A Lossy Bosonic Quantum Channel with Non-Markovian Memory O. V. - PowerPoint PPT Presentation

A Lossy Bosonic Quantum Channel with Non-Markovian Memory O. V. Pilyavets, V. G. Zborovskii and S. Mancini Universit` a di Camerino (Italy) & P. N. Lebedev Physical Institute (Russia) August 30, 2008

Introduction Quantum channels Gaussian channels Lossy bosonic Gaussian channels Characteristics of quantum channel Quantum capacity Classical capacity Rates Homodyne rate Heterodyne rate O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

General definitions Any state can be labeled as ρ or V as all states are Gaussian. V in - covariance matrix for input state V env - covariance matrix for environment state V cl - covariance matrix for classical distribution of coherent amplitude α . V out - covariance matrix for output state of the channel V out - covariance matrix for output state of the channel avaraged over classical distribution (encoding of information) V cl Capacities and rates C n = max states 1 n χ n - classical capacity for n uses of channel C = max n →∞ C n - classical capacity on infinite amount of channel uses Conjectures Capacity for lossy bosonic channel can be achieved on Gaussian states Maximizing of Holevo χ leads to capacity for memory channel too O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

Scheme for 1 use of lossy bosonic channel environment ρ env tot ρ in input encoding of output decoding classical (measurement) tot ) ρ in ( α ) η ρ out ( α )=Tr env ( ρ out information tot ρ out e.g. heterodyne: ρ in ( α )=D( α ) ρ 0 D + ( α ) < α | ρ out | α > P( α ) ↔ V cl Tr(V in +V cl )/(2n)=N+1/2 Energy restriction: in U + out =U( ρ in ⊗ρ env )U + =U ρ tot ρ tot Beam splitter action: O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: notations and known results It is sufficient to consider only diagonal matrices (!). Abitrary covariance matrices in diagonal form for 1 use: � e s � e r � � 0 0 V env = ( N env + 1 / 2) V in = ( N in + 1 / 2) e − s e − r 0 0 Already known capacities: If environment is in ground (vacuum) state: C = g [ η N ] If environment is in termal state: C = g [ η N + (1 − η ) N env ] − g [(1 − η ) N env ] Environment in termal and squeezed state: � � �� C = g η N + (1 − η ) ( N env + 1 / 2) cosh( s ) − 1 / 2 − g [(1 − η ) N env ] g ( x ) = ( x + 1) log 2 ( x + 1) − x log 2 x O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] η =1 η =0.99 3 2 Step by η is 0.05 1 η =0.05 0 0 2 4 6 8 10 12 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] η =1 η =0.99 3 2 Will they continue to grow? 1 η =0.05 0 0 2 4 6 8 10 12 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] η =1 η =0.99 3 2 Step by η is 0.05 1 η =0.05 0 0 2 4 6 8 10 12 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] η =1 η =0.99 3 2 Step by η is 0.05 1 η =0.05 0 0 2 4 6 8 10 12 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] η =1 η =0.99 3 2 why? Step by η is 0.05 1 η =0.05 0 0 2 4 6 8 10 12 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =0.005, N=0.001] η =1 0.01 η =0.9 0.005 Step by η is 0.1 η =0.1 0 0 2 4 6 s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] Optimal value r opt [parameters: N env =2, N=3, η =0.7] η =1 2 η =0.99 3 1.5 2 1 Step by η is 0.05 1 0.5 η =0.05 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 s s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: classical capacity Classical capacity C(s) [parameters: N env =2, N=3] Optimal value r opt [parameters: N env =2, N=3, η =0.7] η =1 2 η =0.99 3 1.5 2 1 Step by η is 0.05 kink? 1 0.5 r opt = s η =0.05 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 s s O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: complete analytical solution Suppose that eigenvalues of V env matrix are e 1 , e 2 . Then, eigenvalues of matrix V cl (which are c 1 , c 2 ) and V in (which are i 1 , i 2 ) can be found from the following relations if both c 1 and c 2 are positive: c 1 = N + 1 2 − 1 � e 1 + 1 � 1 − 1 � ( e 1 − e 2 ) 2 e 2 2 η � e 2 � � c 2 = N + 1 2 − 1 + 1 1 − 1 ( e 2 − e 1 ) 2 e 1 2 η i 1 = 1 � e 1 i 2 = 1 � e 2 , 2 2 e 2 e 1 In this case capacity can be expressed in explicit form and is equal to � � �� C = g η N + (1 − η ) ( N env + 1 / 2) cosh( s ) − 1 / 2 − g [(1 − η ) N env ] O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

1 use capacity: complete analytical solution If c k (according to previous relations) is negative, then c k = 0, c m = 2 N + 1 − i k − 1 / (4 i k ), i m = 1 / (4 i k ), and i k is a solution of the following transcedental equation ( { k , m } = { 1 , 2 } or { k , m } = { 2 , 1 } ): √ a m a k + 1 / 2 √ o m o k + 1 / 2 a m − a k √ a m a k − 1 / 2 = o m i k − o k i m √ a m a k log 2 √ o m o k log 2 √ o m o k − 1 / 2 i k where o 1 = η i 1 + (1 − η ) e 1 a 1 = η ( i 1 + c 1 ) + (1 − η ) e 1 o 2 = η i 2 + (1 − η ) e 2 a 2 = η ( i 2 + c 2 ) + (1 − η ) e 2 No explicit relation for capacity. O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

Is there something new for n uses of the channel? Is there new “physics” there? Can we say that entanglemet is useful for information transmission for many uses of the channel? Let us see... O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

Scheme for n uses of lossy bosonic channel ENVIRONMENT OUTPUT INPUT O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

n uses capacity: analytics In the case of our type of memory it is sufficient to consider only commuting matrices (!). Suppose that eigenvalues of V env matrix are e qk , e pk , k = 1 , ..., n . Then, eigenvalues of matrix V cl (which are c qk , c pk ) and V in (which are i qk , i pk ) can be found from the following relations if for all k both c qk and c pk are positive: c qk = N + 1 2 − 1 + 1 − η � Tr V env � � e qk − e qk 2 η 2 n e pk c pk = N + 1 2 − 1 + 1 − η � Tr V env � � e pk − e pk 2 η 2 n e qk i qk = 1 � e qk i pk = 1 � e pk , 2 e pk 2 e qk O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

n uses capacity: analytics In this case the capacity can be expressed in explicit form. It is equal to n � � Tr V env − 1 �� − 1 � [(1 − η )( √ e qk e pk − 1 / 2)] C n = g η N + (1 − η ) 2 n 2 n k =1 n � � �� Tr V env − 1 1 � [(1 − η )( √ e qk e pk − 1 / 2)] C = g η N + (1 − η ) lim − lim 2 n 2 n n →∞ n →∞ k =1 If c qk or c pk is negative we don’t have explicit relation for the capacity Capacity is always achieved on states V in minimizing uncertainty relation (!) O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

n uses capacity: Ω-model of channel memory Environment matrix: � exp( s Ω) V env = 1 � 0 2 0 exp( − s Ω) where   0 1 . . . . . . . . . . . . 0 1 0 1 . . . . . . . 0     . .   . 1 0 1 . . . 0   Ω = . .  ... ... ...  . .   . .   . .  ... ...  . .  . . 1    0 0 . . . . . . . 1 0 O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory

n uses capacity: Ω-model of channel memory Correlations decay exponentially over channel uses, it is quite “realistic” (correlations are non-Markovian) Energy constraint: Tr ( V in + V cl ) = N + 1 2 n 2 Ω-model allows us to test entanglement Capacity for Ω-model: � η N + 1 � C = g 2(1 − η )( I 0 (2 s ) − 1) O. V. Pilyavets, V. G. Zborovskii and S. Mancini Lossy Bosonic Channel with Non-Markovian Memory