Gaussian Bosonic Channels: conjectures, proofs, and bounds

Transcription

1 IICQI14 Gaussian Bosonic Channels: conjectures, proofs, and bounds Vittorio Giovannetti NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR A. S. Holevo (Steklov, Moskov) R. Garcia-Patron (University Libre Bruxelles) N. J. Cerf (University Libre Bruxelles) S. LLoyd (MIT, Boston) A. Mari (SNS, Pisa) G. De Palma (SNS, Pisa) CM condensed matterand quantumin formation

2 THE RESULT IN A NUTSHELL Pares cum paribus facillime congregantur Cicero, De Senectute CERTAIN FUNCTIONALS* EVALUATED AT THE OUTPUT OF A BOSONIC GAUSSIAN CHANNEL (BGC) ARE OPTIMIZED (SAY MINIMIZED) BY GAUSSIAN INPUT STATES min F( ( )) = min G F( ( G )) *VON NEUMANN ENTROPY RENYI ENTROPIES CONCAVE FUNCTIONALS HOLEVO INFORMATION

3 Outlook 1. Sending classical messages over a quantum channel 2. Bosonic Gaussian Channels (BGCs) 3. The Conjectures 4. Solutions 5. Conclusions and Perspectives

4 1.Sending classical messages over a quantum channel

5 CLASSICAL COMMUNICATION LINE ALICE (the sender) x X A p(y x) NOISY CHANNEL y Y A BOB (the receiver) CLASSICAL INFO SOURCE CLASSICAL OUTPUT

6 CLASSICAL COMMUNICATION LINE ALICE (the sender) CHANNEL USES x 1 x 2 x p(y x) y m M C D m M y 1 y 2 BOB (the receiver) ENCODING x N TRANSFERRING y N DECODING RATE = R = #Bits #channel uses = log 2 M N

7 C = max achievable R = lim 0 lim sup N log2 M N C M,N such that P err (C) < Shannon NOISY CHANNEL CODING THEOREM C = max p(x) H(X : Y ) single letter formula... no regularization needed over N H(X : Y )=H(X)+H(Y ) H(X, Y ) MUTUAL INFORMATION of X, Y H(X) = x p(x) log 2 p(x)

8 Sending classical messages on a Quantum Channel INPUT/OUPUT FORMALISM Linear, Completely Positive, Trace Preserving (LCPT) mapping output state of the carrier = ( ) input state of the carrier

9 As in the classical theory we can define the CAPACITY of the Channels as: C = max achievable R = lim 0 lim sup N log2 M N C M,N such that P err (C) < SEPARABLE ENCODING ENTANGLED ENCODING 0000 C 1 ( ) C( ) ( )/ 2

10 Holevo-Schumacher-Westmoreland (HSW) CHANNEL CODING THEOREM (I) if we restrict the ENCODING to only those which produce SEPARABLE (non entangled) CODEWORDS, then C 1 ( ) max ENS C ( (ENS)) = HOLEVO CAPACITY OF THE CHANNEL MAXIMIZED OVER ALL POSSIBLE ENSEMBLES HOLEVO IEEE 44, 269 (1998) SCHUMACHER and WESTMORELAND PRA 56, 2629 (1998) ENS = { j ; p j : j 2 S(H)} = ensemble of input states of the information carrier (ENS) ={ ( j ); p j : j 2 S(H)} = output ensemble associated with ENS C (ENS) =S( X j p j j ) X p j S( j ) j = HOLEVO INFO of the ensemble ENS

11 Holevo-Schumacher-Westmoreland (HSW) CHANNEL CODING THEOREM (II) if we allows for ANY ENCODING including those which produce ENTANGLED CODEWORDS, then C( )= lim N!1 C 1 ( N ) N REGULARIZATION OVER CHANNEL USES C 1 ( N ) = max ENS C ( N (ENS)) = HOLEVO CAPACITY OF THE CHANNEL N MAXIMIZED OVER ALL POSSIBLE N-dim ENSEMBLES

12 C( ) C 1 ( )

13 C( ) C 1 ( ) ADDITIVITY ISSUE: C is no longer a single expression formula (we have to take the limit over arbitrarily large N).? C( )=C 1 ( ) ADDITIVITY problem of Holevo INFO

14 C( ) C 1 ( ) ADDITIVITY ISSUE: C is no longer a single expression formula (we have to take the limit over arbitrarily large N).? C( )=C 1 ( ) ADDITIVITY problem of Holevo INFO SHOR EQUIVALENCE THEOREM (2004): (i) additivity of the minimum entropy output of a quantum channel (ii) additivity of entanglement of formation (iii) strong super-additivity of the entanglement of formation

15 C( ) C 1 ( ) ADDITIVITY ISSUE: C is no longer a single expression formula (we have to take the limit over arbitrarily large N).? C( )=C 1 ( ) FALSE ADDITIVITY problem of Holevo INFO SHOR EQUIVALENCE THEOREM (2004): (i) additivity of the minimum entropy output of a quantum channel FALSE (ii) additivity of entanglement of formation FALSE (iii) strong super-additivity of the entanglement of formation FALSE Hastings, Nature Physics 5, 255 (2008)

16 C( )= lim N!1 C 1 ( N ) N still, for some special channels it may be the case that the additivity holds...

17 2. Bosonic Gaussian Channels (BGCs)

18 Bosonic Gaussian Channels (BGCs) ALICE BOB h i a j, a k = jk each INPUT SYSTEM is a collection of (say) s independent optical modes annihilation operator of the j-th mode D(z) =exp[a z z a]=exp P s j=1 z j a j z j a j z =(z 1,z 2,,z s ) t displacement (or Weyl) operators a =[a 1,...,a s ] t = 1 s Z d 2s z (z)d( z), (z) =Tr[ D(z)] Symmetrically Ordered Characteristic Function

19 Bosonic Gaussian Channels (BGCs) A state is a Gaussian state iff (z) is a Gaussian function Vacuum state, Coherent states, Squeezed states, Thermal states. A LCPT map is a BGC if it sends Gaussian input states into output Gaussian states ( ) Holevo, Werner PRA 63, 1997 Attenuation (loss), Amplification, Squeezing, Thermalization processes ALICE ALICE Caves, Drummond RMP 1994 BOB BOB

20 Bosonic Gaussian Channels (BGCs) for phase-covariant channels (multimode channels) (z) =Tr[ D(z)]! BOSONIC GAUSSIAN CHANNEL! 0 (z) =Tr[ ( )D(z)] = (K z)exp[ z µz] µ ± 1 2 I KK

21 Bosonic Gaussian Channels (BGCs) Attenuator (or thermal) single mode channel (s=1) 0 (z) = ( p z) e (1 )(N+1/2) z 2 b Thermal reservoir with mean photon number N 0 Beam Splitter transformation ALICE a BOB 2 [0, 1] N=0 E N ( ) =Tr E [U( E )U ] E 0 ( ) =Tr E [U( ØihØ )U ] purely lossy channel (minimal noise attenuator)

22 Bosonic Gaussian Channels (BGCs) Amplifier channel single mode channel (s=1) 0 (z) = ( p applez) e (apple 1)(N+1/2) z 2 Thermal reservoir with mean photon number N 0 b Parametric Amplifier of gain apple 1 ALICE a BOB A N apple ( ) =Tr E [U( E )U ]

23 Bosonic Gaussian Channels (BGCs) Thermal reservoir with mean photon number N 0 ALICE a b Parametric Amplifier of gain apple 1 BOB Ã N apple ( ) =Tr S [U( E )U ] weak complementary of an amplifier channel 0 (z) = ( p apple 1z ) e apple(n+1/2) z 2 THIS IS AN ENTANGLEMENT BREAKING CHANNEL: we can always represent it as a measure and re-prepare channel

24 Bosonic Gaussian Channels (BGCs) ADDITIVE CLASSICAL NOISE CHANNEL random diffusion in phase space 0 (z) = (z) e n z 2 N n ( ) = Z d 2 µp n (µ) D(µ) D (µ)

25 Bosonic Gaussian Channels (BGCs) This set of maps is closed under channel c o n c a t e n a t i o n (semigroup structure) VACUUM VACUUM = apple [ ih ] =(A 0 apple E 0 )[ ih ] VG et al PRA 2004 Garcia-Patron et al. PRL 2012 lossy channel E 0 minimal noise amplifier A 0 apple

26 Bosonic Gaussian Channels (BGCs) CLASSICAL CAPACITY PROBLEM: how much CLASSICAL information can we transfer over these channels? Holevo, Schumacher, Westmoreland (HSW) theorem C( )= lim m!1 C ( )=sup ENS 8 < : S( ( ENS)) 1 m C ( m ) 9 X = p j S( [ j ]) ; j regularization over channel uses (Hastings 2008) ENS = {p j, j } Input energy constraint Tr[a j a j ] apple E maximum mean energy per channel use

27 3. The Conjectures

28 The Conjectures Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles PROVED FOR N=0 (purely lossy channel) VG et al. PRL 2004 Holevo, Werner PRA 63, E N C(E N ; E) =g( E +(1 )N) g((1 )N) C( ; E) 3 2 A N! 1 g(x) =(x + 1) log 2 (x + 1) x log 2 x a)

29 The Conjectures shortcut shortcut PROVED FOR N=0 (purely lossy channel) GIOVANNETTI,GUHA,LLOYD, MACCONE, SHAPIRO,YUEN, PRL 2004 C(E 0 )= lim m!1 sup ENS 6 lim m!1 [S max([e 0 ] m ) S min ([E 0 ] m )]/m 6 S max (E 0 ) [S([E 0 ] m ( ENS )) X p j S([E 0 ] m [ j ])]/m j THERE EXISTS AN ENSEMBLE THAT SATURATES THE BOUND!!!! ENS = coherent states {p( ), i} E 0 ( ih ) = p ih p S(E 0 ( ih )) = 0 gaussian ensemble

30 =) The Conjectures Minimum Output Entropy Conjecture VG et al The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) C( )= lim shortcut m!1 sup ENS [S( m ( ENS )) X p j S( m [ j ])]/m j 6 lim m!1 [S max( m ) S min ( m )]/m Optimal Gaussian ensemble Conjecture If MOE is true then the upper bound coincides with the value attainable by Gaussian encoding The maximization of C can be performed over the set of Gaussian ensembles Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) good luck... CLASSICAL CAPACITY QIT-EQIS ERATO Conference (2003)

31 =) The Conjectures Minimum Output Entropy Conjecture VG et al. PRA 2004 The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) CLASSICAL CAPACITY

32 =) =) The Conjectures Minimum Output Renyi, Weyl Entropy Conjecture =) Majorization Conjecture VG et al. PRA 2004 The output of coherent states majorize all other output states Coherent states minimize the ouput Renyi Entropies Entanglement of Formation Formula for two-mode Gaussian states =) Minimum Output Entropy Conjecture VG et al. PRA 2004 The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) CLASSICAL CAPACITY

33 =) =) =) The Conjectures Minimum Output Renyi, Weyl Entropy Conjecture Coherent states minimize the ouput Renyi Entropies Entanglement of Formation Formula for two-mode Gaussian states =) =) Majorization Conjecture VG et al. PRA 2004 The output of coherent states majorize all other output states Minimum Output Entropy Conjecture VG et al. PRA 2004 The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) =) Entropy Photon-Number Inequality Conjecture Guha, Shapiro 2005 Entropy Power Inequality Conjecture Smith, Koenig 2012 Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles bounds Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) CLASSICAL CAPACITY

34 =) =) =) Minimum Output Renyi, Weyl Entropy Conjecture Coherent states minimize the ouput Renyi Entropies Entanglement of Formation Formula for two-mode Gaussian states The Conjectures =) =) Majorization Conjecture VG et al. PRA 2004 The output of coherent states majorize all other output states Minimum Output Entropy Conjecture VG et al. PRA 2004 The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) =) december 2013 Entropy Photon-Number Inequality Conjecture Guha, Shapiro 2005 Entropy Power Inequality Conjecture Smith, Koenig 2012 Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles bounds Gaussian Additivity Conjecture The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) CLASSICAL CAPACITY

35 =) =) =) The Conjectures Minimum Output Renyi, Weyl Entropy Conjecture Coherent states minimize the ouput Renyi Entropies Entanglement of Formation Formula for two-mode Gaussian states Giovannetti, Garcia-Patron, Cerf, Holevo arxiv: Nature Photonics =) =) Mari, Giovannetti, Holevo arxiv: Nature Communication Majorization Conjecture VG et al. PRA 2004 The output of coherent states majorize all other output states Minimum Output Entropy Conjecture The Von Neumann Entropy at the output of the channel is minimized by coherent input states Giovannetti, Holevo, Garcia- (say the vacuum) Patron arxiv: Comm. Math. Phys. Optimal Gaussian ensemble Conjecture The maximization of C can be performed over the set of Gaussian ensembles Gaussian Additivity Conjecture VG et al. PRA 2004 The output Holevo information is additive (i.e. no regularization over is required) Holevo, Werner PRA 63, 1997 =) =) NOW Entropy Photon-Number Inequality Conjecture Guha, Shapiro 2005 Entropy Power Inequality Conjecture Smith, Koenig 2012 De Palma, Mari, Giovannetti, arxiv: Nature Photonics bounds CLASSICAL CAPACITY

36 The Conjectures Minimum Output Entropy Conjecture The Von Neumann Entropy at the output of the channel is minimized by coherent input states (say the vacuum) Thermal reservoir with mean photon number N INPUT SIGNAL Beam Splitter OUTPUT SIGNAL VG, Guha, Lloyd, Maccone, Shapiro, PRA 2004

37 The Conjectures VG, Guha, whence, Lloyd, Maccone, using (10), Eq. (13) gives (5), while Eq Shapiro, PRA and 2004 (15) respectively give the further bounds a (2) (5) (17) c (16) (4) b (6) (28) (3) (1) (27) (3) of Ref.4 (9) of Ref.5 (8) of Ref.5 m = 1 can be generalized to arbitrary m as follow Minimum Output S min (Nn m )/m > g(n 1), [8n > 1] Entropy Conjecture S min (Nn m )/m > log 2 (2n + 1), S min (Nn m )/m > 1 + log 2 (n), C(N n ) 6 g( N + n) log 2 (2n + 1), C(N n ) 6 g( N + n) 1 log 2 (n), Entropy Power Inequality Conjecture Smith, Koenig 2012 [the generalization of the bound c of [7] is not re here since it converges to Eq. (14) for m!1]. bounds are compared to the lower bound (2) in Fig note how the gap between the upper and lower b closes asymptotically for high noise, n!1. The proof of Eq. (13), and hence of the bound (5 given in Ref. [22] by expanding a generic input s (24) in terms of its multi-mode Husimi distribution fu and VG, Lloyd, applying Maccone, the concavity Shapiro of von Neumann entrop alternative Nat Phot proof follows from inequality a of Ref. [ from (11), using the fact that the channel N n is glement breaking for n > 1 [20]. The proof of Eq. (14) exploits the fact that th Neumann entropy is never smaller than the Rén Electromagnetic tropy of order channel 2 [24, capacity 25] i.e. S( ) > S 2 ( ) := log 2 for practical Thus, for purposes all input density matrices of m chann we have S(Nn m ( )) > S 2 (Nn m ( )) > m log 2 (2n + 1), where the last inequality follows from the fact th minimum Rényi entropy of integer order at the out the channel N is additive and saturated by the v

38 Solution Giovannetti, Holevo, Garcia-Patron arxiv: COMM. MATH. PHYS. The solution in 5 (simple) STEPS

39 STEP I Solution Øi VACUUM Giovannetti, Holevo, Garcia-Patron arxiv: COMM. MATH. PHYS. Øi VACUUM i i = apple [ ih ] =(A 0 apple E 0 )[ ih ] SINCE VACUUM GOES TO THE VACUUM UNDER PURELY LOSSY CHANNEL, PROVING MOE FOR THE AMPLIFIER E 0 [ ØihØ ] = ØihØ Øi VACUUM i apple

40 Solution Giovannetti, Holevo, Garcia-Patron arxiv: COMM. MATH. PHYS. Øi VACUUM i apple A 0 apple[ ih ] THIS IS A PROPER STINESPRING REPRESENTATION FOR THE CHANNEL: there for pure inputs we have à 0 apple[ ih ] STEP II S(Ã0 apple[ ih ]) = S(A 0 apple[ ih ]) STEP III... BUT now we can use once more the LOSSY+minimal NOISE AMPLIFIER decomposition to express à 0 apple[ ih ] =T A 0 apple E 0 0 [ ih ] PHASE CONJUGATION IT DOESN T CHANGE THE SPECTRUM... hence the entropy: WE CAN NEGLET IT!

41 Solution Giovannetti, Holevo, Garcia-Patron arxiv: COMM. MATH. PHYS. Øi VACUUM i apple A 0 apple[ ih ] THIS IS A PROPER STINESPRING REPRESENTATION FOR THE CHANNEL: there for pure inputs we have à 0 apple[ ih ] STEP II S(Ã0 apple[ ih ]) = S(A 0 apple[ ih ]) STEP III... BUT now we can use once more the LOSSY+minimal NOISE AMPLIFIER decomposition to express à 0 apple[ ih ] =T A 0 apple E 0 0 [ ih ] PHASE CONJUGATION IT DOESN T CHANGE THE SPECTRUM... hence the entropy: WE CAN NEGLET IT! LUCKY STRIKE SAME GAIN PARAMETER!! BINGO!!!!

42 Solution Giovannetti, Holevo, Garcia-Patron arxiv: COMM. MATH. PHYS. Step IV E 0 0 ( ih ) = X j p j jih j S(A 0 apple[ ih ]) = S(Ã0 apple[ ih ]) = S(A 0 apple E 0 0 [ ih ]) = S(A 0 apple( X p j jih j ) j X j p j S(A 0 apple[ jih j ]) S(A 0 apple[ ih ]) X j p j S(A 0 k[ jih j ])

43 Solution Giovannetti, Holevo, Garcia-Patron arxiv: Step V ITERATE THE ARGUMENT q times X S(A 0 apple[ ih ]) p j S(A 0 apple[ jih j ]) j X p j jih j =[E 0 ] q ( ih ) 0 j lim [E 0 ] q [ ih ] = ØihØ q!1 0 THE PURELY LOSSY CHANNEL IS MIXING: ITERATING IT MANY TIMES IT BRINGS ALL INPUT STATES TO THE VACUUM... S(A 0 apple[ ih ]) S(A 0 apple[ ØihØ ]) * needs to enforce continuity condition (use the mean energy constraint) * QED

44 Solution Mari, Giovannetti, Holevo arxiv: NATURE COMMUNICATION MAJORIZATION Input Gaussian channel Output

45 Solution Giovannetti, Holevo, Mari arxiv: a consequence: generalization of the Lieb, Solovej inequality Z f(p (z)) d2s z s Z f(p ih (z)) d2s z s p (z) =Tr[ D(z) 0 D (z)] 0 = ØihØ TAKING THIS IS THE HUSIMI DISTRIBUTION f(x) CONCAVE PHASE INVARIANT GAUSSIAN STATE Lieb and Solovej (2012) Lieb (1978)

46 Solution Giovannetti, Garcia-Patron, Cerf, Holevo arxiv: Nature Photonics Øi VACUUM N G apple (k, N) =U k ( N G 0ih0 )U k EoF( (k, N)) apple EoF( (0,N)) = g(k 1) Giedke, Wolf, Kruger, Werner, Cirac PRL 2003 EoF( (k, N)) = X inf p j S(A k ( j ih j )) p j, ji j S(A k ( 0ih0 )) = EoF( (0,N)) Matsumoto, Shimono, Winter, CMP 2004

47 Solution DePalma, Mari, Giovannetti, arxiv: Nature Photonics Entropy Power Inequality Conjecture Smith, Koenig 2012 Given SA and SB input entropies of the device the following inequality holds A B Beam Splitter OUTPUT SIGNAL S( A )=S A S( B )=S B e S C e S A +(1 )e S B PROVED FOR =1/2 IN Smith, Koenig 2012 PROOF EXTENDED FOR ALL AND GENERALIZED TO AMPLIFIER CHANNEL TO e S C applee S A +(apple 1)e S B

48 Solution DePalma, Mari, Lloyd,Giovannetti, arxiv: (multimode) R 2 R 1... scattering region R Y R K exp 1 n S Y X K =1 det M 1 n exp 1 n S

49 Conclusions and Perspectives A BUNCH OF CONJECTURES ON CV SYSTEMS HAVE BEEN RECENTLY SOLVED. THE STRONGEST OF THEM IS THE MAJORIZATION CONJECTURE, e.g. - STRONG CONVERSE FOR GAUSSIAN CHANNELS BARDHAN, GARCIA-PATRON, WILDE, WINTER arxiv: CLASSICAL CAPACITY OF MEMORY GAUSSIAN BOSONIC CHANNELS DEPALMA, MARI, GIOVANNETTI arxiv: OPEN QUESTIONS: i. STILL ON THE CHASE for the ENTROPY PHOTON NUMBER INEQUALITY CONJECTURE... B N C N A +(1 )N B A Beam Splitter OUTPUT SIGNAL photon numbers associated with the Guassian state which have the SAME entropy of the state ii. CONTINUITY... iii. CAPACITY FORMULAS FOR NON PHASE-INVARIANT CHANNELS

50 A solution of the Gaussian optimizer conjecture V. Giovannetti, A. S. Holevo, R. Garcia-Patron arxiv: to appear in Comm Math Phys Quantum state majorization at the output of bosonic Gaussian channels Andrea Mari, Vittorio Giovannetti, Alexander S. Holevo arxiv: Nature Communication Majorization and additivity for multimode bosonic Gaussian channels Vittorio Giovannetti, Alexander S. Holevo, Andrea Mari arxiv: Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture V. Giovannetti, R. Garcia-Patron, N. J. Cerf, A. S. Holevo arxiv: to appear in Nature Photonics Entropy Power Inequality for Bosonic Quantum Systems Giacomo De Palma, Andrea Mari, Vittorio Giovannetti arxiv: to appear in Nature Photonics The multi-mode quantum Entropy Power Inequality Giacomo De Palma, Andrea Mari, Seth Lloyd, Vittorio Giovannetti arxiv: THANK YOU!!!

51 II. Solutions Mari, Giovannetti, Holevo arxiv: repeat the same procedure for arbitrary (strictly) concave functionals of the output states of the channel... STEPS I, II, III, IV as before... F(A 0 apple[ ih ]) = F(A 0 apple E 0 0 [ ih ]) X p j F(A 0 apple[ jih j ]) j i IF MINIMIZE THE FUNCTIONAL SO ALSO MUST DO THE SAME. F E 0 0( ih ) =X j p j jih j BUT IS STRICTLY CONCAVE, HENCE E 0 0[ ih ] MUST BE PURE. THE ONLY STATES WHICH REMAIN PURE UNDER A LOSSY MAP ARE THE COHERENT STATES Aharanov et al. (1966) Asboth et al. (2005) Jiang et al. (2013) QED