Estimating capacities and rates of Gaussian quantum channels - PowerPoint PPT Presentation

Estimating capacities and rates of Gaussian quantum channels Stefano Mancini School of Science University of Camerino, Italy sabato 19 febbraio 2011

Motivations • Most of the performed studies e.g. on classical capacity concern simple settings (memoryless and vacuum environment) • No general methods available for evaluating, e.g. classical capacity • Rates usually derived in a different way with respect to capacity • Consider lossy bosonic channel as a paradigm of Gaussian channels • Introduce a generic model for multiple channel uses and devise a method to evaluate the Holevo function (turns out to be useful for classical capacity as well as for dyne rates) • Maximization problem can be split it into “inner” one and “outer” one based on Pilyavets, Lupo & Mancini, arXiv0907.1532 (provisionally accepted by IT Trans) sabato 19 febbraio 2011

Outline • Gaussian channels • Lossy bosonic channel • Classical capacity and rates • Single channel use (bosonic mode) • The “inner” optimization problem • Solution • Its properties (critical parameters) • Multiple channel uses (bosonic modes) • The “outer” optimization problem • Solution • Its properties (and applications) • Conclusions and outlook sabato 19 febbraio 2011

Gaussian channels They map Gaussian states into Gaussian states; for single use: { a, V } �→ { X T a + d, X T V X + Y } Channel defined by the triad: ( d, X, Y ) For n uses channel defined by a triad: � = ( ⊕ n d, ⊕ n X, ⊕ n Y ) memoryless ( d n , X n , Y n ) = � = ( ⊕ n d, ⊕ n X, ⊕ n Y ) memory sabato 19 febbraio 2011

The lossy channel X = √ η I, Y = (1 − η ) V env V env V in = η V in + (1 − η ) V env V out V mod = η ( V in + V mod ) + (1 − η ) V env V out V in = V in + V mod η � � The eigenvalues of the various matrices will be denoted by e u , i u , i u , m u , o u , o u sabato 19 febbraio 2011

Classical capacity and rates C n := 1 V in ,V mod χ G max n n n � � � � �� o k − 1 o k − 1 � χ G n := g − g 2 2 k =1 g ( x ) := ( x + 1) log( x + 1) − x log x Tr V in ≤ N in + 1 2 n 2 To the logarithmic approximation of g n C log = 1 log o k � max n V in ,V mod o k k =1 R hom = C log n n R het = C log n [ V env → V het env ] n sabato 19 febbraio 2011

Single channel use Theorem The max of Holevo function over Gaussian states is achieved for V in , V mod , V env simultaneously diagonalizable and the optimal V in corresponds to a pure state Corollary If V in , V mod , V env are simultaneously diagonalizable, the maximum of dyne rates is achieved by input pure states Covariance matrices parametrized as � � e s � � N + 1 0 Tr V ≤ N + 1 V = 0 e − s 2 2 2 sabato 19 febbraio 2011

The “inner” optimization problem χ G Maximize 1 With i u > 0 ( i u ⋆ = 1 / (4 i u )) m u , m u ⋆ ≥ 0 i u + 1 + m u + m u ⋆ = 2 N in + 1 4 i u Definition Solution belongs to the 1st stage if m u , m u* =0 are optimal Solution belongs to the 2nd stage if only m u =0 (or m u* ) is optimal Solution belongs to the 3rd stage if m u , m u* >0 are optimal Remark Stages are crossed (from 1st to 3rd) by increasing the input energy sabato 19 febbraio 2011

1st stage capacity equal to zero N in (1 → 2) = 0 2nd stage solution for i u of the transcendent equation � � 1 � � 1 � � � � o − 1 1 o − 1 1 o g ′ − o g ′ = 0 − − 4 i 2 2 o u o u ⋆ 2 o u u o u ⋆ �� � − 1 − η N in (2 → 3) = 1 � N env − e u + 1 � e u ⋆ − 1 e u 2 2 η 3rd stage � � − g ((1 − η ) N env ) C 1 = g η N in + (1 − η ) N env sabato 19 febbraio 2011

Properties of the solution Theorem: C 1 is a concave and increasing function of N in The one-shot capacity for fixed e u , e u* , can be considered as a η black-box returning C 1 upon inputting , while preserving N in the concavity � � N in − → C 1 = C 1 N in → C 1 − Corollary: C 1 is additive Theorem: C 1 is a monotonic function of all its parameters � � except s env η , N in , s env , N env sabato 19 febbraio 2011

Regimes C 1 C 1 η 0 η 0 η ⋆ η ⋆ η η ˜ η ˜ η s env Testo η ⋆ = 1 − 1 Critical parameters at boundaries of regimes, e.g. √ 3 sabato 19 febbraio 2011

Domains N env N in In the domain 1: ˜ η < η < η 0 < η ∗ In the domain 2: ˜ η < η < η ∗ < η 0 ∄ ˜ In the domain 3: η , η � √ ⋆ Critical parameters at boundaries of domains, e.g. 3 3+5 − 1 N in = √ 2 8 3 sabato 19 febbraio 2011

Multiple channel uses E E 2 E 1 E n Different single channel uses come from memory unravelling Lupo & Mancini, PRA 81, 052314 (2010) The action of E could be reduced to that of E 1 , E 1 ,..., E n by finding suitable Gaussian encoding/decoding unitaries k =1 X ( k ) ; D n Y n D T k =1 Y ( k ) ; E T (0 , E n , 0) , (0 , D n , 0) | D n X n E n = ⊕ n n = ⊕ n n E n = I n Always possible for E pure, or thermal squeezed! sabato 19 febbraio 2011

The “outer” optimization problem χ G To maximize it now suffices to consider: n → C (1) = C (1) → C (1) � � N in , 1 − N in , 1 − 1 1 1 → C (2) = C (2) → C (2) � � N in , 2 − N in , 2 − 1 1 1 . . . → C ( n ) = C ( n ) → C ( n ) � � N in ,n − N in ,n − 1 1 1 �� n � Find the distribution of N in ,k k =1 N in ,k = nN in k =1 C ( k ) � n giving the maximum of 1 This “outer” optimization problem can be interpreted as the search for the optimal distribution of modes across stages sabato 19 febbraio 2011

Algorithm ∂ C ( k ) 1 � � Due to the properties of C 1 it’s possible to def. λ max := max N in ,k = 0 < + ∞ ∂ N in ,k k ∂ C ( k ) ∂ C ( k ) � � � � λ 1 → 2 ( k ) = N in ,k (1 → 2) ; λ 2 → 3 ( k ) = N in ,k (2 → 3) 1 1 ∂ N in ,k ∂ N in ,k Testo κ � � � n Look for N in ,k k =1 N in ,k = nN in � Convex separable programming guarantees uniqueness and optimality of the solution together with convergence of the algorithm sabato 19 febbraio 2011

In the stage 1: N in ,k = 0 1 In the stage 2: N out ,k = e ω k /T − 1 N out ,k = o k − 1 / 2 , ω k = o k /o u,k , T = η / λ o k , o u,k can be expressed by means of N in ,k upon solving the “inner” problem � � N in ,k = 1 1 In the stage 3: e λ / η − 1 − (1 − η ) N env ,k η If all modes belong to the 3rd stage n − 1 � � � g ((1 − η ) N env ,k ) C n = g η N in + (1 − η ) N env n k =1 sabato 19 febbraio 2011

Quantum water filling � � e Ω s env � � N env + 1 0 V env = e − Ω s env 0 2 Ω i,j = δ i,j +1 + δ i,j − 1 o κ Test o κ Testo sabato 19 febbraio 2011

Super-additivity Memoryless Memory � � → C (1) = C (1) → C (1) N in − → C 1 = C 1 N in → C 1 � � N in , 1 − N in , 1 − − 1 1 1 � � → C (2) = C (2) → C (2) N in − → C 1 = C 1 N in → C 1 � � N in , 2 − N in , 2 − − 1 1 1 . . . . . . → C ( n ) = C ( n ) → C ( n ) � � � � N in − → C 1 = C 1 N in → C 1 N in ,n − N in ,n − − 1 1 1 For a fixed N env , sufficient condition to have n n � C ( k ) � � < nC 1 N in ,k = nN in � 1 � k =1 k =1 n ⋆ � is η < η ⋆ , N in ,k > nN in k =1 sabato 19 febbraio 2011

� � e Ω s env � � N env + 1 0 V env = e − Ω s env 0 2 Ω i,j = δ i,j +1 + δ i,j − 1 C Testo η = 0 . 9 η = 0 . 5 η = 0 . 1 N in = 1 Testo N env sabato 19 febbraio 2011

Conclusions and outlook • Optimization methods for capacity and rates • Full characterization of the single-mode lossy channel • Concavity (and then additivity) of the one-shot capacity • Full characterization of the multiple use lossy channel • Superadditivity for memory channel related to critical parameters • Application to other Gaussian channels [additive noise, J. Schafer et al. arXiv1011.4118] • Application to other capacities • Open questions: optimality of Gaussian input states; coding theorems for generic memory channels sabato 19 febbraio 2011