Transmission of Classical Information through Gaussian Quantum - PowerPoint PPT Presentation

Transmission of Classical Information through Gaussian Quantum Channels with Memory Oleg V. Pilyavets Advisor: Prof. S. Mancini December 16, 2009

Outline I. Introduction • Basic definitions from quantum channels theory • Definition of lossy bosonic memory channel (LBMC) • Quantum channel capacity: definition II. Lossy bosonic memory channel: capacity • Uncertainty relation and memory model • Calculating of capacity • Purity theorems • Channel capacity for one use • Channel capacity for n uses III. Lossy bosonic memory channel: example with Ω-model of memory • Definition of Ω-model • Third stage case • General case IV. Lossy bosonic memory channel: achievable rates • Definition of achievable rate • Homodyne and heterodyne rates Concluding remarks and summary of results List of publications Oleg V. Pilyavets Information Transmission in Quantum Channels

I. Basic definitions from quantum channels theory Quantum channel T is a (trace-preserving) quantum map : i.e. quantum state − → quantum state, which is completely positive : ✬ ✩ i.e. T ⊗ Id is also quantum map. channel ρ out = T ( ρ in ) ρ in ✲ environment ✫ ✪ If we encode information into quantum state in input, we can decode that information by measuring quantum state at the output of channel. This is classical information transmission by quantum channels: { ρ x } continuous ρ ′ { x ∈ R } − − − − − − − → { π y } alphabet = ⇒ x = ⇒ input states output states measurement channel The aim at decoding (measurement) is to distinguish states with different x . The channel which acts independently on each use ( T = T ⊗ n ) is called me- moryless . Otherwise, we call it memory channel . Oleg V. Pilyavets Information Transmission in Quantum Channels

I. Basic definitions from quantum channels theory Achievable rate of information transmission is the speed at which information can be reliably transferred through channel for fixed encoding and decoding. Maximal achievable rate of information transmission (considering all possible types of encoding and decoding) over channel is called capacity . We take electromagnetic fields E and H (“field quadratures”) as our continuous variables, therefore our channel is called bosonic channel (as we work with bosonic field modes). Achievable rates and capacity for bosonic channel are finite only if there is an energy restriction at channel input. Gaussian channels are those continuous channels which map Gaussian states into Gaussian states (quantum state is Gaussian if, e.g., its representation is given by Gaussian distribution for some variables). Oleg V. Pilyavets Information Transmission in Quantum Channels

I. Definition of LBMC Stinespring’s dilation theorem allows quantum channel to be modelled as E ( ρ ) = Tr E [ U ( ρ ⊗ ρ E ) U † ] Example — (Gaussian) lossy bosonic quantum channel , where losses are introduced by interaction with extra “environment modes” on beam-splitter with transmissivity η : Oleg V. Pilyavets Information Transmission in Quantum Channels

I. Definition of LBMC Schematic representation of multimode ( multiuse ) lossy bosonic channel ( LBC ), where one can introduce memory — correlations between channel uses: ENVIRONMENT OUTPUT INPUT Any Gaussian quantum state ρ can be completely described by covariance matrix V for quadratures x := ( q 1 , . . . , q n , p 1 , . . . , p n ) entering in its Wigner function : � �� � 1 − 1 x − α , V − 1 ( x − α ) ρ ← → W ( x ) = √ exp 2 det V LBC can be reduced to the following relation between covariance matrices for input, environment and output states: V out = η V in + (1 − η ) V env Oleg V. Pilyavets Information Transmission in Quantum Channels

I. Quantum channel capacity: definition Classical symbols to encode are distrubuted as Gaussian with covariance matrix V cl / 2: � � �� 1 α , V − 1 P ( α ) = π n √ det V cl exp − cl α thus, averaged output channel state is V out = η ( V in + V cl ) + (1 − η ) V env n -uses channel capacity C n can be estimated by its Holevo bound χ n maximized over all possible encodings and decodings, as Holevo coding theorem [J.P. Gordon (1964), A.S. Holevo (1973)] states ( S is von Neumann entropy): �� � � � � χ n ρ ( α ) ρ ( α ) C n � max n , χ n = S out P ( α ) d α − S P ( α ) d α out ρ in ,ρ cl Below we conjecture achievability of maximum on the set of Gaussian states and call (for simplicity) this maximum as capacity: χ n C n ≡ max n V in , V cl Oleg V. Pilyavets Information Transmission in Quantum Channels

II. LBMC: uncertainty relation and memory model Quantum state must satisfy HUR: 2 n × 2 n covariance matrix V is admissable iff V + i Ω � 0 where Ω is commutation matrix for canonical variables (quadratures): � 0 n � Id n Ω = − Id n 0 n Ω is called symplectic form . Quantum mechanics makes phase space geometry to be symplectic . ⇒ HUR can be rewritten in terms of symplectic eigenvalues ν k : ν k � 1 / 2 Definition: ν k = ν k ( V ), k = 1 , . . . , n are symplectic eigs of � V qq � � � − V ⊤ V qp − V pp if ± i ν k are eigs of � V = Ω − 1 V = qp V = V ⊤ V pp V qq V qp qp Von Neumann entropy of Gaussian state is function of its symplectic eigs: � � n � ν k − 1 S ( ρ ) = g , where g ( v ) = ( v + 1) log 2 ( v + 1) − v log 2 v 2 k =1 We restrict a class of environment models to study by matrices: � V ( qq ) � 0 where V ( qq ) and V ( pp ) commute V env = , V ( pp ) 0 = ⇒ The problem becomes spectral (all matrices can be taken in diagonal form). Oleg V. Pilyavets Information Transmission in Quantum Channels

II. LBMC: calculating of capacity Thus, mathematical problem we need to solve : � � � � �� n � 1 ν k − 1 ν k − 1 find C = lim n →∞ C n where C n = max g − g n 2 2 i uk , i u ⋆ k k =1 c uk , c u ⋆ k � � with symplectic eigs ν k = ν k ( V out ) and ν k = ν k V out : ν k = √ o qk o pk o uk = η i uk + (1 − η ) e uk ν k = √ a qk a pk a uk = η ( i uk + c uk ) + (1 − η ) e uk Eigs of matrices [ u ∈ { q , p } ; if u = q ⇒ u ⋆ = p ; if u = p ⇒ u ⋆ = q ; k = 1 , ..., n ]: • i uk ↔ V in — input seed state (we encode information in it) • e uk ↔ V env — environment state • c uk / 2 ↔ V cl / 2 — distribution of encoded variable α (“modulation of signal”) • o uk ↔ V out — output state • a uk ↔ V out — output state averaged over encoding (over modulation) The maximum above is taken for fixed e uk , e u ⋆ k , η, N and is constrained by: n � � 2 n Tr ( V in + V cl ) = 1 1 [ i uk + c uk ] = N + 1 • Energy restriction: 2 n 2 k =1 u ∈{ q , p } ν k ( V in ) � 1 i uk > 0 , i uk i u ⋆ k � 1 • HUR: = ⇒ • positivity: c uk � 0 2 4 Oleg V. Pilyavets Information Transmission in Quantum Channels

II. LBMC: purity theorems (purity of V in ) What can be proved without solving above maximization problem: Suppose, we consider LBMC with all covariance matrices to be diagonal. Then: Theorem: Maximum of Holevo bound is always achieved on pure state V in : i uk i u ⋆ k = 1 / 4 [J. Sch¨ afer, D. Daems, E. Karpov, N. J. Cerf, PRA 80, 062313 (2009)] = ⇒ i u ⋆ k is already found , i.e. actual restrictions: c uk , c u ⋆ k � 0 and i uk > 0. Thus, we can use Lagrange multipliers method to find maximum of χ n . Proof: ⊐ i uk i u ⋆ k > 1 / 4 ⇒ i u ⋆ k = i ′ u ⋆ k + δ k , where i ′ u ⋆ k = 1 / (4 i uk ), δ k > 0. Let us change varibales to make V in pure (it preserves energy constraint N ): i ′ i ′ u ⋆ k = i u ⋆ k − δ k uk = i uk c ′ c ′ u ⋆ k = c u ⋆ k + δ k uk = c uk ⇒ ν ′ k = ν , ν ′ k < ν k . Because of g 0 is monotonically growing function of its argument C k ( i ′ uk , i ′ u ⋆ k , c ′ uk , c ′ u ⋆ k ) > C k ( i uk , i u ⋆ k , c uk , c u ⋆ k ) � Oleg V. Pilyavets Information Transmission in Quantum Channels

II. LBMC: channel capacity for one use We will see that Lagrange multipliers method always results in i uk > 0 ⇒ we have to satisfy only to c uk , c u ⋆ k � 0. Below there is a way to find such solution. There are 3 possibilities for 1-use (1-mode) channel depending on energy restriction N and threshold value �� e u � thr ( e u > e u ⋆ ) = 1 − 1 − 1 − η N 2 → 3 0 < N 2 → 3 ( e u ⋆ − e u ) , < ∞ thr 2 e u ⋆ η Both c u , c u ⋆ > 0 — 3rd stage — what holds if N > N 2 → 3 ⇒ capacity can be thr found in enclosed form: C = g [ η N + (1 − η ) M env ] − g [(1 − η ) N env ] Case of c u = 0 , c u ⋆ > 0 — 2nd stage — what holds if N � N 2 → 3 ⇒ capacity thr depends on solution of one transcendent equation for i u . c u = 0 , c u ⋆ = 0 — 1st stage — what holds only if N = 0 (channel is not used for information transmission: C = 0). = ⇒ 1 use capacity for fixed values of e u , e u ⋆ and η can be mentioned as a (concave) function: N − → C = C ( N ) − → C Oleg V. Pilyavets Information Transmission in Quantum Channels