Evaluation of Error probability for Quantum. Gaussian Channels Using Gallager Functions

Transcription

1 ISSN Evaluation of Error probability for Quantum Gaussian Channels Using Gallager Functions Masaki Sohma Quantum Information Science Research Center, Quantum ICT Research Institute, Tamagawa University Tamagawa-gakuen, Machida, Tokyo , Japan Tamagawa University Quantum ICT Research Institute Bulletin, Vol.4, No.1, 11-14, 014 Tamagawa University Quantum ICT Research Institute 014 All rights reserved. No part of this publication may be reproduced in any form or by any means electrically, mechanically, by photocopying or otherwise, without prior permission of the copy right owner.

2 Tamagawa University Quantum ICT Research Institute Bulletin Vol.4 No.1 : 11_14 014) Evaluation of Error Probability for Quantum Gaussian Channels Using Gallager Functions 11 Masaki Sohma Quantum Information Science Research Center, Quantum ICT Research Institute, Tamagawa University Tamagawa-gakuen, Machida, Tokyo , Japan sohma@eng.tamagawa.ac.jp Abstract We review studies about Gallager functions and related quantities for quantum Gaussian channels. In addition we compute a Gallager function to evaluate error probability at transmission rates above channel capacity. I. INTRODUCTION This paper discusses bounds on error probability of quantum Gaussian channels, classical information is conveyed by quantum Gaussian states and a positive operator valued measure is used in the decoding procedure. Let R be the information rate defined by log M/n 1 when we transmit M messages with n use of the channel, that is, a block code of length n. Then the reliability function ER) shows the speed of exponential decay of the error probabilities P e at rates below the capacity: P e exp[ ner)]. Quantum coding theorems for the reliability function were established by Holevo and Burnashev [7], [3]. On the analogy from the classical case, they defined the random coding bound E r R) and the expurgated bound E ex R) and proved that these give lower bounds for reliability function ER) truly in the pure state case. Then Holevo proved the expurgated bound also holds in the mixed state case [4]. Moreover he extended these results to continuous channels with constrained inputs [5]. On the other hand Ogawa and Nagaoka derived a lower bound on the error probability at rates above capacity for a general classical-quantum channel [1]. In this paper, we summarize results about Gallager functions of one-mode quantum Gaussian channels. We introduce a quantum Gaussian channel and show formulas of Gallager functions for it. In addition we compute a lower bound on error probability at rates above capacity for one-mode quantum Gaussian channels with energy constraint on the basis of [1]. II. QUANTUM GAUSSIAN CHANNEL We consider quantum system, such as cavity field with a finite number of modes, described by operators q 1,p 1,...,q r,p r satisfying the Heisenberg CCR [q j,p k ]=iδ jk I, [q j,q k ]=0, [p j,p k ]=0. Let H be the Hilbert space of irreducible representation of CCR. For a real column r-vector z = 1 We use the natural logarithm throughout the paper. x 1,y 1,...,x r,y r ) T, we introduce a unitary operators in H as r V z) = exp i x j q j + y j p j ). j=1 Then the operators V z) satisfy the Weyl-Segal relation V z)v z )=e i z,z )/ V z + z ), 1) the skew symmetric form is given by r z,z )= x jy j x j y j)= z T r z, j=1 with the skew symmetric matrix r. Here V z) gives the representation of the CCR on the symplectic space R r, ). The density operator ρ is called Gaussian, if its quantum characteristic function has the form [ TrρV z) = exp im T z 1 ] zt Az, ) m is a r-dimensional column vector and A is a correlation matrix, which is a real symmetric matrix satisfying A i r 0. 3) In the following our concern is devoted to a single mode Gaussian state r =1) with a mean m =m q,m p ) T and a correlation matrix [ ] γ 0 Aλ, γ) =λ 0 γ,λ,γ >0. 4) Let us consider a classical-quantum one-mode Gaussian channel defined by a mapping Θ : R m ρ m, ρ m is a Gaussian state with a mean m and a correlation matrix 4). Note that ρ 0 describes background noise, comprising quantum noise and ρ m is obtained by applying the displacement operator to ρ 0. We relate codewords w =m 1,..., m n ) of length n to the density operators ρ w = ρ m1 ρ mn and assume the energy constraint as n fm j ) ne, j =1,..., M, 5) i=1 with a energy function fm j )=m q j ) +m p j ) )/.

3 1 We call by code of size M a sequence w 1,X 1 ),..., w M,X M ). Here w k are codewords of length n, satisfying the additive constraint 5) and {X k } is a family of positive operators in H n, satisfying M k=1 X k I. The average error probability for such a code is give by P {w j,x j )}) =1 1 M M Trρ w kx k. 6) k=1 We denote pn, M) the infimum of this error probability with respect to all codes of size M. III. GALLAGER FUNCTIONS A. Bounds for Error Probability We introduce some bounds for the error probability pn, M). According to [5], we have the following bounds: for ɛ>0 and sufficiently large n P e nr,n) exp[ ne r R) ɛ)], R < C, 7) P e nr,n) exp[ ne ex R) ɛ)], R < C. 8) Here C is the capacity of the channel, and E r R) = max µπ,s,p) sr), 0 s 1 0 p π P 1 9) E ex R) = max µπ, s, p) sr), 1 s 0 p π P 1 10) µ and µ are quantum Gallager functions given by µπ,s,p)= log Tr µπ,s,p)= slog e p[fm) E] ρ 1 m e p[fm)+fm ) E] Tr ρ m ρm ) 1 s πdm)πdm ), ) πdm) 11) 1) and P 1 is the set of Gaussian probability distributions π satisfying fm)πdm) < E. 13) B. Gallager Functions for Quantum Gaussian Channels The Gallager functions µπ, s, p) and µπ, s, p) with the a priori Gaussian distribution [ 1 πdm) = π det B exp 1 ] mt B 1 m dm, 16) takes the following forms µπ, s, p) = [ 1 + s) log deti pb) 1 N 1 A)e pe] log [N G 1 A)A +B 1 pi ) 1)], 17) µπ, s, p) =pse + s ] [I log det pb)i pb + sg 1 A)A) 1 B). s 18) Here I is a identity matrix and the functions N s, G s are given as N s A) = f s det A/ ) 1, 19) G s A) = g s det A/ ) I, 0) f s d) =d +1/) s d 1/) s, 1) g s d) = 1 d d +1/)s +d 1/) s d +1/) s d 1/) s, ) and abs ) is defined as follows: for a diagonalizable matrix M = T diagm j )T 1, we put absm = T diag m j )T 1. We compute the Gallager function µπ,s,p) in the single mode case; the computation of µπ,s,p) is summarized in [10]. First we consider the case ρ 0 is a Gaussian state with a correlation matrix A = Aλ, 1) = λi, λ /. 3) Here the optimum a priori Gaussian distribution has the correlation matrix B = diag[e,e]. Then the Gallager function is simplified as follows. In the present case, we have N 1 A) =f 1 λ/ ) 1 4) In addition, applying the Ogawa-Nagaoka lower bound [1] to the continuous case, we obtain for all n [9] P e nr,n) 1 exp ne on R)), R > C. 14) E on R) = max min µπ, s, 0) sr). 15) 1<s 0 π P 1 1/N G 1 A)A +B 1 pi ) 1 ) { = λ/ )g 1 λ/ )+ E 1 pe) + 1 } { λ/ )g 1 λ/ )+ E 1 pe) 1 }. Substituting these into 17), we obtain µπ, s, p) =1 + s) log f 1 λ/ )+p1 + s)e + log [ X s, p) Y s, p) ], 5) 6)

4 13 Xs, p) = λ/ )g 1 λ/ )+1/)1 pe)+e/, Y s, p) = λ/ )g 1 λ/ ) 1/)1 pe)+e/. 7) For a more general correlation matrix 4), the optimum a priori Gaussian distribution has the correlation matrix of the form B = diag[e 1,E ]. Then the function µπ,s,p) has a complicated form but it is simplified when p =0 as µπ, s, 0) =1 + s) log f 1 λ/ ) + log f D/ ), 8) D = detg 1 A)A + B) = λ g 1 λ/ ) +γ E + γ E 1 )λg 1 λ/ ) + E 1 E. 9) C. Optimizations Firstly we review results about the random coding bound in the case ρ 0 is a Gaussian state with correlation matrix 3) [5]. Trying to maximize µπ,s,p) with respect to p we obtain the equation 1/ pλ/ )g 1 λ/ ))Xs, p)s Y s, p) s ) = pxs, p) s + Y s, p) s ). 30) Explicit solutions for this equation are only known for s =0, 1. Thus, contrary to the classical case [11], the maximum in 9) in general has not been found only numerically. Next we compute E on R) in the case ρ 0 is a Gaussian state with a correlation matrix 4). Let us find the optimum a priori distribution π. Putting E 1 = x and E =E x 0 x E), we rewrite D in Eq. 8) as [ γ γ D = x λg 1 λ/ )+E )] [ γ + γ ] + λg 1 λ/ )+E. 31) Since f d) 1 <s<0) is a monotonously decreasing function, the optimum a priori distribution π opt is given by putting x =γ γ )λg 1/ λ/ ) +E. Then we have [ λ µπ opt, s, 0) = log log E + γ + γ ) 1 λ 1 ) 1 ] λg 1 λ/ ) + 1 ) E + γ + γ λg 1 λ/ ) ) 1. 3) Restricting ourselves to the pure state case λ = /), we can simplify this equation as with and hence we have with µπ opt, s, 0) = logµ 1 µ ) 33) µ 1 = E + γ + γ + 1 4, µ = E + γ + γ 1 34) 4, µπ opt, s, 0) sr = R + logls)), 35) Ls) = µ1 e R ) µ e R ). 36) Differentiating Ls) with respect to s, L µ1 ) µ1 ) µ ) µ ) s) = log e R e R log e R e R. 37) Putting L s) =0, we get the optimum value of s as s = 1 log log µ R log µ 1 log µ log µ 1 R 1 =: s optr). Thus we obtain E on R) =µπ opt,s opt R), 0) s opt R)R, 38) for a one-mode quantum Gaussian channel with correlation matrix 4). Fig. 1 shows graphs of s opt R) with respect to R for squeezing parameters γ = 1, 3, 5, 10. Fig. gives graphs of E on R) with respect to R for γ =1, 3, 5, 10. IV. CONCLUSION We have reviewed results about random coding bound for quantum Gaussian channels and computed a lower bound on error probability at transmission rates above capacity on the basis of [1].

5 14 Channel Coding Theorem, IEEE Trans. on Information Theory, vol. 45, no. 7, , 1999 Fig. 1. Dependence of optimum values of s on the transmission rates R [ebits/symbol] for squeezing parameters γ =1, 3, 5, 10. Fig.. Error exponents E onr), which gives a lower bound of error probability as P e nr,n) 1 exp ne on R)), for squeezing parameters γ =1, 3, 5, 10. REFERENCES [1] A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North Holland, 198 [] A.S. Holevo, M. Sohma, O. Hirota, Capacity of quantum Gaussian channels, vol.59, No. 3, 1999 [3] A. S. Holevo, Coding theorems for quantum channels, Tamagawa University Research Review, 4, 1, 1998 [4] A. S. Holevo, Reliability function of general classical-quantum channel, IEEE Trans. on Information Theory, 466), 56-60, 000 [5] A. S. Holevo and M. Sohma and O. Hirota, Error exponents for quantum channels with constrained inputs,reports on Mathematical Physics, 46, 3, , 000 [6] A. S. Holevo and M. Sohma and O. Hirota,Capacity of quantum Gaussian channels,phys. Rev. A, 59, , 1999 [7] M. V. Burnashev and A. S. Holevo, On reliability function of quantum communication channel, Probl. Peredachi Inform., 34, 1-13, 1998 [8] M. Sohma and O. Hirota, Squeezing is good at low information rates.,physical Review A, 65, 0319, 00 [9] M. Sohma and O. Hirota, The Gallager functions and related quantities for quantum Gaussian channel, Proc. of IEEE Int. Symp. on Information Theory, 54, 001 [10] M. Sohma, Transmission of classical information with squeezed states,tamagawa University Quantum ICT Research Institute Bulletin Vol.3 No.1, 013 [11] R. G. Gallager, Information theory and reliable communication,j. Wiley, New York, 1968 [1] T. Ogawa and H. Nagaoka, Strong Converse to the Quantum