '

15 AddiIve Gauss-Markov noise DiagonalizaIon of infinite noise matrix coninuous spectrum on a finite domain OpImal mulimode input state in the original basis is entangled preparaion?

16 Conclusion Gaussian capacity of addiive noise bosonic channels Three types of soluion for one mode Input state that realizes classical capacity is not always the minimum output entropy state Challenge 1: Minimum output entropy for Gaussian channels is achieved on a Gaussian state Challenge 2: Classical capacity of memoryless Gaussian quantum channels is realized by Gaussian states Challenge 3: Classical capacity of memoryless Gaussian quantum channels is addiive

17 Thank you for your ahenion!