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Entropic characteristics of quantum channels and the additivity problem

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Entropic characteristics of quantum channels and the additivity problem

  1. 1. ENTROPIC CHARACTERISTICS OF QUANTUM CHANNELS AND THE ADDITIVITY PROBLEM A. S. Holevo Steklov Mathematical Institute, Moscow
  2. 2. • Introduction: quantum information theory • The classical capacity of quantum channel • Hierarchy of additivity conjectures • Global equivalence • Partial results
  3. 3. INTRODUCTION A brief history of quantum information theory
  4. 4. Information Theory • Born: middle of XX century, 1940-1950s (Shannon,…) • Concepts: random source, entropy, typicality, code, channel, capacity: • Tools: probability theory, discrete math, group theory,… • Impact: digital data processing, data compression, error correction,… ,...ShannonCC =
  5. 5. Quantum Information Theory • Born: second half of XX century Physics of quantum communication, 1950-60s (Gabor, Gordon, Helstrom,…): FUNDAMENTAL QUANTUM LIMITATIONS ON INFORMATIOM TRANSMISSION ? • Mathematical framework: 1970-80s
  6. 6. Quantum Information Theory The early age (1970-1980s) Understanding quantum limits • Concepts: random source, entropy, channel, capacity, coding theorem, …, entanglement • Tools: noncommutative probability, operator algebra, random matrices (large deviations)… • Implications: …, the upper bound for classical capacity of quantum channel: χ-capacity C ≤ Cχ An overview in the book “Statistical structure of quantum theory” (Springer, 2001)
  7. 7. Quantum Information Theory (“Quantum Shannon theory”) The new age (1990-2000s) From quantum limitations to quantum advantages • Q. data compression (Schumacher-Josza,…) • The quantum coding theorem for c-q channels: C = C χ (Holevo; Schumacher-Westmoreland) • Variety of quantum channel capacities/coding theorems (Shor, Devetak, Winter, Hayden,…) Summarized in recent book by Hayashi (Springer, 2006)
  8. 8. Additivity of channel capacity 1 2112 CC CCC nn = += CLASSICAL INFORMATION CLASSICAL INFORMATION 1Φ 2Φ 01001011 11011010 ? ? MEMORYLESS encoding decoding
  9. 9. The χ-CAPACITY and the CLASSICAL CAPACITY of QUANTUM CHANNEL
  10. 10. Finite quantum system ∞<Hdim pointspacephasestatepureClassical xstateClassical ext projectiondim-1statepurepointExtreme setconvexcompact spaceState matrixdensitystateQuantum = = ∈= =Σ∈=Σ == =≥=Σ = )](diag[ ; }:)({)( }1Tr,0:{)( 2 πρ ψψψρ ρρρ ρρρ ρ H H H H
  11. 11. pure!are subsystemsofstatesPartial statePure productTensor 12121,2 12 2121 21 NOT entangled HextHextHHext HH ψψρ ψψψ α αα 1,2 21 Tr )()()( = ⊗= Σ×Σ⊃≠⊗Σ ⊗ ∑ Composite quantum systems – entanglement
  12. 12. Quantum channel Φ 1,2,...nforpositive =⊗ nIdΦ Completely positive (CP) map, Σ(H)→ Σ(H’): ρ ρ’ Φ nId 12ρ ' 12ρ
  13. 13. Product of channels )Id()Id( 212121 ⊗⊗=⊗ ΦΦΦΦ  1Φ 2Id 12ρ ' 12ρ 2Φ 1Id
  14. 14. The minimal output entropy ADDITIVITY ntentanglemeno:yClassicall HHH state)()(onattained HH )(oncontinuousconcave -)H(Entropy ⇒ ×= Φ+Φ=Φ⊗Φ Σ Φ=Φ Σ = )2()1()12( )A()()()( ext ))((min)( logTr 2121 purepurepure pureH H   ρ ρρρ ρ ?
  15. 15. The χ-capacity { }       ∑ Φ−Φ=Φ Φ→→ ∑ x xx x xx p xx HppHC xx ))(())(()( )( , ρρ χ ρρ ρ χ max x ensemble average conditional output entropy output entropy
  16. 16. The Additivity Conjecture )(C)C( )(C 1 lim)C( ,);(C)(C , )(A)(C)(C)(C 21 χ2121 Φ=Φ Φ=Φ ∀Φ=Φ ΦΦ Φ+Φ=Φ⊗Φ ⊗ ∞→ ⊗ χ χ χχ χχχ n CAPACITYCLASSICALthe nn channelsALLfor n n n ?
  17. 17. Separate encodings/ separate decodings )(cc ΦaI Φ CLASSICAL INFORMATION CLASSICAL INFORMATION . . n . separate q. separate q. encodings decodings ACCESSIBLE (SHANNON) INFORMATION Φ
  18. 18. Separate encodings/ entangled decodings )()( acc Φ>Φ ICχ χ Φ CLASSICAL INFORMATION CLASSICAL INFORMATION . . n . separate q. entangled encodings decodings HSW-theorem: χ - CAPACITY! ! Φ )(ΦχC
  19. 19. Entangled encodings/ entangled decodings )(Cn/)(Clim)(C n n Φ=Φ≡Φ ⊗ ∞→ χχ Φ CLASSICAL INFORMATION CLASSICAL INFORMATION . . n . entangled entangled encodings decodings The full CLASSICAL CAPACITY ? ? Φ
  20. 20. HIERARCHY of ADDITIVITY CONJECTURES - minimal output entropy - χ-capacity – convex closure/ constrained χ-capacity/EoF
  21. 21. Additivity of the minimal output entropy )()( )A()()()( ))((min)( 2121 Φ=Φ Φ+Φ=Φ⊗Φ Φ=Φ ⊗ HnH HHH HH n    ρρ ?
  22. 22. Rényi entropies and p-norms 1,pFor )(R)(R)(R oftivityMultiplica p1 p ))((RR )H()(R1,pWhen 1p p R 2p1p21p p1 p1p H p p p p )A()A( )A(ΦΦΦ Φ ΦlogΦmin)( ;logTr 1 1 )( p p )(    ⇒↓ +Φ=⊗ ⇔ − ==Φ ↑↓ > − = → →Σ∈ ρ ρρ ρρ ρ
  23. 23. Rényi entropies for p<1 )(R:0pFor norm!-pNo 1,pFor )H()R1,pWhen 1p0 p R 0 p p p )(rankmin )A()A( ;logTr 1 1 )( p ρ ρρ ρρ ρ Φ=Φ= ⇒↑ ↓↑ <≤ − =  (
  24. 24. The χ-capacity { }         ∑ Φ ∑ −Φ=       ∑ Φ−Φ=Φ = ∑ x xx p x xx x xx p HpH HppHC x xx xx ))((min))((max ))(())((max)( , ρρ ρρ ρρρ ρ χ ensemble average conditional output entropy output entropy convex closure
  25. 25. Convex closure EoF EoFofivitysuperaddit HHH isometrygStinesprin ensembles(finite) VVEHpH Fxx p xx ⇔ +≥ ↑ =ΦΣ= ΦΦΦ⊗Φ Σ= Φ )A()()()( *)())((min)( 2112 2121   ρρρ ρρρ ρρ
  26. 26. Constrained capacity }E:{Aconstraintlinear Asubsetcompact:constraint H-))(H(AC Φ A constTr: )( )( )]([max),( ≤= Σ⊆ Φ=Φ Φ ∈ ρρ ρχ ρρ ρ χ H
  27. 27. Additivity with constraints lyindividual )((C(C AC)A,(C)AA,(C χχ 2χ1χ21χ )A()(A )A()(CA )Φ,Φ) )(CA),(ΦΦΦ χ χ χ2121   ⇓⇓ ⇔ Σ= Φ+=⊗⊗ H ?
  28. 28. Equivalent forms of (CA )χ { } { } )()()()A( ;,)(CA )(CA )(CA 2112 21χ χ χ 22 ρρρ ρρ ΦΦΦ⊗Φ +≥ == HHH- AAarbitrarywith- ;A,Aarbitrarywith- ;A,Alineararbitrarywith- :equivalentarefollowingthe Φ,ΦchannelsFor 11 21 21 21 21  THM
  29. 29. Partial results • Qubit unital channel (King) • Entanglement-breaking channel (Shor) • Depolarizing channel (King) Lieb-Thirring inequality: ))(dim( IId =Φ== 2,H MN x x x ρρ Tr)( ∑=Φ 0;TrTr ≥≥≤ BA,1;pBA(AB) ppp dIpp )(Tr)1()( ρρρ +−=Φ
  30. 30. Recent work on special channels (2003-…) Alicki-Fannes; Datta-Fukuda-Holevo-Suhov; Giovannetti-Lloyd-Maccone-Shapiro-Yen; Hayashi-Imai-Matsumoto-Ruskai-Shimono; King-Nathanson-Ruskai; King-Koldan; Matsumoto-Yura; Macchiavello-Palma; Wolf-Eisert,… ALL ADDITIVE!
  31. 31. Transpose-depolarizing channel ( ) ( ) AH)-(Werner4,783)p3,(d2dp,largeforBREAKS 2,p1forholds unitaries)all-(Usymmetrichighly PIP d ;I d g asymasym T ≥=> ⇒≤< ⊗ − =Φ− − =Φ ))(A),A(()(A 1 2 )( ~ Tr 1 1 )( χp  ρρρρρ Numerical search for counterexamples
  32. 32. Breakthrough 2007 Multiplicativity breaks: • p>2, large d (Winter); • 1<p<2, large d (Hayden); • p=0, large d (Winter); p close to 0. Method: random unitary (non-constructive) It remains 0<p<1 and p=1 (the additivity!) ... And many other questions
  33. 33. Random unitary channels 2.pRRR ddOnd dd I -)( :grandomizin-isyprobabilithighWith unitaryi.i.d.random-U UU n ppp d j j n j j >Φ⊗Φ>Φ+Φ =∞→ ≤Φ Φ =Φ ∞ =∑ ),()()( )log(, )( * ε ρ ε ρρ 1 1
  34. 34. The basic Additivity Conjecture remains open
  35. 35. GLOBAL EQUIVALENCE of additivity conjectures (Shor, Audenaert-Braunstein, Matsumoto- Shimono-Winter, Pomeranski, Holevo- Shirokov)
  36. 36. (EoF)(H ty oferadditivisup Φ )ˆ )Aˆ( ρ Cofadditivity )()(Aχ Φχ )(Hofadditivity Φ  )A( ACofadditivity ),()(CAχ Φχ “Global” proofs involving Shor’s channel extensions Discontinuity of In infinite dimensions )(ΦχC
  37. 37. )(A)(CA globally χχ  →← .A,Asconstraintalland Φ,Φallforholds)(CAthen ,Φ,Φchannelsallforholds)(AIf 21 21χ 21χTHM Proof: Uses Shor’s trick: extension of the original channel which has capacity obtained by the Lagrange method with a linear constraint
  38. 38. Channel extension ETrratetheatbitsdsendsrarelybut ,asactsmostly0,qWhen IE0dqE idle Emeasures:qprobthwi bitsclassicald :q-1probthwi bitsclassicaldofinputsofInputs ρ ΦΦ ΦΦ Φ Φ ΦΦ log ˆ );,,(ˆˆ log:ˆ logˆ ≈ ≤≤= += 1 0
  39. 39. Lagrange Function [ ] [ ] { } capacity-dconstraine cE multiplierLagrange- E)ddE(Clim constd, q, dqLet dinuniformly O(1)qE)dq-q)()(C ρ χ ρρχ λ ρλρχλ λ λ ρρχ ρ χ χ ≥ +=Φ ==∞→→ ++=Φ Φ Φ Φ Tr:)(max Tr)(max),log/,(ˆ log0 Trlog()(1maxˆ
  40. 40. ∞=Hdim
  41. 41. • Set of states is separable metric space, not locally compact • Entropy is “almost always” infinite and everywhere discontinuous BUT • Entropy is lower semicontinuous • Entropy is finite and continuous on “useful” compact subset of states (of bounded “mean energy”) ∞=Hdim
  42. 42. The χ-capacity { }       ∑ Φ−Φ=Φ Φ→→ ∑ x xx x xx p xx HppHC x xx ))(())((sup)( )( , ρρ ρρ ρ χ ensemble average conditional output entropy output entropy Generalized ensemble (GE)=Borel probability measure on state space
  43. 43. )dim,(CA)dim,(A χχ ∞≤⇒∞< HH s.constraintarbitraryand channelsldimensiona-infinite allforholds)(CAthen channels,ldimensiona-finite allforholds)(AIf χ χTHM In particular, for all Gaussian channels with energy constraints
  44. 44. Gaussian channels Canonical variables (CCR) Gaussian environment Gaussian states Gaussian states Energy constraint PROP For arbitrary Gaussian channel with energy constraint an optimal generalized ensemble (GE) exists. CONJ Optimal GE is a Gaussian probability measure supported by pure Gaussian states with fixed correlation matrix. (GAUSSIAN CHANNELS HAVE GAUSSIAN OPTIMIZERS?) Holds for c-c, c-q, q-c Gaussian channels ------------------------------------------------------------ RRE RKKRRR T ee ε= +=→ '
  45. 45. CLASSES of CHANNELS
  46. 46. Complementary channels (AH, Matsumoto et al.,2005) Observation: additivity holds for very classical channels; for very quantum channels Example: Id 0 0 00 )(Tr)( ~ )( ρ ρρρ ρρρρρρ =Φ⊗=Φ
  47. 47. Complementary channels dilationgStinesprinThe VV)( *VV)( :Visometry B VA C B C A A CBA *Tr ~ Tr ~ ρρ ρρ H H HHH =Φ =Φ ⊗→ Φ Φ
  48. 48. Complementary channels 21 21 2121 ~ ~ , ~ )A()A( ,)A()A(THEOREM ~~ )()( ) ~ ()( ΦΦ⇔ ΦΦ Φ⊗ΦΦ⊗Φ = Φ=Φ ΦΦ forholds)(resp. forholds)(resp. tois HH HH     arycomplement ρρ
  49. 49. Entanglement-breaking channels dephasing""-channelsaryComplement breaking)-ntentanglemeischannel(the 0BABA))(Id( )HH(arbitraryand2,...dfor(ii) quantum);classicalquantumischannel(the MNwithtionrepresentaaisthere(i) :equivalentare conditionsfollowingTheShor)Ruskai,ki,(P.Horodec MN kk k kk12d d12 xx x x x ≥⊗=⊗Φ ⊗Σ∈= →→ ≥≥ =Φ ∑ ∑ ,; 0,0(*) THEOREM (*)Tr)( ρ ρ ρρ
  50. 50. Entanglement-breaking channels -- additivity 1 ppp 21 channelsdephasingforholdproperties additivitytheallarity,complementBy BA,BA(AB) :inequalityThirring-Liebthe onbasingKing,by-p Shorbydestablishe)fact,(in arbitrarybreaking,-ntentanglemeFor Φ ≥≤ > ΦΦ ~ 0;TrTr ,1),(A )A()(A),A( p χ 
  51. 51. Symmetric channels nonunitalqubit, Kingbyprovedp dIpp channelngdepolarizi(ii) 2);H( channelsunitalqubitsymmetricbinary(i) )(H-(I/d))H()(C :thene,irreducibl-Uif Gg*;)V(V*)ρUU χ g gggg )A( )(A,1),(A )(Tr)1()( dim )A()(A Φ( χp χ    > +−=Φ = = ΦΦ=Φ ⇔ ∈=Φ ρρρ ρ ?

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