Upgrade to Pro — share decks privately, control downloads, hide ads and more …

古典エントロピーと比べて学ぶ量子エントロピー

 古典エントロピーと比べて学ぶ量子エントロピー

Etsuji Nakai

April 15, 2019
Tweet

More Decks by Etsuji Nakai

Other Decks in Science

Transcript

  1. ݹయΤϯτϩϐʔͱൺ΂ֶͯͿྔࢠΤϯτϩϐʔ
    தҪ ӻ࢘
    2018 ೥ 10 ݄ 2 ೔

    View full-size slide

  2. 2
    0.1 Կͷ࿩͔ͱ͍͏ͱ
    ͜͜൒೥΄ͲྔࢠίϯϐϡʔςΟϯάͷษڧΛਐΊ͍ͯͯɺ·ͣ͸ɺఆ൪ͷڭՊॻ [1]
    Λಡഁͨ͠ͷͰ͕͢ɺྔࢠ৘ใཧ࿦ʹؔ͢Δষ͚ͩ͸ɺͳ͔ͳ͔खڧͯ͘ཧղ͕௥͍͖ͭ
    ·ͤΜͰͨ͠ɻͦ͜Ͱɺݹయ৘ใཧ࿦ͷڭՊॻ [2] ΛಡΈ௚ͯ͠ɺྔࢠ৘ใཧ࿦ʹಛԽ͠
    ͨڭՊॻ [3] ΛಡΜͰʜʜͱ΍͖ͬͯͨͷͰ͕͢ɺͦͷதͰɺྔࢠΤϯτϩϐʔʢϊΠϚ
    ϯΤϯτϩϐʔʣͷੑ࣭ʹڵຯΛ࣋ͭΑ͏ʹͳΓ·ͨ͠ɻ
    ͦ΋ͦ΋ݹయతͳ৘ใΤϯτϩϐʔʢγϟϊϯΤϯτϩϐʔʣ΋ͦͷఆ͕ٛ࣋ͭҙຯ͸
    ࣗ໌Ͱ͸ͳ͘ɺ৘ใཧ࿦ʹؔ͢Δ͞·͟·ͳૢ࡞తղऍʢOperational Interpretationʣʹ
    ΑΓɺͦͷ໾ׂ͕໌Β͔ʹͳΓ·͢ɻҰํɺྔࢠΤϯτϩϐʔ͸ɺঢ়ଶີ౓ԋࢉࢠΛʮෳ
    ਺ͷྔࢠঢ়ଶʢ७ਮঢ়ଶʣͷ౷ܭతΞϯαϯϒϧʯͱղऍͨ͠ࡍͷ౷ܭతͳෆ֬ఆੑͱղ
    ऍ͢Δ͜ͱ͕Ͱ͖·͕͢ɺݹయΤϯτϩϐʔʹྨࣅͷੑ࣭Λ࣋ͭͱͱ΋ʹɺݹయΤϯτϩ
    ϐʔʹ͸ͳ͍ݻ༗ͷੑ࣭΋͋Γ·͢ɻ͞ΒʹɺݹయΤϯτϩϐʔͱྨࣅͷੑ࣭Λূ໌͢Δ
    ࡍʹɺྔࢠ࿦ʹݻ༗ͷಛघͳٞ࿦͕ඞཁͱͳΔ৔߹΋͋Γ·͢ɻ
    ྔࢠΤϯτϩϐʔʹ͍ͭͯ΋ɺ࠷ऴతʹ͸ɺྔࢠ৘ใཧ࿦ʹ͓͚Δૢ࡞తղऍΛ௨͠
    ͯɺͦͷ໾ׂɺ͋Δ͍͸ɺఆٛͷଥ౰ੑ͕໌Β͔ʹͳΔ΋ͷͱߟ͑ΒΕ·͢ɻ͔͠͠ͳ͕
    Βɺ·ͣ͸ɺݹయΤϯτϩϐʔͱൺֱͯ͠ɺͦͷྨࣅੑɺ͋Δ͍͸ɺ૬ҧੑΛຯΘ͏͜ͱ
    Ͱɺྔࢠܥʹಛ༗ͷߏ଄ɺ΋͘͠͸ɺྔࢠ৘ใཧ࿦ʹ͓͍ͯྔࢠΤϯτϩϐʔ͕Ռͨ͢໾
    ׂͷҰ୺͕֞ؒݟ͑Δ͔΋஌Ε·ͤΜɻͦͷΑ͏ͳૉ๿ͳظ଴ͷ΋ͱʹɺຊߘͰ͸ɺݹయ
    Τϯτϩϐʔͱൺֱ͠ͳ͕ΒɺྔࢠΤϯτϩϐʔͷ͞·͟·ͳੑ࣭Λ਺ֶతʹಋ͍͍͖ͯ
    ·͢ɻ
    ͳ͓ɺݹయ৘ใཧ࿦ͷ஌ࣝ͸લఏͱͯ͠ɺݹయΤϯτϩϐʔͷੑ࣭ʹؔ͢Δূ໌͸ߦ͍
    ·ͤΜͷͰɺඞཁͳࡍ͸ɺݹయ৘ใཧ࿦ͷڭՊॻ [2] ͳͲΛࢀߟʹ͍ͯͩ͘͠͞ɻ·ͨɺ
    ྔࢠ৘ใཧ࿦ͷجૅͱͳΔɺঢ়ଶີ౓ԋࢉࢠʹΑΔྔࢠঢ়ଶͷهड़ͱྔࢠܥͷଌఆʹؔ͢
    ΔҰൠ࿦ɺͦͯ͠ɺྔࢠνϟωϧͷߟ͑ํʹ͍ͭͯ͸ɺओཁͳ݁࿦Λ࠷ॳʹ੔ཧ͓͖ͯ͠
    ·͢ɻৄࡉʹ͍ͭͯ͸ɺલड़ͷڭՊॻ [1][3] ͳͲΛࢀߟʹ͍ͯͩ͘͠͞ɻ
    0.2 લఏ஌ࣝͷ੔ཧ
    ͸͡Ίʹɺྔࢠ৘ใཧ࿦ʹඞཁͱͳΔྔࢠྗֶͷجૅ஌ࣝΛ·ͱΊ·͢ɻ͋͘·Ͱɺ͜
    ͷޙͷٞ࿦ͷલఏΛ੔ཧ͢Δ͜ͱ͕໨తͰ͢ͷͰɺৄࡉͳઆ໌͸ׂѪ͍ͯ͠·͢ɻ

    View full-size slide

  3. 0.2 લఏ஌ࣝͷ੔ཧ 3
    0.2.1 ঢ়ଶີ౓ԋࢉࢠʹΑΔྔࢠঢ়ଶͷهड़
    ྔࢠܥͷಛఆͷঢ়ଶʢ७ਮঢ়ଶʣ͸ɺෳૉ਺ C ্ͷώϧϕϧτۭؒʹଐ͢Δ୯ҐϕΫ
    τϧ |ψ⟩ ʹΑͬͯهड़͞Ε·͢ɻ͜ΕΛঢ়ଶϕΫτϧͱݺͼ·͢ɻҰൠʹ͸ɺແݶ࣍ݩͷ
    ώϧϕϧτۭ͕ؒඞཁͱͳΓ·͕͢ɺྔࢠίϯϐϡʔςΟϯάͷจ຺Ͱ͸ɺ΄ͱΜͲͷ৔
    ߹ɺ༗ݶ࣍ݩʹݶఆͯ͠ߟ͑·͢ɻ·ͨɺҰఆͷ཭ࢄత֬཰෼෍ p(x) ʹैͬͯɺෳ਺ͷ
    ྔࢠঢ়ଶ |ψx
    ⟩ ΛूΊͨू߹ʢΞϯαϯϒϧʣΛࠞ߹ঢ়ଶͱݺͼ·͢ɻૢ࡞తʹ͸ɺࠞ߹
    ঢ়ଶ͸࣍ͷΑ͏ʹ࡞Γग़͢͜ͱ͕Ͱ͖·͢ɻ·ͣɺΞϦε͸ɺ֬཰෼෍ p(x) ʹै͏֬཰
    ม਺͔ΒαϯϓϧΛ 1 ͭऔಘͯ͠ɺͦͷ݁ՌʹԠͯ͡ɺରԠ͢Δྔࢠঢ়ଶ |ψx
    ⟩ ΛϘϒʹ
    ౉͠·͢ɻ͜ͷ࣌ɺϘϒ͕ड͚औͬͨܥ͸ɺΞϯαϯϒϧ {p(x), |ψx
    ⟩} ʹରԠ͢Δࠞ߹
    ঢ়ଶͱͳΓ·͢ɻ
    ҰൠʹɺΞϯαϯϒϧ {p(x), |ψx
    ⟩} ʹରԠ͢Δࠞ߹ঢ়ଶ͸ɺ࣍ͷঢ়ଶີ౓ԋࢉࢠʹΑͬ
    ͯهड़͞Ε·͢ɻ
    ρ =

    x
    p(x)|ψx
    ⟩⟨ψx
    | (1)
    ྔࢠঢ়ଶ |ψx
    ⟩ ͕ଐ͢ΔώϧϕϧτۭؒΛ H ͱ͢Δͱɺρ ͸ɺH ͔Β H ΁ͷઢܗԋࢉ
    ࢠʹͳ͓ͬͯΓɺ͜ͷઢܗԋࢉࢠͷۭؒΛ L(H) ͱ͍͏ه߸Ͱද͠·͢ɻঢ়ଶີ౓ԋࢉࢠ
    ͸ɺͦͷఆٛΑΓɺτϨʔε͕ 1 Ͱɺ൒ਖ਼ఆ஋ͷΤϧϛʔτԋࢉࢠʹͳΓ·͢ɻτϨʔε
    ʹ͍ͭͯ͸ɺ࣍ͷܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻ
    Tr ρ =

    x
    p(x)

    i
    ⟨i|ψx
    ⟩⟨ψx
    |i⟩ =

    x
    p(x)

    i
    ⟨ψx
    |i⟩⟨i|ψx

    =

    x
    p(x)⟨ψx
    |ψx
    ⟩ =

    x
    p(x) = 1
    ͜͜Ͱɺ{|i⟩} ͸ɺ⟨i|j⟩ = δij
    Λຬͨ͢ਖ਼ن௚ަجఈΛද͠·͢ɻ·ͨɺঢ়ଶີ౓ԋࢉ
    ࢠ͸ΤϧϛʔτԋࢉࢠͰ͋Δ͜ͱ͔Βɺਖ਼ن௚ަܥʹΑΔεϖΫτϧ෼ղ͕ՄೳͰɺ
    ρ =

    i
    pi
    |i⟩⟨i| (2)
    ͱॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ͜͜Ͱɺ|i⟩ ͸ɺ⟨i|j⟩ = δij
    Λຬͨ͢ਖ਼ن௚ަجఈΛද͠·
    ͢ɻͨͩ͠ɺਖ਼ن௚ަجఈͷͱΓํ͸ɺρ ʹґଘͯ͠มΘΔ఺ʹ஫ҙ͕ඞཁͰ͢ɻ·ͨɺ
    Ұൠʹ (2) ʹݱΕΔݻ༗஋ pi
    ͸ɺઌʹΞϯαϯϒϧΛߏ੒͢ΔͨΊʹಋೖͨ͠ (1) ͷ֬
    ཰෼෍ p(x) ͱ͸ҟͳΓ·͕͢ɺpi
    ΋·ͨɺ֬཰෼෍ͱͯ͠ղऍ͢Δ͜ͱ͕Ͱ͖·͢ɻͳ

    View full-size slide

  4. 4
    ͥͳΒɺρ ͕൒ਖ਼ఆ஋Ͱ͋Δ͜ͱ͔Β pi
    ≥ 0 Ͱ͋Γɺρ ͷτϨʔε͕ 1 Ͱ͋Δ͜ͱ͔Βɺ

    i
    pi
    = 1
    ͕੒Γཱ͔ͭΒͰ͢ɻͭ·Γɺޓ͍ʹ௚ߦ͢Δͱ͸ݶΒͳ͍ɺ೚ҙͷྔࢠঢ়ଶ |ψx
    ⟩ Λू
    Ίͨࠞ߹ঢ়ଶ͸ɺҟͳΔ֬཰෼෍ʹैͬͯɺਖ਼ن௚ަܥ |i⟩ ΛूΊͨࠞ߹ঢ়ଶͱඞͣಉҰ
    ࢹ͢Δ͜ͱ͕Ͱ͖ΔͷͰ͢ɻ͜͜Ͱݴ͏ʮಉҰࢹʯͱ͍͏ͷ͸ɺϘϒ͔Βݟͨࡍʹɺ(1)
    ͱ (2) ͸ݪཧతʹ۠ผ͢Δ͜ͱ͕Ͱ͖ͳ͍ͱ͍͏͜ͱͰ͢*1ɻ
    ͳ͓ɺਖ਼ن௚ަجఈͷ͢΂ͯͷঢ়ଶΛۉ౳ʹؚΉΞϯαϯϒϧͷ৔߹ɺ֤ঢ়ଶ͕ಘΒΕ
    Δ֬཰͸ɺd = dim H ͱͯ͠ɺ
    1
    d
    Ͱ༩͑ΒΕΔͷͰɺରԠ͢Δঢ়ଶີ౓ԋࢉࢠ͸ɺ
    π =

    i
    1
    d
    |i⟩⟨i| =
    1
    d
    I
    Ͱ༩͑ΒΕ·͢ɻ͜͜ʹɺI ∈ L(H) ͸ɺ߃౳ԋࢉࢠΛද͠·͢ɻ
    0.2.2 ߹੒ܥͷྔࢠঢ়ଶ
    ώϧϕϧτۭؒ HA
    ͷঢ়ଶϕΫτϧͰهड़͞ΕΔܥ A ͱɺώϧϕϧτۭؒ HB
    ͷঢ়ଶ
    ϕΫτϧͰهड़͞ΕΔܥ B ͕ଘࡏ͢Δ࣌ɺ͜ΕΒΛ 1 ͭͷܥͱΈͳͨ͠ঢ়ଶ͸ɺ͜ΕΒͷ
    ςϯιϧੵͷۭؒ HA
    ⊗ HB
    ʹଐ͢Δঢ়ଶϕΫτϧʹΑͬͯهड़͞Ε·͢ɻͨͱ͑͹ɺܥ
    A ͱܥ B ͷ૬ޓ࡞༻Λःஅͨ͠ঢ়ଶͰɺͦΕͧΕͷྔࢠঢ়ଶΛ |ψA
    ⟩ɺ͓Αͼɺ|ψB
    ⟩ ͱ͠
    ͯ༻ҙͨ͠৔߹ɺ͜ΕΒΛ߹੒ͨ͠ܥͷঢ়ଶ͸ςϯιϧੵ |ψA
    ⟩ ⊗ |ψB
    ⟩ Ͱ༩͑ΒΕ·͢ɻ
    ͨͩ͠ɺΑΓҰൠʹ͸ɺෳ਺ͷςϯιϧੵͷઢܗ݁߹ΛͱΔ͜ͱ΋ՄೳͰ͢ɻHA
    ͱ
    HB
    ͷਖ਼ن௚ަجఈΛͦΕͧΕ |iA
    ⟩ɺ|jB
    ⟩ ͱ͢Δͱɺςϯιϧੵۭؒ HA
    ⊗ HB
    ͸ɺਖ਼
    ن௚ަجఈ |iA
    ⟩ ⊗ |jB
    ⟩ ͰுΒΕΔͷͰɺ߹੒ܥ AB ͷঢ়ଶϕΫτϧ͸ɺҰൠʹɺ࣍ͷΑ
    ͏ʹల։͢Δ͜ͱ͕Ͱ͖·͢ɻ
    |ψAB
    ⟩ =

    i,j
    cij
    |iA
    ⟩ ⊗ |jB

    ͜ͷ࣌ɺܥ AB ͷ೚ҙͷঢ়ଶʢHA
    ⊗ HB
    ͷ೚ҙͷཁૉʣ |ψAB
    ⟩ ʹ͍ͭͯɺܥ A ͱܥ
    B ͷਖ਼ن௚ަجఈΛ͏·͘औΔͱɺ࣍ͷΑ͏ʹల։Ͱ͖Δ͜ͱ͕஌ΒΕ͍ͯ·͢ɻ
    |ψAB
    ⟩ =

    i
    λi
    |iA
    ⟩ ⊗ |iB
    ⟩ (λi
    ≥ 0) (3)
    *1 ͜ͷޙͰઆ໌͢ΔΑ͏ʹɺ͜ͷܥʹରͯ͠Ϙϒ͕ߦ͏ଌఆ݁Ռ͸ɺঢ়ଶີ౓ԋࢉࢠͷΈʹΑܾͬͯ·Γ·
    ͢ɻ͕ͨͬͯ͠ɺϘϒ͸ɺଌఆʹΑͬͯ 2 छྨͷΞϯαϯϒϧ {p(x), |ψx⟩} ͱ {pi, |i⟩} Λ۠ผ͢Δ͜
    ͱ͸Ͱ͖·ͤΜɻ͜ͷ͋ͨΓ͔Βɺྔࢠܥ͕࣋ͭʮ৘ใʯͷਂ෵͕֞ؒ͞ݟ͑ͯ͘ΔΑ͏ͳʜʜɻ

    View full-size slide

  5. 0.2 લఏ஌ࣝͷ੔ཧ 5
    ͜ΕΛ߹੒ܥʹର͢Δγϡϛοτ෼ղͱݺͼ·͢ɻ্هͷਖ਼ن௚ަجఈ͸ɺγϡϛοτ
    ෼ղΛద༻͢Δঢ়ଶ |ψAB
    ⟩ ʹΑͬͯҟͳΔ఺ʹ஫ҙ͍ͯͩ͘͠͞ɻ࿨ͷ্ݶ͸໌ࣔͯ͠
    ͍·ͤΜ͕ɺ౰વͳ͕ΒɺHA
    ͱ HB
    ͷͦΕͧΕͷ࣍ݩͷখ͍͞ํΛ௒͑Δ͜ͱ͸͋Γ·
    ͤΜɻ·ͨɺ|ψAB
    ⟩ ͕୯ҐϕΫτϧͰ͋Δ͜ͱ͔Βɺ
    ⟨ψAB
    |ψAB
    ⟩ =

    i
    λ2
    i
    = 1
    ͕੒Γཱͪ·͢ɻ
    ߹੒ܥ AB ʹ͍ͭͯ΋ɺܥ AɺB ୯ಠͷ৔߹ͱಉ༷ʹࠞ߹ঢ়ଶΛߟ͑Δ͜ͱ͕Ͱ͖·
    ͢ɻҰൠʹɺΞϯαϯϒϧ {p(x), |ψx
    AB
    ⟩} ʹରԠ͢Δঢ়ଶີ౓ԋࢉࢠ͸ɺ࣍Ͱ༩͑ΒΕ
    ·͢ɻ
    ρAB
    =

    x
    p(x)|ψx
    AB
    ⟩⟨ψx
    AB
    |
    ͜͜Ͱɺ߹੒ܥ AB ʹ͍ͭͯɺগ͠໘ന͍ঢ়گΛߟ͑·͢ɻࠓɺ2 ਓͷ؍ଌऀɺΞϦε
    ͱϘϒ͕͍Δͱͯ͠ɺΞϦε͸ܥ A ͷΈɺͦͯ͠ɺϘϒ͸ܥ B ͷΈΛ؍ଌͰ͖Δ΋ͷͱ
    ͠·͢ɻ͜ͷ࣌ɺΞϦε͕؍ଌ͢Δܥ A ͷঢ়ଶ͸ɺ࣍ͷঢ়ଶີ౓ԋࢉࢠͰهड़͞Ε·͢ɻ
    ρA
    = TrB
    ρAB
    ͜͜ͰɺTrB
    ͸ɺ෦෼ۭؒ HB
    ʹ͍ͭͯͷΈτϨʔεΛऔΔ͜ͱΛද͓ͯ͠Γɺ۩ମ
    తʹ͸ɺHB
    ͷਖ਼ن௚ަجఈΛ |jB
    ⟩ ͱͯ͠ɺ
    ρA
    = TrB
    ρAB
    =

    j
    ⟨jB
    |ρAB
    |jB

    Ͱܭࢉ͞Ε·͢ɻ্هͷܭࢉʹ͓͚Δ ⟨jB
    | ͱ |jB
    ⟩ ͸ɺ
    ਖ਼֬ʹ͸ɺ
    ߃౳ԋࢉࢠ IA
    ∈ L(HA
    )
    Λ༻͍ͯɺIA
    ⊗ ⟨jB
    |ɺ͓ΑͼɺIA
    ⊗ |jB
    ⟩ ͱॻ͘΂͖΋ͷͰ͕͢ɺ͜ͷΑ͏ͳܭࢉʹ͓͚
    Δ߃౳ԋࢉࢠ͸هࡌΛলུ͢Δ͜ͱ͕͋Γ·͢ɻ͜Εͱಉ༷ʹɺϘϒ͕؍ଌ͢Δܥ B ͷ
    ঢ়ଶ͸ɺ࣍ͷঢ়ଶີ౓ԋࢉࢠͰهड़͞Ε·͢ɻ
    ρB
    = TrA
    ρAB
    ͳ͓ɺ͜ͷΑ͏ͳঢ়گઃఆʹ͓͍ͯ͸ɺϘϒ͕ܥ B ʹର͢ΔଌఆΛߦ͏͜ͱͰɺܥ A
    ͷঢ়ଶ͕มԽ͢ΔՄೳੑ͕͋Γ·͕͢ɺޙ΄Ͳઆ໌͢ΔΑ͏ʹɺϘϒͷଌఆ݁Ռʹؔ͢Δ
    ৘ใΛΞϦε͕ೖख͠ͳ͍ݶΓɺΞϦε͔Βݟͨܥ A ͷঢ়ଶ͸ɺঢ়ଶີ౓ԋࢉࢠ ρA
    Ͱ
    هड़͞ΕΔ͜ͱʹมΘΓ͋Γ·ͤΜɻ͜ͷΑ͏ʹɺ؍ଌର৅֎ͷܥ B ʹ͍ͭͯτϨʔε
    ΛͱΔ͜ͱΛʮܥ B ΛτϨʔεΞ΢τ͢Δʯͱݴ͍·͢ɻ

    View full-size slide

  6. 6
    0.2.3 ྔࢠܥʹର͢Δଌఆॲཧ
    ώϧϕϧτۭؒ H ্ͷܥʹର͢Δ؍ଌՄೳͳ෺ཧྔ͸ɺҰൠʹɺΤϧϛʔτԋࢉࢠ
    O ∈ L(H) Ͱهड़͞Ε·͢ɻO ΛεϖΫτϧ෼ղͯ͠ɺ
    O =

    i
    Oi
    |i⟩⟨i|
    ͱදࣔͨ͠ࡍʹɺ{Oi
    } ͸ଌఆ݁Ռͱͯ͠ಘΒΕΔ஋ͷू߹ͱͳΓ·͢ɻ·ͨɺঢ়ଶີ౓
    ԋࢉࢠ ρ Ͱද͞ΕΔঢ়ଶʹ͍ͭͯଌఆΛߦͬͨࡍʹɺOi
    ͱ͍͏஋͕؍ଌ͞ΕΔ֬཰͸ɺ
    Pr [Oi
    ] = ⟨i|ρ|i⟩ = Tr (|i⟩⟨i|ρ) (4)
    Ͱܭࢉ͞Ε·͢ɻ͜͜Ͱɺ|i⟩⟨i| ͸ɺO ͷݻ༗஋ Oi
    ʹରԠ͢Δݻ༗ۭؒ΁ͷࣹӨԋࢉࢠ
    ʹͳ͍ͬͯ·͢ͷͰɺ(4) ͷؔ܎͸ɺࣹӨԋࢉࢠ Pi
    = |i⟩⟨i| Λ༻͍ͯɺ
    Pr [Oi
    ] = Tr (Pi
    ρ)
    ͱද͢͜ͱ΋Ͱ͖·͢ɻ͜ͷ࣌ɺࣹӨԋࢉࢠͷू߹ {Pi
    } ͸࣍ͷ৚݅Λຬͨ͠·͢ɻ
    Pi
    Pj
    = 0, P2
    i
    = Pi
    ,

    i
    Pi
    = 1 (5)
    Ұൠʹɺ͜ΕΒͷ৚݅Λຬࣹͨ͢Өԋࢉࢠ {Pi
    } ʹΑΔଌఆΛࣹӨଌఆͱݺͼ·͢ɻ·
    ͨɺଌఆޙʹͦͷ؍ଌ஋Λ֬ೝͤͣʹഁغͨ͠৔߹ɺܥͷঢ়ଶີ౓ԋࢉࢠ͸࣍ʹมԽ͠
    ·͢ɻ
    ρ′ =

    i
    Pi
    ρPi
    (6)
    (6) ͷؔ܎͸ɺ࣍ͷΑ͏ʹಋ͘͜ͱ͕Ͱ͖·͢ɻ·ͣɺҰൠʹ७ਮঢ়ଶ |ψ⟩ ʹࣹӨଌఆ
    Λߦ͍ɺ؍ଌ஋ Oi
    ͕ಘΒΕͨ৔߹ɺଌఆޙͷঢ়ଶ͸ɺ
    Pi
    |ψ⟩

    ⟨ψ|Pi
    |ψ⟩
    =
    Pi
    |ψ⟩

    Pr [Oi
    ]
    ʹมԽ͠·͢ɻ͕ͨͬͯ͠ɺଌఆޙͷঢ়ଶີ౓ԋࢉࢠ͸ɺ
    Pi
    |ψ⟩⟨ψ|Pi
    Pr [Oi
    ]
    ͱܾ·Γ·͢ɻҰํɺଌఆ݁ՌΛ֬ೝ͠ͳ͍৔߹ɺ࠷ऴతͳঢ়ଶ͸ɺ֬཰ Pr [Oi
    ] Ͱ্ه
    ͷঢ়ଶ͕ಘΒΕΔΞϯαϯϒϧʹͳΔͷͰɺରԠ͢Δঢ়ଶີ౓ԋࢉࢠ͸ɺ

    i
    Pr [Oi
    ]
    Pi
    |ψ⟩⟨ψ|Pi
    Pr [Oi
    ]
    =

    i
    Pi
    |ψ⟩⟨ψ|Pi

    View full-size slide

  7. 0.2 લఏ஌ࣝͷ੔ཧ 7
    ͱܾ·Γ·͢ɻ͞Βʹɺ͸͡Ίͷঢ়ଶ͕Ξϯαϯϒϧ {pi
    , |ψi
    ⟩} ͔ΒͳΔࠞ߹ঢ়ଶ
    ρ =

    i
    pi
    |ψ′
    i
    ⟩⟨ψ′
    i
    |
    ͷ৔߹ɺ্هͷ݁Ռ͕֬཰ pi
    ͰಘΒΕΔͨΊɺ࠷ऴతͳঢ়ଶີ౓ԋࢉࢠ͸ɺ
    ρ′ =

    i
    pi
    Pi
    |ψ′
    i
    ⟩⟨ψ′
    i
    |Pi
    =

    i
    Pi
    ρPi
    ͱܾ·Γ·͢ɻ
    ͳ͓ɺྔࢠྗֶͷٞ࿦Ͱ͸ɺ্ड़ͷࣹӨଌఆͷΈΛऔΓѻ͏͜ͱ͕΄ͱΜͲͰ͕͢ɺ
    ྔࢠ৘ใཧ࿦ͷ࿮૊ΈͰ͸ɺࣹӨଌఆΛ֦ுͨ͠ɺPOVM ʢPositive Operator-Valued
    Measureʣͱݺ͹ΕΔଌఆॲཧ΋༻͍ΒΕ·͢ɻ͜Ε͸ɺ

    i
    Λi
    = I
    ͱ͍͏׬શܥͷ৚݅Λຬͨ͢൒ਖ਼ఆ஋ͷઢܗԋࢉࢠ Λi
    ∈ L(H) ͷू߹Λ༻͍ͯද͞ΕΔ
    ଌఆॲཧͰɺi ൪໨ͷ؍ଌ஋͕ಘΒΕΔ֬཰͸ɺ
    Tr (Λi
    ρ)
    Ͱ༩͑ΒΕ·͢ɻ
    ݱ࣮ͷܥ A ʹରͯ͠ POVM ʹରԠ͢ΔଌఆΛߦ͏ʹ͸ɺଌఆ༻ͷ৽ͨͳܥ R Λ༻ҙ
    ͯ͠ɺ߹੒ܥ AR ʹҰఆͷखଓ͖Λద༻͢Δඞཁ͕͋Γ·͢ɻ۩ମతͳํ๏ʹ͍ͭͯ͸ɺ
    [1][3] ͳͲͷඪ४తͳڭՊॻΛࢀর͍ͯͩ͘͠͞ɻ·ͨɺPOVM ͷ৔߹ɺଌఆޙͷܥͷ
    ঢ়ଶΛ໌ࣔతʹॻ͖ද͢͜ͱ͸Ͱ͖·ͤΜɻPOVM ͸ɺͦΕͧΕͷ؍ଌ஋͕ಘΒΕΔ֬
    ཰ͷΈʹڵຯ͕͋Γɺଌఆޙͷܥ͸ഁغ͢Δͱ͍͏લఏͰ༻͍Δख๏ʹͳΓ·͢ɻͳ͓ɺ
    (5) ͷ৚݅ΑΓɺࣹӨଌఆ͸ɺPOVM ͷಛผͳ৔߹ʹ͋ͨΔ͜ͱ͕Θ͔Γ·͢ɻ
    ͜͜Ͱ࠷ޙʹɺલ߲Ͱٞ࿦ͨ͠ɺ߹੒ܥ AB ʹ͍ͭͯɺΞϦε͸ܥ A ͷΈɺϘϒ͸ܥ
    B ͷΈΛ؍ଌ͢Δͱ͍͏ঢ়گΛ΋͏Ұ౓ߟ͑·͢ɻࠓɺܥ AB ͸ঢ়ଶີ౓ԋࢉࢠ ρAB
    Ͱ
    هड़͞ΕΔͱͯ͠ɺϘϒ͕ܥ B ʹରͯ͠ɺࣹӨଌఆ PBi
    Λߦͬͨͱ͠·͢ɻϘϒͷଌఆ
    ݁ՌΛ֬ೝ͠ͳ͍৔߹ɺଌఆޙͷܥ AB ͷঢ়ଶີ౓ԋࢉࢠ͸࣍ʹͳΓ·͢ɻ
    ρ′
    AB
    =

    i
    PBi
    ρAB
    PBi
    ͜͜Ͱɺܥ B ΛτϨʔεΞ΢τ͢ΔͱɺτϨʔεͷઢܗੑͱ८ճੑʢTr (ABC) =

    View full-size slide

  8. 8
    Tr (CAB)ʣΛ༻͍ͯɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ
    TrB
    ρ′
    AB
    =

    i
    TrB
    (PBi
    ρAB
    PBi
    ) =

    i
    TrB
    (
    P2
    Bi
    ρAB
    )
    = TrB
    (

    i
    PBi
    ρAB
    )
    = TrB
    ρAB
    ࠷ޙͷ݁ՌΛݟΔͱɺϘϒ͕ଌఆΛߦ͏લʹɺܥ B ΛτϨʔεΞ΢τͨ͠৔߹ͱಉ͡
    ݁Ռʹͳ͓ͬͯΓɺϘϒͷଌఆॲཧʹΑͬͯɺΞϦε͔Βݟͨܥ A ͷঢ়ଶ͸มԽ͠ͳ͍
    ͜ͱ͕Θ͔Γ·͢ɻ
    0.2.4 ܥͷ७ਮԽʢPurificationʣ
    ߹੒ܥ AB ͕ HA
    ⊗ HB
    ্ͷঢ়ଶϕΫτϧ |ψ⟩ Ͱද͞ΕΔ७ਮঢ়ଶʹ͋Δ΋ͷͱ͠·
    ͢ɻ͜ͷ࣌ɺܥ B ΛτϨʔεΞ΢τͯ͠ɺܥ A ͷঢ়ଶີ౓ԋࢉࢠΛٻΊΔͱɺ
    ρA
    = TrB
    (|ψ⟩⟨ψ|)
    ͱͳΓ·͕͢ɺ͜Ε͸ඞͣ͠΋७ਮঢ়ଶͱ͸ݶΓ·ͤΜɻݴ͍׵͑Δͱɺܥશମ͕७ਮঢ়
    ଶʹ͋Γɺ౷ܭతͳෆ֬ఆੑΛ࣋ͨͳ͍ͱͯ͠΋ɺܥͷҰ෦ͷΈΛ؍ଌ͢Δ؍ଌऀʹ͸ɺ
    ౷ܭతͳෆ֬ఆੑΛؚΉࠞ߹ঢ়ଶ͕༩͑ΒΕΔ͜ͱʹͳΓ·͢ɻͦͯ͠ɺ໘ന͍͜ͱʹɺ
    ೚ҙͷࠞ߹ঢ়ଶ͸ɺ͜Εͱಉ͡खଓ͖ͰಘΒΕΔ͜ͱ͕ࣔ͞Ε·͢ɻ
    ࠓɺܥ A ͸ɺঢ়ଶີ౓ԋࢉࢠ ρA
    ∈ L(H) Ͱهड़͞ΕΔࠞ߹ঢ়ଶʹ͋Δͱͯ͠ɺρA
    Λ
    ࣍ͷΑ͏ʹεϖΫτϧ෼ղ͠·͢ɻ
    ρA
    =

    i
    pi
    |iA
    ⟩⟨iA
    | (7)
    ͜͜ͰɺHA
    ͱಉ͡ɺ΋͘͠͸ɺͦΕҎ্ͷ࣍ݩΛ࣋ͭώϧϕϧτۭؒ HE
    Ͱهड़͞Ε
    Δ؀ڥܥ E Λ༻ҙͯ͠ɺHA
    ⊗ HE
    ্ͷঢ়ଶϕΫτϧ
    |ψAE
    ⟩ =

    i

    pi
    |iA
    ⟩ ⊗ |iE

    Λߟ͑·͢ɻ͜͜ʹɺ|iE
    ⟩ ͸ɺHE
    ͷ೚ҙͷਖ਼ن௚ަܥͱ͠·͢ɻ͜ͷ࣌ɺ|ψAE
    ⟩ Ͱه
    ड़͞ΕΔ७ਮঢ়ଶʹରͯ͠ɺܥ E ΛτϨʔεΞ΢τͯ͠ɺܥ A ʹର͢Δঢ়ଶີ౓ԋࢉࢠ

    View full-size slide

  9. 0.2 લఏ஌ࣝͷ੔ཧ 9
    ρA
    Λܭࢉͯ͠Έ·͢ɻ
    ρA
    = TrE
    (|ψAE
    ⟩⟨ψAE
    |)
    =

    k
    ⟨kE
    |



    i,j

    pi

    pj
    |iA
    ⟩ ⊗ |iE
    ⟩⟨jA
    | ⊗ ⟨jE
    |

     |kE

    =

    k

    i,j

    pi

    pj
    |iA
    ⟩⟨kE
    |iE
    ⟩⟨jA
    |⟨jE
    |kE

    =

    k

    i,j

    pi

    pj
    |iA
    ⟩⟨jA
    |δik
    δjk
    =

    i
    pi
    |iA
    ⟩⟨iA
    |
    ࠷ޙͷ݁ՌΛݟΔͱɺ(7) ͱಉ͡ঢ়ଶີ౓ԋࢉࢠ͕ಘΒΕͨ͜ͱ͕Θ͔Γ·͢ɻ͜ͷΑ
    ͏ʹɺࠞ߹ঢ়ଶͷܥ A ʹରͯ͠ɺ؀ڥܥ E ΛՃ͑ͯɺ߹੒ܥ AE ʹ͓͚Δ७ਮঢ়ଶΛߏ
    ੒͢Δख๏Λܥͷ७ਮԽʢPurificationʣͱݺͼ·͢*2ɻ
    0.2.5 ঢ়ଶมԽͱྔࢠνϟωϧ
    ֎෦ͷ؍ଌऀʹΑΔଌఆॲཧΛؚ·ͳ͍ɺྔࢠܥͷࣗવͳঢ়ଶมԽ͸ɺϢχλϦʔม׵
    ʹΑͬͯද͞Ε·͢ɻͨͱ͑͹ɺώϧϕϧτۭؒ H ͷ্ͷঢ়ଶϕΫτϧ |ψ⟩ Ͱهड़͞Ε
    Δ७ਮঢ়ଶʹ͍ͭͯɺԿΒ͔ͷঢ়ଶมԽΛҾ͖ىͨ͜͠৔߹ɺมԽޙͷঢ়ଶ͸ɺϢχλ
    Ϧʔԋࢉࢠ U ∈ L(H) Λ༻͍ͯɺ
    |ψ′⟩ = U|ψ⟩
    ͱද͞Ε·͢ɻ͜Ε͸ɺঢ়ଶີ౓ԋࢉࢠ ρ = |ψ⟩⟨ψ| Ͱදݱ͢Δͱɺ
    ρ′ = UρU† (8)
    ͱͳΓ·͢ɻΞϯαϯϒϧ {p(x), |ψx
    ⟩} ʹରԠ͢Δࠞ߹ঢ়ଶ
    ρ =

    x
    p(x)|ψx
    ⟩⟨ψx
    |
    ͷ৔߹͸ɺͦΕͧΕͷঢ়ଶ |ψx
    ⟩ ͕ಉҰͷϢχλϦʔԋࢉࢠ U Ͱม׵͞ΕΔ͜ͱ͔Βɺม
    Խޙͷঢ়ଶີ౓ԋࢉࢠ͸ɺ΍͸Γɺ(8) Ͱ༩͑ΒΕ·͢ɻ
    ࣍ʹɺ͜ͷߟ͑ํΛ߹੒ܥʹద༻ͯ͠Έ·͢ɻࠓɺ؍ଌର৅ͷܥ A ͸ঢ়ଶີ౓ԋࢉࢠ
    ρA
    ∈ L(HA
    ) Ͱද͞ΕΔঢ়ଶʹ͋Δͱ͠·͢ɻ͜Εʹɺ؀ڥܥ E ͷ७ਮঢ়ଶ |0E
    ⟩ ∈ HE
    *2 ౷ܭྗֶʹ͓͍ͯɺ౷ܭతͳΞϯαϯϒϧ͕ੜ͡ΔཁҼΛ͜ͷΑ͏ͳྔࢠྗֶతͳݱ৅ͱͯ͠هड़͠Α
    ͏ͱߟ͑Δݚڀऀ΋͍ΔΑ͏Ͱ͢ɻ

    View full-size slide

  10. 10
    Λ݁߹͢Δͱɺঢ়ଶີ౓ԋࢉࢠ
    ρA
    ⊗ |0E
    ⟩⟨0E
    | ∈ L(HA
    ⊗ HE
    )
    Ͱهड़͞ΕΔঢ়ଶ͕ಘΒΕ·͢ɻ͜͜Ͱɺܥ AE શମʹҰఆͷૢ࡞ΛߦͬͯɺϢχλϦʔ
    ԋࢉࢠ UAE
    ∈ L(HA
    ⊗ HE
    ) Ͱද͞ΕΔม׵Λࢪͨ͠ͱ͠·͢ɻ͜ͷ࣌ɺܥ A ͷΈΛ؍
    ଌ͢ΔΞϦε͔Βݟͨ৔߹ɺܥ A ͷঢ়ଶ͸ͲͷΑ͏ʹมԽ͢ΔͰ͠ΐ͏͔ʁ
    ม׵ޙͷܥ AE શମͷঢ়ଶ͸ɺUAE
    (ρA
    ⊗ |0E
    ⟩⟨0E
    |)U†
    AE
    ʹͳΓ·͕͢ɺΞϦε͸ܥ E
    Λ؍ଌ͢Δ͜ͱ͕Ͱ͖ͳ͍ͨΊɺܥ E ΛτϨʔεΞ΢τͨ͠΋ͷ͕ɺΞϦε͔Βݟͨܥ
    A ͷঢ়ଶ ρ′
    A
    ʹͳΓ·͢ɻ͜Ε͸ɺ࣍ͷΑ͏ʹ·ͱΊΔ͜ͱ͕Ͱ͖·͢ɻ
    ρ′
    A
    = TrE
    {
    UAE
    (ρA
    ⊗ |0E
    ⟩⟨0E
    |)U†
    AE
    }
    =

    i
    ⟨iE
    |UAE
    |0E
    ⟩ρA
    ⟨0E
    |U†
    AE
    |iE

    =

    Ki
    ρA
    K†
    i
    ͜͜ʹɺKi
    ͸ɺKi
    = ⟨iE
    |UAE
    |0E
    ⟩ ∈ L(HA
    ) Ͱఆٛ͞ΕΔԋࢉࢠͰɺ࣍ͷ׬શܥͷ৚
    ݅Λຬͨ͠·͢ɻ

    i
    K†
    i
    Ki
    =

    i
    ⟨0E
    |U†
    AE
    |iE
    ⟩⟨iE
    |UAE
    |0E

    = ⟨0E
    |U†
    AE
    UAE
    |0E
    ⟩ = I
    ྔࢠ৘ใཧ࿦Ͱ͸ɺ͜ͷߟ͑ํΛΑΓҰൠԽͯ͠ɺ

    i
    K†
    i
    Ki
    = I
    Λຬͨ͢೚ҙͷԋࢉࢠͷ૊ {Ki
    } Λ༻͍ͯɺ
    ρ′ =

    i
    Ki
    ρK†
    ͱද͞ΕΔঢ়ଶมԽΛҾ͖ى͜͢ʮ࢓૊ΈʯΛྔࢠνϟωϧͱݺͼ·͢ɻ͜ͷ৔߹ɺKi
    ͸
    ҟͳΔώϧϕϧτۭؒɺͨͱ͑͹ɺHA
    ͱ HB
    ͷؒͷઢܗࣸ૾ L(HA
    , HB
    ) Ͱ΋ߏ͍·
    ͤΜɻ΋͏গ͠۩ମతʹݴ͏ͱɺΞϦε͕ॴ༗͢Δܥ A ͷྔࢠঢ়ଶ ρA
    ∈ L(HA
    ) ΛԿΒ
    ͔ͷํ๏ͰϘϒʹૹ෇ͨ͠ͱͯ͠ɺϘϒ͕ܥ B ͱͯ͠ड͚औΔྔࢠঢ়ଶΛ ρB
    ∈ L(HB
    )
    ͱ͠·͢*3ɻ͜ͷ࣌ɺϘϒ͕ड͚औΔঢ়ଶ͸ૹ෇ʹ༻͍ͨํ๏ʹΑͬͯมΘΓ·͕͢ɺ͜
    *3 ͜͜Ͱݴ͏ʮૹ෇ʯ͸ɺ෺ཧతʹҠૹ͢Δͱ͍͏͚ͩͰ͸ͳ͘ɺྔࢠྗֶΛ༻͍ͨ৴߸ॲཧʹΑͬͯɺΞ
    Ϧεͷखݩʹ͋Δྔࢠঢ়ଶ ρA
    ʹج͍ͮͯɺϘϒͷखݩʹ৽͍͠ྔࢠঢ়ଶ ρB
    Λߏ੒͢ΔΑ͏ͳ৔߹΋
    ؚΈ·͢ɻ

    View full-size slide

  11. 0.2 લఏ஌ࣝͷ੔ཧ 11
    ͷૹ෇ϓϩηεΛ L(HA
    , HB
    ) ʹଐ͢Δԋࢉࢠͷ૊ {Ki
    } Λ༻͍ͯɺ
    ρB
    =

    i
    Ki
    ρA
    K†
    ͱද͢΋ͷͱߟ͍͑ͯͩ͘͞ɻ͜ͷΑ͏ʹͯ͠ྔࢠνϟωϧΛఆٛ͢Δԋࢉࢠͷ૊ {Ki
    }
    Λ Klaus ԋࢉࢠͱݺͼ·͢ɻ
    ·ͨɺ೚ҙͷ Klaus ԋࢉࢠ Ki
    ∈ L(HA
    , HB
    ) ͷ૊͕༩͑ΒΕͨ࣌ʹɺ͜ΕʹରԠ͢Δ
    ෺ཧతͳૢ࡞Λ؀ڥܥ E Λ༻͍ͯ࠶ߏ੒͢Δ͜ͱ͕Ͱ͖·͢ɻ۩ମతʹ͸ɺ
    U =

    i
    Ki
    ⊗ |iE
    ⟩ ∈ L(HA
    , HBE
    )
    Ͱఆٛ͞ΕΔઢܗԋࢉࢠΛߟ͑Δͱɺ͜Ε͸ɺ࣍ͷੑ࣭Λຬͨ͠·͢ɻ
    U†U =
    (

    i
    K†
    i
    ⊗ ⟨iE
    |
    ) 


    j
    Kj
    ⊗ |jE



    =

    i,j
    K†
    i
    Kj
    δij
    =

    i
    K†
    i
    Ki
    = I
    ͜ͷ৚݅Λຬͨ͢ԋࢉࢠΛҰൠʹɺΞΠιϝτϦʔԋࢉࢠͱݺͼ·͢ɻ͜Ε͸ɺϢχλ
    ϦʔԋࢉࢠΛҟͳΔώϧϕϧτۭؒͷؒͷઢܗࣸ૾ʹ֦ுͨ͠΋ͷͰɺࠓͷ৔߹ɺU ͸ɺ
    ܥ A ͔Βܥ BE ΁ͷࣗવͳঢ়ଶมԽΛද͠·͢ɻͦ͜Ͱɺঢ়ଶີ౓ԋࢉࢠ ρA
    Ͱද͞Ε
    Δܥ A ͷঢ়ଶΛ্هͷΞΠιϝτϦʔԋࢉࢠͰม׵ͨ͠ޙʹɺܥ E ΛτϨʔεΞ΢τ͠
    ͯɺܥ A ͔Βܥ B ΁ͷঢ়ଶมԽΛٻΊͯΈ·͢ɻ
    ρB
    = TrE
    (UρA
    U†)
    = TrE




    i,j
    (
    Ki
    ⊗ |iE

    )
    ρA
    (
    K†
    j
    ⊗ ⟨jE
    |
    )



    =

    i,j
    Ki
    ρA
    K†
    j
    δij
    =

    i
    Ki
    ρA
    K†
    i
    ͜ͷ݁Ռ͸ɺKlaus ԋࢉࢠ Ki
    Λ༻͍ͨྔࢠνϟωϧʹΑΔม׵ͱҰக͍ͯ͠·͢ɻ͜
    ͷΑ͏ʹɺ೚ҙͷྔࢠνϟωϧ͸ɺ؀ڥܥ E ΛؚΉࣗવͳঢ়ଶมԽͱ࣮ͯ͠૷͢Δ͜ͱ
    ͕Ͱ͖ΔͷͰ͢ɻ

    View full-size slide

  12. 12
    0.3 ྔࢠΤϯτϩϐʔ
    ֬཰ม਺ X ͕཭ࢄత֬཰෼෍ p(x) ʹै͏࣌ɺ֬཰ม਺ X ͷݹయΤϯτϩϐʔʢγϟ
    ϊϯΤϯτϩϐʔʣ͸ɺ
    H(X) = −

    x
    p(x) log p(x)
    Ͱఆٛ͞Εͯɺ֬཰෼෍ p(x) ʹΑΒͣ H(X) ≥ 0 Λຬͨ͠·͢ɻ·ͨɺH(X) = 0 ͕
    ੒ཱ͢Δͷ͸ɺಛఆͷ x ʹ͍ͭͯͷΈ p(x) = 1 ͱͳΔ৔߹ʹݶΓ·͢ɻ͜Εͱಉ༷ʹɺ
    Ξϯαϯϒϧ {p(x), |ψx
    ⟩} ͔ΒͳΔࠞ߹ঢ়ଶͷྔࢠܥʹ͍ͭͯɺ౷ܭతͳෆ֬ఆੑΛද
    ͢ྔͱͯ͠ɺྔࢠΤϯτϩϐʔʢϊΠϚϯΤϯτϩϐʔʣΛఆٛ͠·͢ɻͨͩ͠ɺྔࢠܥ
    ͷ৔߹͸ɺ
    ʮ0.2.1 ঢ়ଶີ౓ԋࢉࢠʹΑΔྔࢠঢ়ଶͷهड़ʯͰݟͨΑ͏ʹɺෳ਺ͷΞϯαϯ
    ϒϧ͕ಉҰͷঢ়ଶີ౓ԋࢉࢠʹରԠ͢Δ͜ͱ͔Βɺਖ਼ن௚ަجఈͰల։ͨ͠ࡍͷ෼෍Λ༻
    ͍·͢ɻ۩ମతʹ͸ɺঢ়ଶີ౓ԋࢉࢠ
    ρ =

    x
    p(x)|ψx
    ⟩⟨ψx
    |
    ΛεϖΫτϧ෼ղͯ͠ɺ
    ρ =

    i
    pi
    |i⟩⟨i| (9)
    ͱදͨ͠ࡍͷݻ༗஋ pi
    Λ༻͍ͯɺ͜ͷܥ A ͷྔࢠΤϯτϩϐʔʢQuantum entropyʣΛ
    ࣍Ͱఆٛ͠·͢ɻ
    H(A) = −

    i
    pi
    log pi
    (10)
    લड़ͷΑ͏ʹɺpi
    ͸֬཰෼෍ͱͯ͠ͷ৚݅Λຬ͍ͨͯ͠ΔͷͰɺݹయΤϯτϩϐʔͱಉ
    ༷ʹɺ೚ҙͷঢ়ଶີ౓ԋࢉࢠ ρ ʹରͯ͠ H(A) ≥ 0 ͕੒ཱ͠·͢ɻͦͯ͠ɺ͜ͷఆ͔ٛ
    Β໌Β͔ͳΑ͏ʹɺ७ਮঢ়ଶͷྔࢠΤϯτϩϐʔ͸ 0 ʹͳΓ·͢ɻ
    ·ͨɺ(9) ͷঢ়ଶີ౓ԋࢉࢠ ρ ʹ͍ͭͯɺͦͷର਺ log ρ Λ
    log ρ =

    i
    log pi
    |i⟩⟨i|
    Ͱఆٛ͢Δͱɺ(10) ͷఆٛ͸࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ
    H(A) = −Tr (ρ log ρ)
    ͪ͜Β͸ɺεϖΫτϧ෼ղͷදهʹґଘ͠ͳ͍ͱ͍͏ར఺͕͋Γ·͕͢ɺ࣮ࡍͷܭࢉΛ
    ߦ͏ࡍ͸ɺ(10) ͷදࣜͷํ͕ศརͳ͜ͱ΋͋Γ·͢ɻͳ͓ɺҰൠʹɺ೚ҙͷ 1 ม਺ؔ਺ f

    View full-size slide

  13. 0.3 ྔࢠΤϯτϩϐʔ 13
    ͱΤϧϛʔτԋࢉࢠ H ʹ͍ͭͯɺf(H) ͷ஋͸ɺH ͷεϖΫτϧ෼ղ
    H =

    i
    xi
    |i⟩⟨i|
    Λ༻͍ͯɺ
    f(H) =

    i
    f(xi
    )|i⟩⟨i|
    Ͱఆٛ͞Ε·͢ɻ
    ଓ͍ͯɺ߹੒ܥͷྔࢠΤϯτϩϐʔʹ͍ͭͯߟ͑·͢ɻ߹੒ܥ AB ͷঢ়ଶ͕ঢ়ଶີ౓
    ԋࢉࢠ ρAB
    Ͱද͞ΕΔͱ͖ɺܥ AB ͷྔࢠΤϯτϩϐʔ͸ɺ
    H(AB) = −Tr (ρAB
    log ρAB
    )
    Ͱ༩͑ΒΕ·͢ɻ͞Βʹɺ෦෼ܥ AɺB ͷঢ়ଶີ౓ԋࢉࢠ͸ɺͦΕͧΕɺ
    ρA
    = TrB
    ρAB
    ρB
    = TrA
    ρAB
    Ͱ༩͑ΒΕͯɺܥ Aɺܥ B ͷྔࢠΤϯτϩϐʔ͸ɺ
    H(A) = −Tr (ρA
    log ρA
    )
    H(B) = −Tr (ρB
    log ρB
    )
    Ͱܭࢉ͞Ε·͢ɻݹయΤϯτϩϐʔͷੈքͰݴ͏ͱɺ͜Ε͸ɺ2 ͭͷ֬཰ม਺ X, Y ʹର
    ͢Δಉ࣌෼෍ p(x, y) ͷΤϯτϩϐʔ H(X, Y ) ͱɺͦΕͧΕͷ֬཰ม਺ʹର͢Δपล෼
    ෍ p(x), p(y) ͷΤϯτϩϐʔ H(X), H(Y ) Λߟ͍͑ͯΔ͜ͱʹ͋ͨΓ·͢ɻ
    ͔͠͠ͳ͕ΒɺݹయΤϯτϩϐʔͱͷܾఆతͳҧ͍ͱͯ͠ɺ෦෼ܥͷྔࢠΤϯτϩϐʔ
    ͸ɺܥશମͷྔࢠΤϯτϩϐʔΑΓ΋খ͍͞ͱ͸ݶΒͳ͍ɺͱ͍͏఺͕͋Γ·͢ɻͨͱ͑
    ͹ɺݹయΤϯτϩϐʔͷ৔߹ɺ৚݅෇͖Τϯτϩϐʔ H(Y | X) = H(X, Y ) − H(X) ʹ
    ͍ͭͯɺ
    H(Y | X) ≥ 0
    ͕੒Γཱͭ͜ͱ͔Βɺৗʹɺ
    H(X) ≤ H(X, Y )
    ͕੒Γཱͪ·͢ɻͭ·Γɺपล෼෍Λͱͬͯʮ؍ଌൣғʯΛݶఆ͢Ε͹ɺ౷ܭతͳෆ֬ఆ
    ੑ͸ඞͣݮগ͠·͢ɻҰํɺྔࢠܥͷ৔߹ɺ͜ͷΑ͏ͳؔ܎͸ඞͣ͠΋੒Γཱͪ·ͤΜɻ
    ͨͱ͑͹ɺ߹੒ܥ AB ͕७ਮঢ়ଶͰ͋Ε͹ɺH(AB) = 0 ͕੒Γཱͪ·͕͢ɺ෦෼ܥ Aɺ

    View full-size slide

  14. 14
    B ͸Ұൠʹ͸ࠞ߹ঢ়ଶͱͳΔͷͰɺH(A) > 0ɺ΋͘͠͸ɺH(B) > 0 ͱͳΔՄೳੑ͕͋
    Γ·͢ɻͭ·Γɺྔࢠܥʹ͓͍ͯ͸ɺ؍ଌൣғΛݶఆ͢Δ͜ͱͰɺ౷ܭతͳෆ֬ఆੑ͕૿
    Ճ͢ΔՄೳੑ͕͋ΔͷͰ͢ɻ
    ͞Βʹɺܥ AB ͕७ਮঢ়ଶͷ৔߹ɺ෦෼ܥ AɺB ͷྔࢠΤϯτϩϐʔ͕Ұக͢Δͱ͍
    ͏ɺஶ͍͠ੑ࣭͕͋Γ·͢ɻ͜Ε͸ɺ࣍ͷఆཧͷূ໌ʹ͋ΔΑ͏ʹɺ߹੒ܥͷঢ়ଶϕΫτ
    ϧʹγϡϛοτ෼ղΛద༻͢Δ͜ͱͰ༰қʹࣔ͢͜ͱ͕Ͱ͖·͢ɻ
    ఆཧ 1 ߹੒ܥ AB ͕७ਮঢ়ଶͷ࣌ɺܥ ABɺ͓Αͼɺ෦෼ܥ AɺB ͷྔࢠΤϯτϩϐʔ
    ʹ͍ͭͯɺ࣍ͷؔ܎͕੒Γཱͭɻ
    H(AB) = 0, H(A) = H(B)
    [ূ໌] ܥ AB ͸७ਮঢ়ଶͰ͋Δ͜ͱ͔ΒɺH(AB) = 0 ͸ࣗ໌ʹ੒Γཱͭɻ࣍ʹɺܥ AB
    ͷঢ়ଶϕΫτϧ |ψAB
    ⟩ Λγϡϛοτ෼ղͨ͠΋ͷΛ
    |ψAB
    ⟩ =

    i

    pi
    |iA
    ⟩ ⊗ |iB

    ͱ͢Δɻ͜ͷ࣌ɺܥ A ͷঢ়ଶີ౓ԋࢉࢠΛܭࢉ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕΔɻ
    ρA
    = TrB
    (|ψAB
    ⟩⟨ψAB
    |)
    = TrB
    (

    i,j

    pi

    pj
    |iA
    ⟩⟨jA
    | ⊗ |iB
    ⟩⟨jB
    |)
    =

    k

    i,j

    pi

    pj
    |iA
    ⟩⟨jA
    |⟨kB
    |iB
    ⟩⟨jB
    |kB

    =

    k

    i,j

    pi

    pj
    |iA
    ⟩⟨jA
    |δik
    δjk
    =

    i
    pi
    |iA
    ⟩⟨iA
    |
    ಉ༷ʹɺܥ B ͷঢ়ଶີ౓ԋࢉࢠΛܭࢉ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕΔɻ
    ρB
    =

    i
    pi
    |iB
    ⟩⟨iB
    |
    ͕ͨͬͯ͠ɺܥ A ͱܥ B ͸ɺಉҰͷ֬཰෼෍ pi
    Λݻ༗஋ͱ͢ΔεϖΫτϧ෼ղΛ࣋ͬ
    ͓ͯΓɺ͜ΕΑΓɺH(A) = H(B) ͕੒Γཱͭɻ ˙
    ಉ༷ͷٞ࿦͸ɺ3 ͭҎ্ͷܥ͔ΒͳΔ߹੒ܥʹ͍ͭͯ΋੒Γཱͪ·͢ɻͨͱ͑͹ɺ߹੒
    ܥ ABC ͕७ਮঢ়ଶͰ͋Ε͹ɺܥ A ͱܥ BCɺ΋͘͠͸ɺܥ B ͱܥ AC ͳͲɺෳ਺ͷ૊
    Έ߹ΘͤͰ 2 ͭͷܥʹ෼ׂ͢Δ͜ͱ͕Ͱ͖·͢ɻ͍ͣΕͷ৔߹΋ɺ෼ׂͨ͠ 2 ͭͷܥͷΤ

    View full-size slide

  15. 0.3 ྔࢠΤϯτϩϐʔ 15
    ϯτϩϐʔ͸Ұகͯ͠ɺ࣍ͷؔ܎͕ͦΕͧΕ੒Γཱͪ·͢ɻ
    H(A) = H(BC)
    H(B) = H(AC)
    H(C) = H(AB)
    ଓ͍ͯɺྔࢠܥʹ͓͚Δɺ
    ʮಠཱͳ֬཰ม਺ʯʹ૬౰͢Δ֓೦Λಋೖ͠·͢ɻҰൠʹɺ2
    ͭͷ֬཰ม਺ X, Y ͕ಠཱͰ͋Δ৔߹ɺX ͱ Y ͷಉ࣌෼෍ p(x, y) ʹ͍ͭͯ
    p(x, y) = p(x)p(y)
    ͕੒Γཱͪɺ͜ͷ݁Ռɺಉ࣌෼෍ͷΤϯτϩϐʔ͸ɺͦΕͧΕͷपล෼෍ͷΤϯτϩϐʔ
    ͷ࿨ʹͳΓ·ͨ͠ɻ
    H(X, Y ) = H(X) + H(Y )
    ྔࢠܥʹ͓͍ͯɺ͜ΕʹରԠ͢Δͷ͕࣍ͷఆཧʹͳΓ·͢ɻ
    ఆཧ 2 ߹੒ܥ AB ͷঢ়ଶີ౓ԋࢉࢠ ρAB
    ͕ɺܥ A ͱܥ B ͷঢ়ଶີ౓ԋࢉࢠ ρA
    , ρB
    ͷςϯιϧੵͰ༩͑ΒΕΔɺ͢ͳΘͪɺ
    ρAB
    = ρA
    ⊗ ρB
    ͕੒Γཱͭ࣌ɺ
    H(AB) = H(A) + H(B)
    ͕੒ཱ͢Δɻ
    [ূ໌] ܥ A ͱܥ B ͷঢ়ଶີ౓ԋࢉࢠΛεϖΫτϧ෼ղͯ͠ɺ
    ρA
    =

    i
    pi
    |iA
    ⟩⟨iA
    |
    ρB
    =

    i
    qi
    |iB
    ⟩⟨iB
    |
    ͱද͢ͱɺܥ AB ͷঢ়ଶີ౓ԋࢉࢠ͸ɺ
    ρAB
    =

    i,j
    pi
    qj
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |
    Ͱ༩͑ΒΕΔɻ͜Ε͸ɺਖ਼ن௚ަجఈ |iA
    ⟩ ⊗ |jB
    ⟩ Λ༻͍ͨ ρAB
    ͷεϖΫτϧ෼ղΛ༩

    View full-size slide

  16. 16
    ͍͑ͯΔͷͰɺ͜ΕΑΓɺܥ AB ͷΤϯτϩϐʔ͸ɺ࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    H(AB) = −

    i,j
    pi
    qj
    log(pi
    qj
    )
    = −

    i,j
    pi
    qj
    log pi


    i,j
    pi
    qj
    log qj
    = −

    i
    pi
    log pi


    j
    qj
    log qj
    = H(A) + H(B)
    ˙
    ޙ΄Ͳɺ
    ʮ0.7 ͦͷଞͷੑ࣭ʯʹ͓͍ͯɺྔࢠΤϯτϩϐʔͷ Concavity Λূ໌͠·͢
    ͕ɺͦͷࡍʹɺ
    ʮݹయʕྔࢠܥʯͱݺ͹ΕΔಛघͳ߹੒ܥΛར༻͠·͢ɻ͜͜Ͱ͸ɺͦͷ४
    උͱͯ͠ɺݹయʕྔࢠܥʹ͍ͭͯઆ໌͓͖ͯ͠·͢ɻࠓɺܥ A ͷਖ਼ن௚ަجఈΛ |iA
    ⟩ ͱ
    ͯ͠ɺ७ਮঢ়ଶͷ૊ {|iA
    ⟩⟨iA
    |} Λ༻ҙ͠·͢ɻ͞Βʹɺ͜ΕʹରԠͯ͠ɺܥ B ʹ͓͚Δ
    ࠞ߹ঢ়ଶͷ૊ {ρBi
    } Λ༻ҙ͠·͢ɻͦͯ͠ɺ֬཰෼෍ pi
    ʹै͏֬཰ม਺ͷαϯϓϧΛऔ
    ಘͯ͠ɺͦͷ஋ʹԠͯ͡ɺࠞ߹ܥ AB ͷঢ়ଶ |iA
    ⟩⟨iA
    | ⊗ ρBi
    Λ༻ҙ͠·͢ɻ͜ͷΑ͏ʹ͠
    ͯಘΒΕΔΞϯαϯϒϧ {pi
    , |iA
    ⟩⟨iA
    | ⊗ ρBi
    } ͸ɺ࣍ͷঢ়ଶີ౓ԋࢉࢠͰهड़͞Ε·͢ɻ
    ρAB
    =

    i
    pi
    |iA
    ⟩⟨iA
    | ⊗ ρBi
    ͜ͷΑ͏ͳঢ়ଶΛݹయʕྔࢠܥͱݺͼ·͢ɻ͜ͷ࣌ɺܥ A ͸ɺਖ਼ن௚ަجఈΛ༻͍ͨ
    ࣹӨଌఆʹΑΓɺܥͷঢ়ଶΛਖ਼֬ʹ൑ఆ͢Δ͜ͱ͕Ͱ͖·͢ɻͭ·Γɺ֬཰ม਺͔ΒಘΒ
    Εͨαϯϓϧͷ஋Λਖ਼֬ʹ஌Δ͜ͱ͕Ͱ͖ΔͷͰɺݹయతͳ֬཰෼෍ pi
    ͱಉ౳ͷ৘ใྔ
    Λ࣋ͪ·͢ɻ࣮ࡍɺܥ B ʹ͍ͭͯτϨʔεΞ΢τͯ͠ɺܥ A ͷঢ়ଶີ౓ԋࢉࢠΛٻΊΔ
    ͱɺTrB
    ρBi
    = 1 Λ༻͍ͯɺ
    ρA
    =

    i
    pi
    |iA
    ⟩⟨iA
    | (11)
    ͱͳΓ·͢ͷͰɺࣹӨԋࢉࢠ Pi
    = |iA
    ⟩⟨iA
    | ʹΑΔࣹӨଌఆͰܥͷঢ়ଶ͕൑ผͰ͖·͢ɻ
    ·ͨɺ(11) ͷදࣜΑΓɺܥ A ͷྔࢠΤϯτϩϐʔ͸ɺ֬཰෼෍ pi
    ͷݹయΤϯτϩϐʔ
    ʹҰக͢Δ͜ͱ͕Θ͔Γ·͢ɻ
    H(ρA
    ) = −

    i
    pi
    log pi
    Ұํɺܥ A ʹ͍ͭͯτϨʔεΞ΢τͯ͠ɺܥ B ͷঢ়ଶີ౓ԋࢉࢠΛٻΊΔͱɺ࣍ͷΑ

    View full-size slide

  17. 0.4 ྔࢠ૬ରΤϯτϩϐʔ 17
    ͏ʹɺෳ਺ͷࠞ߹ঢ়ଶΛ͞Βʹࠞ߹ͨ͠ྔࢠܥ͕ಘΒΕ·͢ɻ
    ρB
    =

    i
    pi
    ρBi
    0.4 ྔࢠ૬ରΤϯτϩϐʔ
    ྔࢠΤϯτϩϐʔʹؔ࿈͢Δ֓೦ͱͯ͠ɺ͜ͷޙͷઅͰղઆ͢Δɺ৚݅෇͖ྔࢠΤϯτ
    ϩϐʔ΍ྔࢠ૬ޓ৘ใྔͳͲ͕͋Γ·͕͢ɺ͜ΕΒͷੑ࣭ͷଟ͘͸ɺྔࢠ૬ରΤϯτϩ
    ϐʔΛ༻͍ͯಋ͘͜ͱ͕Ͱ͖·͢ɻྔࢠ૬ରΤϯτϩϐʔ͸ɺݹయ৘ใཧ࿦ʹ͓͚Δ૬ର
    ΤϯτϩϐʔʢKL μΠόʔδΣϯεʣʹରԠ͢Δ΋ͷͰ͢ͷͰɺ·ͣ͸ɺKL μΠόʔ
    δΣϯεͷੑ࣭Λ֬ೝ͓͖ͯ͠·͢ɻ
    Ұൠʹɺ֬཰ม਺ X ʹର͢Δ 2 छྨͷ֬཰෼෍ p(x) ͱ q(x) ʹ͍ͭͯɺKL μΠόʔ
    δΣϯε͸࣍ࣜͰఆٛ͞Ε·͢ɻ
    D(p || q) =

    x
    p(x) log
    p(x)
    q(x)
    =

    x
    p(x) log p(x) −

    x
    p(x) log q(x) (12)
    ͜Ε͸ɺ೚ҙͷ p(x), q(x) ʹ͍ͭͯɺඇෛʢਖ਼ɺ·ͨ͸ɺ0ʣͷ஋ΛऔΓɺಛʹ 0 ʹͳ
    Δͷ͸ɺ2 ͭͷ֬཰෼෍͕Ұகͯ͠ɺp(x) = q(x) ͕੒Γཱͭ৔߹ʹݶΓ·͢ɻ
    D(p || q) ≥ 0
    D(p || q) = 0 ⇔ p(x) = q(x)
    ͜ΕΒͷؔ܎͔ΒɺKL μΠόʔδΣϯε͸ɺ֬཰෼෍ p(x) ͱ q(x) ͷྨࣅੑΛଌΔࢦ
    ඪͱͳΓ·͢ɻ
    ྔࢠ૬ରΤϯτϩϐʔʢQuantum relative entropyʣ͸ɺ(12) ͷදࣜΛྔࢠܥʹ֦ு
    ͨ͠΋ͷͰ͢ɻώϧϕϧτۭؒ H ্ͷ 2 छྨͷঢ়ଶີ౓ԋࢉࢠ ρ ͱ σ ʹରͯ͠ɺྔࢠ
    ૬ରΤϯτϩϐʔ D(ρ || σ) ͸ɺ࣍ࣜͰఆٛ͞Ε·͢ɻ
    D(ρ || σ) = Tr(ρ log ρ) − Tr(ρ log σ)
    ͦͯ͠ɺྔࢠ૬ରΤϯτϩϐʔ΋·ͨɺρ ͱ σ ͷྨࣅੑΛଌΔࢦඪͱͳ͓ͬͯΓɺ࣍ͷ
    ఆཧ͕੒Γཱͪ·͢ɻ

    View full-size slide

  18. 18
    ఆཧ 3 ೚ҙͷঢ়ଶີ౓ ρ, σ ʹରͯ͠ɺྔࢠ૬ରΤϯτϩϐʔ͸ඇෛ஋ΛͱΔɻ
    D(ρ || σ) ≥ 0
    ·ͨɺྔࢠ૬ରΤϯτϩϐʔ͕ 0 ʹͳΔͷ͸ɺρ = σ ͷ৔߹ʹݶΔɻ
    D(ρ || σ) = 0 ⇔ ρ = σ
    [ূ໌] ρ ͱ σ Λ࣍ͷΑ͏ʹεϖΫτϧ෼ղ͢Δɻ
    ρ =

    i
    pi
    |i⟩⟨i|
    σ =

    j
    qj
    |j⟩⟨j|
    ͜͜Ͱɺ{|i⟩} ͱ {|j⟩} ͸ɺҰൠʹ͸ҟͳΔਖ਼ن௚ަجఈͰ͋ΓɺҎ߱͸ɺఴࣈ i, j Ͱ
    ۠ผ͢Δ΋ͷͱ͢Δɻ͜ͷ࣌ɺྔࢠ૬ରΤϯτϩϐʔ͸ɺ࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    D(ρ || σ) =

    i
    pi
    log pi
    − Tr




    i,j
    pi
    |i⟩⟨i|(log qj
    )|j⟩⟨j|



    =

    i
    pi
    log pi


    i,j
    pi
    (log qj
    )⟨i | j⟩⟨j | i⟩
    =

    i
    pi

    log pi


    j
    Pij
    log qj

     (13)
    ͜͜ʹɺ
    Pij
    = ⟨i | j⟩⟨j | i⟩ = |⟨i | j⟩|2 (14)
    Ͱ͋ΓɺఆٛΑΓɺ࣍ͷੑ࣭Λຬͨ͢ɻ
    Pij
    ≥ 0 (15)

    i
    Pij
    = 1 (16)

    j
    Pij
    = 1 (17)
    ͜͜Ͱɺ೚ҙͷ i ʹ͍ͭͯɺ(17) ͕੒Γཱͭ͜ͱ͔Βɺର਺ؔ਺্͕ʹತͰ͋Δ͜ͱΛ
    ༻͍ͯɺJensen ͷෆ౳ࣜ

    j
    Pij
    log qj
    ≤ log



    j
    Pij
    qj

     (18)

    View full-size slide

  19. 0.4 ྔࢠ૬ରΤϯτϩϐʔ 19
    ͕੒Γཱͭɻ͞Βʹɺ
    ri
    =

    j
    Pij
    qj
    (19)
    ͱஔ͘ͱɺ(15)(16) ΑΓɺ
    ri
    ≥ 0
    ͓Αͼ ∑
    i
    ri
    =

    i,j
    Pij
    qj
    =

    j
    qj
    = 1
    ͕੒ΓཱͭͷͰɺri
    ͸֬཰෼෍ͷ৚݅Λຬ͍ͨͯ͠Δɻ
    ͜͜Ͱɺ(13) ʹ (18)(19) Λ୅ೖ͢Δͱɺ͕࣍ಘΒΕΔɻ
    D(ρ || σ) ≥

    i
    pi
    (log pi
    − log ri
    )
    ͜ͷ࠷ޙͷදࣜ͸ɺ֬཰෼෍ pi
    ͱ ri
    ͷݹయ૬ରΤϯτϩϐʔ D(p || r) ʹҰக͓ͯ͠
    Γɺݹయ૬ରΤϯτϩϐʔͷੑ࣭ΑΓɺ
    D(p || r) =

    i
    pi
    (log pi
    − log ri
    ) ≥ 0 (20)
    ͕੒Γཱͭɻ͜ΕͰɺ
    D(ρ || σ) ≥ 0
    ͕ࣔ͞Εͨɻ
    ͜͜Ͱɺ౳߸͕੒ཱ͢Δͷ͸ɺ(18) ͱ (20) ͷͦΕͧΕͰ౳߸͕੒ཱ͢Δ৔߹ʹҰக͢
    Δɻ·ͣɺ(18) ͷ౳߸͕੒ཱ͢Δͷ͸ɺ೚ҙͷ i ʹ͍ͭͯɺ͋Δͻͱͭͷ j ʹ͍ͭͯͷ
    ΈɺPij
    = 1 Ͱ͋Γɺͦͷଞͷ j ʹ͍ͭͯ Pij
    = 0 ͱͳΔ৔߹Ͱ͋ΔɻPij
    ͷఆٛ (14) ʹ
    ໭Δͱɺ͜Ε͸ɺ{|i⟩} ͱ {|j⟩} ͕ू߹ͱͯ͠Ұக͢Δ͜ͱΛҙຯ͓ͯ͠Γɺqi
    ͷॱংΛద
    ౰ʹೖΕସ͑ͯɺॱংΛؚΊͯҰக͢Δͱͯ͠΋ҰൠੑΛࣦΘͳ͍ɻ͜ͷ৔߹ɺPij
    = δij
    ͕੒Γཱͭɻ࣍ʹɺ(20) ͷ౳߸͕੒ཱ͢Δͷ͸ɺ֬཰෼෍ pi
    ͱ ri
    ͕Ұக͢Δ࣌Ͱ͋Δ
    ͕ɺ͜Ε͸ɺࠓͷ৔߹ɺPij
    = δij
    ͱ͍͏৚͔݅Βɺpi
    = qi
    Λҙຯ͓ͯ͠Γɺ͜ΕΑΓɺ
    ρ = σ ͕ಘΒΕΔɻ ˙
    ྔࢠ૬ରΤϯτϩϐʔ͕ඇෛͰ͋Δͱ͍͏ෆ౳ࣜ͸ɺKlein ͷෆ౳ࣜͱ΋ݺ͹Ε·͢ɻ
    ͜ͷޙͷઅͰݟΔΑ͏ʹɺKlein ͷෆ౳ࣜ͸ɺྔࢠΤϯτϩϐʔʹؔ࿈͢Δ͞·͟·ͳෆ
    ౳ࣜͷূ໌ʹར༻͞Ε·͢ɻͨͱ͑͹ɺ߹੒ܥ AB ͷঢ়ଶີ౓ԋࢉࢠ ρAB
    ͔ΒɺτϨʔ

    View full-size slide

  20. 20
    εΞ΢τʹΑΓɺܥ AɺB ͦΕͧΕͷঢ়ଶີ౓ԋࢉࢠΛٻΊ·͢ɻ
    ρA
    = TrB
    ρAB
    ρB
    = TrA
    ρAB
    ࣍ʹɺρAB
    ͱςϯιϧੵ ρA
    ⊗ ρB
    ͷྔࢠ૬ରΤϯτϩϐʔ D(ρAB
    || ρA
    ⊗ ρB
    ) Λ
    ܭࢉ͢Δͱɺܥ ABɺܥ Aɺܥ B ͦΕͧΕͷྔࢠΤϯτϩϐʔ͕ࣗવʹݱΕͯɺ࣍ͷ
    Subadditivity ͱݺ͹ΕΔؔ܎͕ࣔ͞Ε·͢ɻ
    ఆཧ 4 ߹੒ܥ AB ͱͦͷ෦෼ܥ AɺB ͷྔࢠΤϯτϩϐʔʹ͍ͭͯɺ࣍ͷෆ౳͕ࣜ੒
    Γཱͭɻ
    H(AB) ≤ H(A) + H(B)
    ౳߸͕੒ཱ͢Δͷ͸ɺܥ AB ͷঢ়ଶີ౓ԋࢉࢠ ρAB
    ͕ɺܥ AɺB ͷঢ়ଶີ౓ԋࢉࢠ
    ρA
    , ρB
    ͷςϯιϧੵʹҰக͢Δ৔߹ʹݶΔɻ
    H(AB) = H(A) + H(B) ⇔ ρAB
    = ρA
    ⊗ ρB
    [ূ໌] ͸͡Ίʹɺ࣍ͷؔ܎Λࣔ͢ɻ
    log(ρA
    ⊗ ρB
    ) = (log ρA
    ) ⊗ IB
    + IA
    ⊗ (log ρB
    ) (21)
    ࠓɺρA
    ͱ ρB
    ͷͦΕͧΕΛεϖΫτϧ෼ղͨ͠΋ͷΛ
    ρA
    =

    i
    pi
    |iA
    ⟩⟨iA
    | (22)
    ρB
    =

    j
    qj
    |jB
    ⟩⟨jB
    |
    ͱ͢ΔͱɺρA
    ⊗ ρB
    ͸࣍ͷΑ͏ʹද͞ΕΔɻ
    ρA
    ⊗ ρB
    =

    i,j
    pi
    qj
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |
    ͜Ε͸ɺρA
    ⊗ ρB
    Λਖ਼ن௚ަجఈ |iA
    ⟩ ⊗ |jB
    ⟩ Λ༻͍ͯεϖΫτϧ෼ղͨ͠΋ͷʹଞͳ
    Βͣɺ͜ΕΑΓɺ࣍ͷܭࢉ͕੒Γཱͭɻ
    log(ρA
    ⊗ ρB
    ) =

    i,j
    log(pi
    qj
    )|iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |
    =

    i,j
    log pi
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    | +

    i,j
    log pi
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |
    =

    i
    log pi
    |iA
    ⟩⟨iA
    | ⊗ IB
    + IA


    j
    log qj
    |jB
    ⟩⟨jB
    |
    = (log ρA
    ) ⊗ IB
    + IA
    ⊗ (log ρB
    )

    View full-size slide

  21. 0.4 ྔࢠ૬ରΤϯτϩϐʔ 21
    ͜ΕͰ (21) ͕ࣔ͞Εͨɻ͜ΕΛ༻͍ΔͱɺρAB
    ͱ ρA
    ⊗ ρB
    ͷྔࢠ૬ରΤϯτϩϐʔ
    ͸ɺ࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    D(ρAB
    || ρA
    ⊗ ρB
    ) = Tr(ρAB
    log ρAB
    ) − Tr {ρAB
    log(ρA
    ⊗ ρB
    )}
    = −H(AB) − Tr {ρAB
    (log ρA
    ) ⊗ IB
    }
    −Tr {ρAB
    IA
    ⊗ (log ρB
    )} (23)
    ࠷ޙͷදࣜͷୈ 2 ߲ͱୈ 3 ߲͸ɺ࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δɻ
    −Tr {ρAB
    (log ρA
    ) ⊗ IB
    } = H(A) (24)
    −Tr {ρAB
    IA
    ⊗ (log ρB
    )} = H(B) (25)
    ࣮ࡍɺρAB
    Λਖ਼ن௚ަجఈ |iA
    ⟩ ⊗ |jB
    ⟩ Λ༻͍ͯεϖΫτϧ෼ղͨ͠΋ͷΛ
    ρAB
    =

    i,j
    rij
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |
    ͱ͢Δͱɺ
    ρA
    = TrB
    ρAB
    =

    j
    (

    i
    rij
    )
    |iA
    ⟩⟨iA
    |
    ͕ಘΒΕͯɺ͜ΕΛ (22) ͱൺֱ͢Δͱɺ

    j
    rij
    = pi
    ͱͳΔ͜ͱ͕Θ͔Δɻ͜ΕΑΓɺ࣍ͷܭࢉ͕੒Γཱͭɻ
    −Tr {ρAB
    (log ρA
    ) ⊗ IB
    } = −Tr




    i,j
    rij
    |iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |

    i′
    (log pi′
    )|i′
    A
    ⟩⟨i′
    A
    | ⊗ IB



    = −Tr




    i,j
    rij
    (log pi
    )|iA
    ⟩⟨iA
    | ⊗ |jB
    ⟩⟨jB
    |



    = −

    i,j
    rij
    log pi
    = −

    i
    pi
    log pi
    = H(A)
    ͜ΕͰ (24) ͕ࣔ͞Εͨɻ(25) ʹ͍ͭͯ΋ಉ༷Ͱ͋Δɻ(23) ʹ (24)(25) Λ୅ೖ͢Δͱɺ
    ࠷ऴతʹ࣍ͷؔ܎͕ಘΒΕΔɻ
    D(ρAB
    || ρA
    ⊗ ρB
    ) = H(A) + H(B) − H(AB) (26)

    View full-size slide

  22. 22
    ͜Εʹ Klein ͷෆ౳ࣜΛద༻͢Δͱɺఆཧͷओு಺༰͕ಘΒΕΔɻ ˙
    ্هͷূ໌ͷ࠷ޙʹಘΒΕͨྔ H(A) + H(B) − H(AB) ͸ɺ
    ʮ0.6 ྔࢠ૬ޓ৘ใྔʯͰ
    આ໌͢Δɺྔࢠ૬ޓ৘ใྔ I(A ; B) ʹҰக͍ͯ͠·͢ɻͭ·ΓɺSubaditivity ͸ɺྔࢠ
    ૬ޓ৘ใྔ͕ඇෛͰ͋Δ͜ͱͱಉ஋ʹͳΓ·͢ɻ
    ͞Βʹ΋͏ҰͭɺKlein ͷෆ౳ࣜͷԠ༻ྫͱͯ͠ɺࣹӨଌఆʹΑͬͯྔࢠΤϯτϩϐʔ
    ͕૿Ճ͢Δͱ͍͏ࣄ࣮Λࣔ͠·͢ɻ·ͣɺྔࢠΤϯτϩϐʔ͸ɺ౷ܭతͳෆ֬ఆੑΛଌΔ
    ྔͰ͢ͷͰɺ७ਮঢ়ଶʹରͯ͠ɺ؍ଌʹΑͬͯ౷ܭతͳෆ֬ఆੑΛಋೖ͢ΔͱɺྔࢠΤϯ
    τϩϐʔ͸૿Ճ͠·͢ɻͨͱ͑͹ɺ७ਮঢ়ଶ ρ = |ψ⟩⟨ψ| ʹࣹӨଌఆ Pi
    Λࢪ͢ͱɺଌఆ
    ݁ՌΛ؍ଌ͠ͳ͍৔߹ɺଌఆޙͷঢ়ଶ͸ɺ࣍ͷࠞ߹ঢ়ଶͱͳΓ·͢ɻ
    ρ′ =

    i
    Pi
    ρPi
    ७ਮঢ়ଶͷྔࢠΤϯτϩϐʔ͸ 0 Ͱ͋Δ͜ͱ͔Βɺ͜ͷ৔߹͸ɺࣗ໌ʹ
    H(ρ′) ≥ H(ρ) (27)
    ͕੒Γཱͪ·͢ɻҰൠʹɺ७ਮঢ়ଶʹݶఆ͠ͳ͍ɺ೚ҙͷঢ়ଶ ρ ʹ͍ͭͯɺ(27) ͕੒Γ
    ཱͪ·͢ɻ
    ఆཧ 5 Pi
    ͸ࣹӨଌఆԋࢉࢠͷ׬શܥͰ͋Γɺ࣍ͷؔ܎Λຬͨ͢΋ͷͱ͢Δɻ
    P2
    i
    = Pi
    ,

    i
    Pi
    = 1
    ͜ͷ࣌ɺ೚ҙͷঢ়ଶີ౓ԋࢉࢠ ρ ʹରͯ͠ɺ
    ρ′ =

    i
    Pi
    ρPi
    ͱͯ͠ɺ
    H(ρ′) ≥ H(ρ)
    ͕੒Γཱͭɻ
    [ূ໌] ·ͣɺP2
    i
    = 1 ΑΓɺ
    ρ′Pi
    = Pi
    ρ′ = ρ′
    ͕੒Γཱͪɺρ′ ͱ Pi
    ͸Մ׵ʹͳΔɻ͜ΕΑΓɺH(ρ′) = −Tr (ρ′ log ρ′) Λ࣍ͷΑ͏ʹม

    View full-size slide

  23. 0.5 ৚݅෇͖ྔࢠΤϯτϩϐʔ 23
    ܗ͢Δ͜ͱ͕Ͱ͖Δɻ
    Tr (ρ′ log ρ′) = Tr
    (

    i
    Pi
    ρPi
    log ρ′
    )
    = Tr
    (

    i
    Pi
    ρ(log ρ′)Pi
    )
    = Tr
    (

    i
    P2
    i
    ρ log ρ′
    )
    = Tr
    (

    i
    Pi
    ρ log ρ′
    )
    = Tr (ρ log ρ′)
    2 ͭ໨ͷ౳߸Ͱ͸ɺρ′ ͱ Pi
    ͷՄ׵ੑΛ༻͍͓ͯΓɺ3 ͭ໨ͷ౳߸Ͱ͸ɺτϨʔεͷ८
    ճੑΛ༻͍͍ͯΔɻ4 ͭ໨ͷ౳߸͸ɺP2
    i
    = Pi
    ΑΓ੒Γཱͪɺ࠷ޙʹ

    i
    Pi
    = 1 Λ༻͍
    ͍ͯΔɻ͜ΕΑΓɺ࣍ͷؔ܎͕੒Γཱͭɻ
    H(ρ′) = −Tr (ρ log ρ′) (28)
    ࣍ʹɺρ′ ͱ ρ ʹ͍ͭͯɺKlein ͷෆ౳ࣜΛద༻͢Δͱɺ(28) ͷؔ܎Λ༻͍ͯɺ͕࣍੒
    Γཱͭɻ
    0 ≤ D(ρ || ρ′) = Tr (ρ log ρ − ρ log ρ′) = −H(ρ) + H(ρ′)
    ͜ΕͰɺ
    H(ρ′) ≥ H(ρ)
    ͕ࣔ͞Εͨɻ ˙
    ͳ͓ɺࣹӨଌఆԋࢉࢠΛ Pi
    = |i⟩⟨i| ͱͯ͠ɺ|i⟩ Λجఈͱ͢ΔߦྻදࣔΛ༻͍ͨ৔߹ɺ
    ଌఆޙͷঢ়ଶີ౓ԋࢉࢠ ρ′ ͷߦྻ੒෼͸ɺଌఆલͷঢ়ଶ ρ ͷߦྻද͓͍ࣔͯɺඇର֯੒
    ෼Λ͢΂ͯ 0 ʹͨ͠΋ͷʹҰக͠·͢ɻͭ·Γɺ͋Δঢ়ଶີ౓ԋࢉࢠ ρ ͷߦྻදࣔʹ͓
    ͍ͯɺͦͷඇର֯੒෼Λ 0 ʹͨ͠ঢ়ଶີ౓ԋࢉࢠ ρ′ Λߟ͑Δͱɺ͔ͳΒͣɺ
    H(ρ′) ≥ H(ρ)
    ͕੒Γཱͪ·͢ɻ
    0.5 ৚݅෇͖ྔࢠΤϯτϩϐʔ
    ͜͜Ͱ͸·ͣɺݹయΤϯτϩϐʔʹ͓͚Δ৚݅෇͖ΤϯτϩϐʔΛઆ໌͠·͢ɻ·ͣɺ
    2 ͭͷ֬཰ม਺ X ͱ Y ͷಉ࣌෼෍Λ p(x, y) ͱ͢Δ࣌ɺY Λ͋Δ஋ y ʹݻఆͨ͠ࡍͷ
    X ͷ৚݅෇͖֬཰෼෍ p(x | Y = y) ʹ͍ͭͯΤϯτϩϐʔΛܭࢉ͠·͢ɻ
    H(X | Y = y) = −

    x
    p(x | y) log p(x | y)

    View full-size slide

  24. 24
    ࣍ʹ Y ʹ͍ͭͯͷظ଴஋ΛऔΓ·͢ɻ͜Ε͕৚݅෇͖ΤϯτϩϐʔͰ͢ɻ
    H(X | Y ) =

    y
    p(y)H(X | Y = y)
    = −

    x,y
    p(y)p(x | y) log p(x | y) (29)
    ৚݅෇͖֬཰෼෍ͷެࣜ p(x | y) =
    p(x, y)
    p(y)
    Λ༻͍Δͱɺ͜Ε͸࣍ͷΑ͏ʹॻ͖௚͢͜
    ͱ͕Ͱ͖·͢ɻ
    H(X | Y ) = −

    x,y
    p(x, y) log p(x, y) +

    x,y
    p(x, y) log p(y)
    = −

    x,y
    p(x, y) log p(x, y) +

    y
    p(y) log p(y)
    = H(X, Y ) − H(Y ) (30)
    ৚݅෇͖Τϯτϩϐʔ͸ɺH(X | Y = y) ≥ 0 ͷظ଴஋Ͱ͢ͷͰɺH(X | Y ) ≥ 0 ͕੒
    Γཱͪ·͢ɻ
    ࣍ʹɺ৚݅෇͖ྔࢠΤϯτϩϐʔΛఆٛ͠·͕͢ɺྔࢠܥʹ͓͍ͯ͸ɺಉ࣌෼෍ɺ͋
    Δ͍͸ɺ৚݅෇͖֬཰෼෍ͷ֓೦͕ͳ͍ͨΊɺ(29) ʹ૬౰͢ΔఆٛΛ༩͑Δ͜ͱ͕Ͱ͖
    ·ͤΜɻͦ͜Ͱɺ(30) ͷؔ܎ʹ஫໨ͯ͠ɺ࣍Ͱ৚݅෇͖ྔࢠΤϯτϩϐʔʢConditional
    quantum entropyʣΛఆٛ͠·͢ɻ
    H(A | B) = H(AB) − H(B) (31)
    ʮ0.3 ྔࢠΤϯτϩϐʔʯͰ͸ɺ߹੒ܥ AB ͕७ਮঢ়ଶͷ৔߹ͳͲΛߟ͑Δͱɺඞͣ͠
    ΋ H(AB) ≥ H(B) ͱ͸ͳΒͳ͍͜ͱΛࢦఠ͠·ͨ͠ɻ͜Ε͸ɺ৚݅෇͖Τϯτϩϐʔ
    ͷݴ༿Ͱݴ͏ͱɺ৚݅෇͖ྔࢠΤϯτϩϐʔ͸ɺෛͷ஋ΛͱΔՄೳੑ͕͋Δ͜ͱΛࣔͯ͠
    ͍·͢ɻ
    ͳ͓ɺݹయΤϯτϩϐʔͰ͸ɺ৚݅෇͖Τϯτϩϐʔʹ͍ͭͯɺ࣍ͷνΣΠϯϧʔϧ͕
    ੒Γཱͭ͜ͱ͕஌ΒΕ͍ͯ·͢ɻ
    H(X1
    , X2
    , · · · , Xn
    ) = H(X1
    ) + H(X2
    | X1
    ) + H(X3
    | X1
    , X2
    ) +
    · · · + H(Xn
    | X1
    , · · · , Xn−1
    )
    ͜Ε͸ɺX1
    , · · · , Xn
    ͷಉ࣌෼෍ʹؔ͢ΔΤϯτϩϐʔ͸ɺ
    ʮ͢Ͱʹ༻͍ͨ΋ͷΛ৚݅ʹ
    ෇͚Ճ͍͑ͯ͘ʯͱ͍͏ϧʔϧͷ΋ͱʹɺX1
    ͔ΒॱʹͦΕͧΕͷΤϯτϩϐʔΛՃ͑Δ
    ͜ͱͰಘΒΕΔ͜ͱΛ͓ࣔͯ͠Γɺ(30) ͷؔ܎͔Βؼೲతʹಋ͘͜ͱ͕Ͱ͖·͢ɻྔࢠ
    Τϯτϩϐʔʹ͓͍ͯ΋ɺ(31) ͷఆ͔ٛΒɺಉ༷ͷؔ܎Λࣔ͢͜ͱ͕Ͱ͖·͢ɻ

    View full-size slide

  25. 0.6 ྔࢠ૬ޓ৘ใྔ 25
    ఆཧ 6 ߹੒ܥ X1
    X2
    · · · Xn
    ͷྔࢠΤϯτϩϐʔʹ͍ͭͯɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    H(X1
    X2
    · · · Xn
    ) = H(X1
    ) + H(X2
    | X1
    ) + H(X3
    | X1
    X2
    ) +
    · · · + H(Xn
    | X1
    · · · Xn−1
    )
    [ূ໌] n = 1 ͷ৔߹͸ɺ৚݅෇͖ྔࢠΤϯτϩϐʔͷఆٛ
    H(X2
    | X1
    ) = H(X1
    X2
    ) − H(X1
    )
    ΑΓɺࣗ໌ʹ੒Γཱͭɻ࣍ʹɺn = k ͷ৔߹ʹ੒ཱ͢ΔͱԾఆͯ͠ɺn = k + 1 ͷ৔߹Λ
    ߟ͑Δɻ߹੒ܥ X1
    · · · Xk+1
    Λ X1
    , X2
    , · · · , Xk
    Xk+1
    ͷ k ݸͷܥʹ෼ׂ͢ΔͱɺԾఆΑ
    Γɺ͕࣍੒Γཱͭɻ
    H(X1
    X2
    · · · Xk
    ) = H(X1
    ) + H(X2
    | X1
    ) + · · · + H(Xk
    Xk+1
    | X1
    · · · Xk−1
    )
    ͕ͨͬͯ͠ɺ
    H(Xk
    Xk+1
    | X1
    · · · Xk−1
    ) = H(Xk
    | X1
    · · · Xk−1
    ) + H(Xk+1
    | X1
    · · · Xk
    )
    ͕੒Γཱͯ͹Α͍ɻ͜Ε͸ɺXk
    = B, Xk+1
    = C, X1
    · · · Xk−1
    = A ͱஔ͘ͱɺ
    H(BC | A) = H(B | A) + H(C | AB)
    ͱಉ஋Ͱ͋Γɺ࣍ͷܭࢉ͔Β੒Γཱͭ͜ͱ͕Θ͔Δɻ
    H(B | A) + H(C | AB) = H(AB) − H(A) + H(ABC) − H(AB)
    = H(ABC) − H(A) = H(BC | A)
    ˙
    0.6 ྔࢠ૬ޓ৘ใྔ
    ݹయ৘ใཧ࿦ʹ͓͍ͯɺ֬཰ม਺ X ͱ Y ͷ૬ޓ৘ใྔ͸ɺ࣍ࣜͰఆٛ͞Ε·͢ɻ
    I(X ; Y ) = H(X) + H(Y ) − H(X, Y ) (32)
    ͜Ε͸ɺ৚݅෇͖ΤϯτϩϐʔΛ༻͍ͯɺ࣍ͷΑ͏ʹॻ͖௚͢͜ͱ΋Ͱ͖·͢ɻ
    I(X ; Y ) = H(X) − H(X | Y )
    = H(Y ) − H(Y | X)

    View full-size slide

  26. 26
    ͜ͷදࣜΛݟΔͱɺ૬ޓ৘ใྔ I(X ; Y ) ͸ɺ֬཰ม਺ X ͷෆ֬ఆੑ͕ɺ֬཰ม਺
    Y ͷ஋Λ஌Δ͜ͱʹΑͬͯɺฏۉతʹͲΕ΄Ͳݮগ͢Δ͔Λද͢͜ͱ͕Θ͔Γ·͢ɻ࣮
    ࡍɺX ͱ Y ͕ಠཱͰ͋Ε͹ɺI(X ; Y ) = 0 ͱͳΔ͜ͱ͸ɺఆ͔ٛΒ͙͢ʹΘ͔Γ·͢ɻ
    I(X ; Y ) ͸ X ͱ Y ʹ͍ͭͯରশͳͷͰɺ֬཰ม਺ Y ͷෆ֬ఆੑ͕ɺ֬཰ม਺ X ͷ஋
    Λ஌Δ͜ͱʹΑͬͯɺฏۉతʹͲΕ΄Ͳݮগ͢Δ͔Λද͢ͱݟͯ΋ಉ͡Ͱ͢ɻ
    ·ͨɺX ͱ Y ͷ૬ޓ৘ใྔ͸ɺ࣮ࡍͷಉ࣌෼෍ p(x, y) ͱɺX ͱ Y ͕ಠཱͱԾఆ͠
    ͨ৔߹ͷಉ࣌෼෍ p(x)p(y) ͷ KL μΠόʔδΣϯεʹҰக͠·͢ɻ
    I(X ; Y ) = D(p(x, y) || p(x)p(y)) (33)
    ͜ΕΑΓɺ૬ޓ৘ใྔ͸ඇෛ஋ΛऔΓɺX ͱ Y ͕ಠཱͳ৔߹ʹݶͬͯɺ0 ʹͳΔ͜ͱ
    ͕Θ͔Γ·͢ɻ
    I(X ; Y ) ≥ 0
    I(X ; Y ) = 0 ⇔ p(x, y) = p(x)p(y)
    ྔࢠ৘ใཧ࿦ʹ͓͚Δྔࢠ૬ޓ৘ใྔʢQuantum mutual informationʣ͸ɺ(32) ʹର
    Ԡͯ͠ɺ
    I(A ; B) = H(A) + H(B) − H(AB)
    Ͱఆٛ͞Ε·͢ɻ(33) ʹରԠ͢Δؔ܎ࣜ΋੒Γཱͪɺ࣍ͷఆཧ͕੒ཱ͠·͢ɻ
    ఆཧ 7 ߹੒ܥ AB ͷঢ়ଶີ౓ԋࢉࢠΛ ρAB
    ͱ͢Δͱɺ෦෼ܥ AɺB ͷঢ়ଶີ౓ԋࢉ
    ࢠ͸ɺ
    ρA
    = TrB
    ρAB
    ρB
    = TrA
    ρAB
    Ͱ༩͑ΒΕΔɻ͜ͷ࣌ɺ࣍ͷؔ܎͕੒Γཱͭɻ
    I(A ; B) = D(ρAB
    || ρA
    ⊗ ρB
    )
    I(A ; B) ≥ 0
    I(A ; B) = 0 ⇔ ρAB
    = ρA
    ⊗ ρB
    [ূ໌] ఆཧ 4 ͷ (26) ʹྔࢠ૬ޓ৘ใྔͷఆٛΛ୅ೖ͢Δ͜ͱͰɺ
    I(A ; B) = D(ρAB
    || ρA
    ⊗ ρB
    )
    ͕ಘΒΕΔɻ͜Εʹఆཧ 3 Λద༻͢Δ͜ͱͰɺ࢒Γͷؔ܎͕ಘΒΕΔɻ ˙

    View full-size slide

  27. 0.6 ྔࢠ૬ޓ৘ใྔ 27
    ৚݅෇͖ΤϯτϩϐʔΛ༻͍Δ͜ͱͰɺ৚݅෇͖૬ޓ৘ใྔΛఆٛ͢Δ͜ͱ΋Ͱ͖·
    ͢ɻݹయ৘ใཧ࿦ͷ৔߹͸ɺ
    I(X ; Y | Z) = H(X | Z) + H(Y | Z) − H(X, Y | Z) (34)
    ͱͯ͠ఆٛ͞Ε·͕͢ɺ͜Ε͸ɺZ = z ͱܾ·ͬͨ৔߹ͷ৚݅෇͖֬཰෼෍ p(x, y | z)
    ʹΑΔ૬ޓ৘ใྔ I(X ; Y | Z = z) Λ Z ʹ͍ͭͯظ଴஋Λऔͬͨ΋ͷͱߟ͑Δ͜ͱ͕Ͱ
    ͖·͢ɻ
    I(X ; Y | Z) =

    z
    p(z)I(X ; Y | Z = z) (35)
    ͜ͷදࣜΑΓɺI(X ; Y | Z) ≥ 0 ͕੒Γཱͭ͜ͱ͕Θ͔Γɺ͞Βʹɺ͜Εͱಉ஋ͳ࣍ͷ
    ෆ౳͕ࣜ੒Γཱͪ·͢ɻ
    H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
    ͜ΕΛݹయΤϯτϩϐʔʹؔ͢Δ Strong subadditivity ͱݴ͍·͢ɻ͜ͷؔ܎͸ɺ2 छ
    ྨͷू߹ R = {X, Y } ͱ S = {Y, Z} Λߟ͑Δͱɺू߹࿦ͷه߸Λܗࣜతʹ༻͍ͯɺ
    H(R ∪ S) + H(R ∩ S) ≤ H(R) + H(S)
    ͱॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ͞Βʹ·ͨɺ͜Ε͸ɺ࣍ͷؔ܎ͱ΋ಉ஋ʹͳΓ·͢ɻ
    I (X ; (Y, Z)) ≥ I(X ; Z) (36)
    ͜ͷ࠷ޙͷؔ܎͸ɺಉ࣌෼෍ (Y, Z) ͔Β Y ͷ৘ใΛऔΓআ͘ͱɺX ͱͷ૬ޓ৘ใྔ
    ͕ݮগ͢Δͱ͍͏͜ͱΛ͍ࣔͯ͠·͢ɻͦΕͰ͸ɺ͜Εͱಉؔ͡܎ΛྔࢠΤϯτϩϐʔ
    ʹؔͯࣔ͢͜͠ͱ͸Ͱ͖ΔͰ͠ΐ͏͔ʁɹ·ͣɺ৚݅෇͖ྔࢠ૬ޓ৘ใྔʢConditional
    quantum mutual informatinʣ͸ɺ(34) ʹରԠ͢ΔܗͰɺ
    I(A ; B | C) = H(A | C) + H(B | C) − H(AB | C)
    ͱఆٛ͞Ε·͢ɻ͔͠͠ͳ͕Βɺ৚݅෇͖ྔࢠΤϯτϩϐʔͷͱ͜ΖͰઆ໌ͨ͠Α͏ʹɺ
    ྔࢠܥͰ͸ɺಉ࣌֬཰෼෍Λ༻͍ͨٞ࿦͕Ͱ͖ͳ͍ͨΊɺ(35) ͷΑ͏ͳؔ܎Λ༻͍ͯɺྔ
    ࢠ৚݅෇͖૬ޓ৘ใྔ͕ඇෛʹͳΔͱओு͢Δ͜ͱ͸Ͱ͖·ͤΜɻ࣮͸ɺྔࢠΤϯτϩ
    ϐʔͷ৔߹͸ɺݹయΤϯτϩϐʔͱ͸·ͬͨ͘ҟͳΔಓےͰɺStrong subadditivity ͕ࣔ
    ͞Ε·͢ɻͦͷେݩͱͳΔͷ͕ɺ࣍ͷఆཧͰ͢ɻ
    ఆཧ 8 ߹੒ܥ AB ͷ 2 छྨͷঢ়ଶີ౓ԋࢉࢠ ρAB
    , σAB
    ʹ͍ͭͯɺͦΕͧΕɺܥ B Λ
    τϨʔεΞ΢τͨ͠΋ͷΛ ρA
    , σA
    ͱ͢Δ࣌ɺ࣍ͷෆ౳͕ࣜ੒Γཱͭɻ
    D(ρA
    || σA
    ) ≤ D(ρAB
    || σAB
    )

    View full-size slide

  28. 28
    ɹ
    ͜ͷఆཧͷূ໌͸ɺ࣮͸ɺͦΕ΄Ͳ؆୯Ͱ͸͋Γ·ͤΜɻ਺ֶతʹ͔ͳΓࠐΈೖͬͨٞ࿦
    ͕ඞཁͱͳΓ·͢ɻ͜͜Ͱ͸ɺ͜Ε͕੒Γཱͭ͜ͱΛೝΊͯɺྔࢠΤϯτϩϐʔͷ Strong
    subadditivity Λࣔ͠·͢ɻͳ͓ɺྔࢠ૬ରΤϯτϩϐʔ D(ρ || σ) ͸ɺρ ͱ σ ͷྨࣅ౓
    ΛଌΔࢦඪͰͨ͠ͷͰɺ্هͷఆཧ͸ɺ߹੒ܥ AB ʹ͓͍ͯɺܥ B ͷ৘ใΛࣦ͏ͱɺ2 ͭ
    ͷঢ়ଶͷྨࣅ౓͕ߴ͘ͳΔɺͭ·Γɺ2 ͭͷঢ়ଶ͸ΑΓ۠ผ͠ʹ͘͘ͳΔͱ͍͏ࣄΛҙຯ
    ͠·͢ɻ
    ͸͡Ίʹɺ্هͷఆཧΛ༻͍ͯ (36) ʹରԠ͢Δɺ࣍ͷఆཧΛࣔ͠·͢ɻ
    ఆཧ 9 ߹੒ܥ ABC ͷ෦෼ܥʹؔ͢Δྔࢠ૬ޓ৘ใྔʹ͍ͭͯɺ
    ࣍ͷෆ౳͕ࣜ੒Γཱͭɻ
    I(A ; BC) ≥ I(A ; C)
    [ূ໌] ߹੒ܥ ABC ͷঢ়ଶີ౓ԋࢉࢠΛ ρABC
    ͱͯ͠ɺ͜ΕΛܥ A ͱܥ BC ʹ෼཭ͨ͠
    ࡍͷͦΕͧΕͷঢ়ଶີ౓ԋࢉࢠͷςϯιϧੵΛ
    σABC
    = ρA
    ⊗ ρBC
    (37)
    ͱஔ͘ɻ͜ͷ࣌ɺρABC
    ͱ σABC
    ͷͦΕͧΕʹ͍ͭͯɺܥ B ΛτϨʔεΞ΢τͨ͠΋ͷ
    Λ ρAB
    , σAB
    ͱ͢Δͱɺఆཧ 8 ΑΓɺ࣍ͷؔ܎͕੒Γཱͭɻ
    D(ρABC
    || σABC
    ) ≥ D(ρAC
    || σAC
    )
    (37) ͷఆٛΛ୅ೖ͢Δͱɺ͕࣍ಘΒΕΔɻ
    D(ρABC
    || ρA
    ⊗ ρBC
    ) ≥ D(ρAC
    || ρA
    ⊗ ρC
    )
    ఆཧ 7 ͷؔ܎
    I(A ; B) = D(ρAB
    || ρA
    ⊗ ρB
    )
    ΑΓɺ͜Ε͸ࣔ͢΂͖ؔ܎ʹҰக͍ͯ͠Δɻ ˙
    ྔࢠ૬ޓ৘ใྔͷఆٛΛ༻͍ͯɺఆཧ 9 ͷෆ౳ࣜΛྔࢠΤϯτϩϐʔͰॻ͖௚͢ͱɺ࣍
    ͷྔࢠΤϯτϩϐʔʹؔ͢Δ Strong subadditivity ͕ಘΒΕ·͢ɻ
    H(ABC) + H(C) ≤ H(AC) + H(BC) (38)
    ͋Δ͍͸ɺ৚݅෇͖ྔࢠ૬ޓ৘ใྔ͕ඇෛͰ͋Δͱ͍͏ɺ࣍ͷ৚݅ͱ΋ಉ஋ʹͳΓ·͢ɻ
    I(A ; B | C) ≥ 0 (39)

    View full-size slide

  29. 0.6 ྔࢠ૬ޓ৘ใྔ 29
    ͜͜·Ͱ͸ɺݹయΤϯτϩϐʔͷ৔߹ͱ΄΅ಉ݁͡࿦Ͱ͕͢ɺྔࢠܥʹ͓͍ͯ͸ɺ͜Ε
    Λ͞Βʹมܗͯ͠ɺ࣍ͷఆཧΛಘΔ͜ͱ͕Ͱ͖·͢ɻ
    ఆཧ 10 ߹੒ܥ ABC ͷ෦෼ܥʹ͍ͭͯɺ࣍ͷؔ܎͕੒Γཱͭɻ
    H(AC) + H(BC) ≥ H(A) + H(B)
    [ূ໌] ߹੒ܥ ABC ʹܥ D ΛՃ͑ͯ७ਮԽ͢Δͱɺఆཧ 1 ΑΓɺ
    H(AC) = H(BD), H(ABC) = H(D)
    ͕੒Γཱͭɻ͜ΕΛ Strong subadditivity
    H(ABC) + H(C) ≤ H(AC) + H(BC)
    ʹ୅ೖ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕΔɻ
    H(D) + H(C) ≤ H(BD) + H(BC)
    ͜͜ͰɺB ˠ CɺD ˠ AɺC ˠ B ͱه߸Λஔ͖׵͑Δͱɺࣔ͢΂͖ؔ܎ʹҰக͢Δɻ
    ˙
    ্هͷ݁Ռ͸ɺ৚݅෇͖ྔࢠΤϯτϩϐʔΛ༻͍ͯɺ࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖
    ·͢ɻ
    H(C | A) + H(C | B) ≥ 0 (40)
    ݹయΤϯτϩϐʔͷ৔߹ɺ৚݅෇͖Τϯτϩϐʔ͸ඇෛ஋ΛऔΔͷͰɺ͜Ε͸ࣗ໌ʹ੒
    Γཱͭؔ܎Ͱ͕͢ɺྔࢠΤϯτϩϐʔͷ৔߹͸ɺܾͯࣗ͠໌ͳ݁ՌͰ͸͋Γ·ͤΜɻྔࢠ
    ܥʹ͓͍ͯ͸ɺ৚݅෇͖ྔࢠΤϯτϩϐʔ͕ෛʹͳΔ͜ͱ΋͋Δ͔ΒͰ͢ɻ(40) ͸ɺܥ
    C ʹରͯ͠ɺ2 छྨͷ৚݅෇͖ྔࢠΤϯτϩϐʔͷ࿨͸ɺෛʹ͸ͳΒͳ͍͜ͱΛҙຯ͠·
    ͢ɻͭ·ΓɺҰํ͕ෛʹͳͬͨ৔߹ɺ΋͏Ұํ͸ɺͦΕΛଧͪফ͢Ҏ্ʹେ͖ͳ஋ʹͳΔ
    ͷͰ͢ɻ
    ৚݅෇͖ྔࢠΤϯτϩϐʔ H(C | A) ͕ෛʹͳΔྫͱͯ͠ɺࠞ߹ঢ়ଶͷܥ A ʹܥ C Λ
    Ճ͑ͯ७ਮԽ͢Δɺ͢ͳΘͪɺ߹੒ܥ AC Λ७ਮঢ়ଶʹ͢Δͱ͍͏৔߹͕ߟ͑ΒΕ·͢ɻ
    (40) ͷ݁Ռ͸ɺܥ A ͱܥ B ʹɺಉ͡ܥ C ΛՃ͑ͯɺ྆ํΛಉ࣌ʹ७ਮঢ়ଶʹ͸Ͱ͖ͳ
    ͍͜ͱΛҙຯ͓ͯ͠Γɺ͜ͷΑ͏ͳྔࢠܥͷੑ࣭Λ monogamy*4ͱݴ͍·͢ɻ
    *4 ೔ຊޠͰݴ͏ͱʮҰ෉Ұ්ੑʯ
    ɻ

    View full-size slide

  30. 30
    0.7 ͦͷଞͷੑ࣭
    ͜͜Ͱ͸ɺ͜Ε·Ͱʹઆ໌͍ͯ͠ͳ͔ͬͨɺͦͷଞͷओཁͳੑ࣭Λ঺հ͠·͢ɻ͸͡Ί
    ʹɺྔࢠΤϯτϩϐʔ͸ɺݹయΤϯτϩϐʔͱಉ༷ͷ Concavity Λ࣋ͭ͜ͱΛࣔ͠·͢ɻ
    ·ͣɺݹయΤϯτϩϐʔͷ Concavity ͱ͸ɺ࣍ͷΑ͏ͳੑ࣭Ͱ͢ɻࠓɺෳ਺ͷ֬཰෼෍
    qi
    (x) (i = 1, 2, · · · ) ͕͋ΓɺͦΕͧΕͷݹయΤϯτϩϐʔΛ H(qi
    ) ͱද͠·͢ɻ͜ͷ࣌ɺ
    ͜ΕΒͷ֬཰෼෍Λࠞͥ߹Θͤͯɺ৽ͨͳ֬཰෼෍
    p(x) =

    i
    pi
    qi
    (x)
    Λ࡞Γ·͢ɻ܎਺ pi
    ΋·ͨ֬཰෼෍Ͱɺ

    i
    pi
    = 1 Λຬͨ͠·͢ɻ͜ͷ࣌ɺp(x) ʹରԠ
    ͢ΔݹయΤϯτϩϐʔΛ H(p) ͱͯ͠ɺ࣍ͷෆ౳͕ࣜ੒Γཱͪ·͢ɻ͜Ε͕ɺݹయΤϯτ
    ϩϐʔͷ Concavity Ͱ͢ɻ
    H(p) ≥

    i
    pi
    H(qi
    )
    ௚ײతʹݴ͏ͱɺෳ਺ͷ֬཰෼෍ʹରͯ͠ɺ͜ΕΒΛ͞ΒʹҰఆͷ֬཰෼෍Ͱࠞͥ߹Θ
    ͤΔͨΊɺશମͱͯ͠ͷෆ֬ఆੑ͸͞Βʹ૿Ճ͢Δͱ͍͏͜ͱͰ͢ɻྔࢠΤϯτϩϐʔʹ
    ͓͍ͯ΋ɺ͜Εʹྨࣅͷෆ౳͕ࣜ੒Γཱͪ·͢ɻ
    ఆཧ 11 ෳ਺ͷঢ়ଶີ౓ԋࢉࢠ ρi
    ∈ L(H) (i = 1, 2, · · · ) ͕͋ΓɺͦΕͧΕͷྔࢠΤϯτ
    ϩϐʔΛ H(ρi
    ) ͱද͢ɻ·ͨɺ͜ΕΒΛ֬཰෼෍ pi
    ʹैͬͯࠞͥ߹Θͤͨɺ৽ͨͳঢ়ଶ
    ີ౓ԋࢉࢠΛ
    ρ =

    i
    pi
    ρi
    ͱͯ͠ɺ͜ΕʹରԠ͢ΔྔࢠΤϯτϩϐʔΛ H(ρ) ͱද͢ɻ͜ͷ࣌ɺ࣍ͷෆ౳͕ࣜ੒Γ
    ཱͭɻ
    H(ρ) ≥

    i
    pi
    H(ρi
    )
    [ূ໌] ঢ়ଶີ౓ԋࢉࢠ ρi
    (i = 1, 2, · · · ) ͕هड़͢Δܥ A ʹ৽ͨʹܥ X ΛՃ͑ͯɺ࣍ͷݹ
    యʕྔࢠܥ XA Λ༻ҙ͢Δɻ
    ρXA
    =

    i
    pi
    |iX
    ⟩⟨iX
    | ⊗ ρi

    View full-size slide

  31. 0.7 ͦͷଞͷੑ࣭ 31
    ͜ͷ࣌ɺ෦෼ܥ A ͱ X ͷঢ়ଶີ౓ԋࢉࢠ͸࣍ʹͳΔɻ
    ρX
    = TrA
    ρXA
    =

    i
    pi
    |iX
    ⟩⟨iX
    |
    ρA
    = TrX
    ρXA
    =

    i
    pi
    ρi
    ্هͷ ρA
    ͸ఆཧͰ༩͑ΒΕͨ ρ ʹҰக͓ͯ͠Γɺܥ A ͷྔࢠΤϯτϩϐʔ H(A) ʹ
    ͍ͭͯɺ
    H(A) = H(ρ)
    ͕੒Γཱͭɻ͞ΒʹɺρX
    , ρXA
    Λ༻͍ͯɺܥ X ͱܥ XAɺͦΕͧΕͷྔࢠΤϯτϩϐʔ
    Λܭࢉ͢Δɻ্هͷ ρX
    ͸ਖ਼ن௚ަجఈ |iX
    ⟩ ʹΑΔεϖΫτϧ෼ղΛ༩͍͑ͯΔͷͰɺ
    ܥ X ͷྔࢠΤϯτϩϐʔ͸࣍Ͱ༩͑ΒΕΔɻ
    H(X) =

    i
    pi
    log pi
    ܥ XA ʹ͍ͭͯ͸ɺρi
    ͷεϖΫτϧ෼ղΛ
    ρi
    =

    j
    qi
    j
    |jA
    ⟩⟨jA
    |
    ͱͯ͠ɺ
    ρXA
    =

    i,j
    pi
    qi
    j
    |iX
    ⟩⟨iX
    | ⊗ |jA
    ⟩⟨jA
    |
    ͕੒Γཱͭɻ͜Ε͸ɺ|iX
    ⟩ ⊗ |jA
    ⟩ Λਖ਼ن௚ަجఈͱ͢Δ ρXA
    ͷεϖΫτϧ෼ղΛ༩͑ͯ
    ͓Γɺ͜ΕΑΓɺܥ XA ͷྔࢠΤϯτϩϐʔ͸࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    H(XA) =

    i,j
    (pi
    qi
    j
    ) log
    (
    pi
    qi
    j
    )
    =

    i,j
    (pi
    qi
    j
    ) log pi
    +

    i,j
    (pi
    qi
    j
    ) log qi
    j
    =

    i
    pi
    log pi
    +

    i
    pi

    j
    qi
    j
    log qi
    j
    = H(X) +

    i
    pi
    H(ρi
    )
    ͜͜Ͱɺఆཧ 4ʢSubaditivityʣΑΓ੒Γཱͭ
    H(XA) ≤ H(X) + H(A)

    View full-size slide

  32. 32
    ʹ্هͷ݁ՌΛ୅ೖ͢Δͱɺ
    H(X) +

    i
    pi
    H(ρi
    ) ≤ H(X) + H(ρ)
    ͱͳΓɺ͜ΕΑΓɺఆཧͷओு
    H(ρ) ≥

    i
    pi
    H(ρi
    )
    ͕ಘΒΕΔɻ ˙
    ࣍ʹɺఆཧ 4 Ͱࣔͨ͠ྔࢠΤϯτϩϐʔͷ Subaditivity ʹྨࣅͨ͠ɺ࣍ͷࡾ֯ෆ౳ࣜΛ
    ࣔ͠·͢ɻ͜Ε͸ɺSubaditivity ʹɺ७ਮԽͷςΫχοΫΛద༻͢Δ͜ͱͰಘΒΕ·͢ɻ
    ఆཧ 12 ߹੒ܥ AB ͱͦͷ෦෼ܥ AɺB ͷྔࢠΤϯτϩϐʔʹ͍ͭͯɺ࣍ͷෆ౳͕ࣜ੒
    Γཱͭɻ
    H(AB) ≥ |H(A) − H(B)|
    [ূ໌] ܥ AB ʹܥ C ΛՃ͑ͯ७ਮԽͨ͠߹੒ܥ ABC Λߟ͑Δͱɺ
    H(AC) = H(B), H(C) = H(AB)
    ͕੒Γཱͭɻఆཧ 4 ΑΓ੒Γཱͭ
    H(AC) ≤ H(A) + H(C)
    ʹ͜ΕΒΛ୅ೖ͢Δͱɺ
    H(B) ≤ H(A) + H(AB)
    ͢ͳΘͪɺ
    H(AB) ≥ H(B) − H(A)
    ͕ಘΒΕΔɻA ͱ B ͷରশੑʹΑΓɺA ͱ B ΛೖΕସ͑ͨٞ࿦͔Βɺ
    H(AB) ≥ H(A) − H(B)
    ΋੒ΓཱͭͷͰɺ͜ΕΒΛ͋ΘͤΔͱఆཧͷओு
    H(AB) ≥ |H(A) − H(B)|
    ͕ಘΒΕΔɻ ˙
    ࠷ޙʹɺఆཧ 8 ͷԠ༻ͱͯ͠ɺ೚ҙͷྔࢠνϟωϧʹΑΔঢ়ଶมԽ͸ܥͷྨࣅੑΛߴΊ
    Δͱ͍͏ɺ࣍ͷఆཧΛࣔ͠·͢ɻ

    View full-size slide

  33. 0.7 ͦͷଞͷੑ࣭ 33
    ఆཧ 13 ܥ A ͷঢ়ଶΛܥ B ͷঢ়ଶʹม׵͢Δྔࢠνϟωϧ N ͕͋Γɺ
    ܥ A ͷ 2 छྨͷ
    ঢ়ଶີ౓ԋࢉࢠ ρ, σ ʹ͍ͭͯɺͦΕͧΕΛ N ͰมԽͨ͠ঢ়ଶີ౓ԋࢉࢠΛ N(ρ), N(σ)
    ͱ͢Δɻ͜ͷ࣌ɺྔࢠ૬ରΤϯτϩϐʔʹ͍ͭͯɺ࣍ͷෆ౳͕ࣜ੒Γཱͭɻ
    D(N(ρ) || N(σ)) ≤ D(ρ || σ)
    [ূ໌] ʮ0.2.5 ঢ়ଶมԽͱྔࢠνϟωϧʯͰࣔͨ͠Α͏ʹɺྔࢠνϟωϧ N ʹΑΔม׵
    ͸ɺΞΠιϝτϦʔԋࢉࢠ U ∈ L(A, HBE
    ) ʹΑΔม׵ޙʹɺ؀ڥܥ E ΛτϨʔεΞ΢
    τ͢Δͱ͍͏खଓ͖ʹҰக͢Δɻ
    N(ρ) = TrE
    (UρU†)
    N(σ) = TrE
    (UσU†)
    ͦ͜Ͱ·ͣɺΞΠιϝτϦʔԋࢉࢠ U ʹΑΔม׵͸ɺྔࢠ૬ରΤϯτϩϐʔΛมԽ͞
    ͤͳ͍͜ͱΛࣔ͢ɻࠓɺρ ͱ σ ΛͦΕͧΕεϖΫτϧ෼ղͨ͠΋ͷΛ
    ρ =

    i
    pi
    |iA
    ⟩⟨iA
    |
    σ =

    j
    qj
    |jA
    ⟩⟨jA
    |
    ͱ͢Δͱɺ͜ΕΒͷྔࢠ૬ରΤϯτϩϐʔ͸ɺ࣍Ͱܭࢉ͞ΕΔ*5ɻ
    D(ρ || σ) = Tr(ρ log ρ) − Tr(ρ log σ)
    =

    i
    pi
    log pi
    − Tr




    i,j
    (pi
    log qj
    )|iA
    ⟩⟨iA
    |jA
    ⟩⟨jA
    |



    =

    i
    pi
    log pi


    i,j
    |⟨iA
    |jA
    ⟩|2pi
    log qj
    (41)
    Ұํɺ্هͷ ρ ͱ σ Λ U Ͱม׵͢Δͱ͕࣍ಘΒΕΔɻ
    UρU† =

    i
    pi
    U|iA
    ⟩⟨iA
    |U†
    UσU† =

    j
    qj
    U|jA
    ⟩⟨jA
    |U†
    *5 Ұൠʹɺ|iA⟩ ͱ |jA⟩ ͸ɺҟͳΔਖ਼ن௚ަجఈͱͳΔɻ

    View full-size slide

  34. 34
    ΞΠιϝτϦʔԋࢉࢠͷੑ࣭ U†U = I ΑΓɺ{U|iA
    ⟩}ɺ͓Αͼɺ {U|jA
    ⟩} ͸ɺͲͪΒ
    ΋ਖ਼ن௚ަܥͰ͋ΓɺU ʹΑΔม׵Ͱ ρ, σ ͷݻ༗஋ pi
    , qi
    ͸มԽͤͣɺ
    log
    (
    UρU†
    )
    =

    i
    (log pi
    )U|iA
    ⟩⟨iA
    |U†
    log
    (
    UσU†
    )
    =

    j
    (log qj
    )U|jA
    ⟩⟨jA
    |U†
    ͕੒Γཱͭɻ͜ΕΑΓɺม׵ޙͷྔࢠ૬ରΤϯτϩϐʔ͸ɺ࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    D(UρU† || UσU†) = Tr
    {
    (UρU†) log
    (
    UρU†
    )}
    − Tr
    {
    (UρU†) log
    (
    UσU†
    )}
    =

    i
    pi
    log pi
    − Tr




    i,j
    (pi
    log qj
    )U|iA
    ⟩⟨iA
    |U†U|jA
    ⟩⟨jA
    |U†



    =

    i
    pi
    log pi


    i,j
    |⟨iA
    |jA
    ⟩|2pi
    log qj
    (42)
    (41)(42) ΑΓɺ͔֬ʹɺ
    D(ρ || σ) = D(UρU† || UσU†)
    ͕੒Γཱͭɻ
    ࣍ʹɺ؀ڥܥ E ΛτϨʔεΞ΢τ͢Δૢ࡞Λߟ͑Δͱɺఆཧ 8 ΑΓɺ
    D(TrE
    (UρU†) || TrE
    (UσU†)) ≤ D(UρU† || UσU†)
    ͕੒Γཱͭɻ
    Ҏ্ͷ݁ՌΛ·ͱΊΔͱɺ
    D(N(ρ) || N(σ)) = D(TrE
    (UρU†) || TrE
    (UσU†))
    ≤ D(UρU† || UσU†)
    = D(ρ || σ)
    ͱͳΓɺ͜ΕͰఆཧͷओு͕ࣔ͞Εͨɻ ˙
    0.8 ʢࢀߟʣStrong subadditivity ͱಉ஋ͳ৚݅
    ຊߘͰ͸ɺఆཧ 8 ʹج͍ͮͯఆཧ 9 Λࣔ͠ɺ͜Ε͕ (38) ͷ Strong subsadditivity ʹ
    Ұக͢Δ͜ͱΛࣔ͠·ͨ͠ɻ͋Δ͍͸·ͨɺఆཧ 8 ͷԠ༻ͱͯ͠ɺఆཧ 13 Λࣔ͠·ͨ͠ɻ
    ࣮͸ɺ͜ΕΒΛؚΉҎԼͷ৚݅͸͢΂ͯಉ஋ʹͳΔ͜ͱ͕஌ΒΕ͍ͯ·͢ɻ

    View full-size slide

  35. 0.8 ʢࢀߟʣStrong subadditivity ͱಉ஋ͳ৚݅ 35
    ఆཧ 14 ࣍ͷ৚݅͸ɺ͢΂ͯಉ஋ʹͳΔɻ
    1. D(ρA
    || σA
    ) ≤ D(ρAB
    || σAB
    ) : ఆཧ 8
    2. D(N(ρ) || N(σ)) ≤ D(ρ || σ) : ఆཧ 13
    3.

    i
    pi
    D(ρi
    || σi
    ) ≥ D
    (

    i
    pi
    ρi
    ||

    i
    pi
    σi
    )
    (pi
    > 0,

    i
    pi
    = 1)
    4. I(A ; BC) ≥ I(A ; C) : ఆཧ 9
    5. H(ABC) + H(C) ≤ H(AC) + H(BC) : (38)
    6. I(A ; B | C) ≥ 0 : (39)
    7. H(AC) + H(BC) ≥ H(A) + H(B) : ఆཧ 10
    4.ʙ 7. ͕ಉ஋Ͱ͋Δ͜ͱ͸ɺຊจதͰ΋આ໌ͨ͠Α͏ʹɺఆ͔ٛΒͷ௚઀ܭࢉͰ֬ೝͰ
    ͖·͢ɻ3. ͸ɺ͜Ε·Ͱʹઆ໌͍ͯ͠·ͤΜͰ͕ͨ͠ɺྔࢠ૬ରΤϯτϩϐʔͷ Jointly
    convexity ͱݺ͹ΕΔੑ࣭Ͱ͢ɻ1.ʙ 3. ͷͲΕ͔ 1 ͕ͭ୯ಠͰূ໌Ͱ͖Ε͹ɺ͔ͦ͜Β
    Strong subadditivity Λಋ͚Δ͜ͱʹͳΓ·͢ɻจݙʹΑͬͯɺStrong subadditivity ͷ
    ূ໌ʹ͞·͟·ͳόϦΤʔγϣϯ͕ݟΒΕΔͷ͸ɺ͜Ε͕ཧ༝ͱͳΓ·͢ɻͨͱ͑͹ɺ[3]
    Ͱ͸ɺ্هͷ 1. Λ͸͡Ίʹূ໌͍ͯ͠·͢ɻ͋Δ͍͸ɺ[1] Ͱ͸ɺLieb ͷఆཧ͔Βɺ্ه
    ͷ 3. Λূ໌͢Δͱ͍͏ಓےΛ࠾༻͍ͯ͠·͢ɻ
    ͜͜Ͱ͸ɺࢀߟͱͯ͠ɺ3. Λ΋ͱʹͯ͠ɺ5. ͷ Strong subadditivity Λಋ͘ํ๏Λࣔ
    ͠·͢ɻ
    ఆཧ 15 ྔࢠ૬ରΤϯτϩϐʔͷ Jointly convexity

    i
    pi
    D(ρi
    || σi
    ) ≥ D
    (

    i
    pi
    ρi
    ||

    i
    pi
    σi
    )
    (pi
    > 0,

    i
    pi
    = 1)
    ΑΓɺྔࢠΤϯτϩϐʔͷ Strong subadditivity
    H(ABC) + H(C) ≤ H(AC) + H(BC)
    ͕ಘΒΕΔɻ

    View full-size slide

  36. 36
    [ূ໌] ߹੒ܥ ABC ͷঢ়ଶີ౓ԋࢉࢠ ρABC
    ʹରͯ͠ɺ࣍ͷؔ਺ T Λఆٛ͢Δɻ
    T(ρABC
    ) = H(A) − H(AC) + H(B) − H(BC)
    ܥ ABC ͕७ਮঢ়ଶͷ৔߹ɺH(A) = H(BC), H(AC) = H(B) ͱͳΔ͜ͱ͔Βɺ
    T(ρABC
    ) = 0 ͱͳΔɻ
    Ұํɺ෦෼ܥ AC ʹ͍ͭͯɺρAC
    ͱ ρA

    1
    d
    IC
    (d = dim HC
    ) ͱ͍͏ 2 छྨͷঢ়ଶີ
    ౓ԋࢉࢠͷྔࢠ૬ରΤϯτϩϐʔΛܭࢉ͢Δͱɺ࣍ͷܭࢉ͕੒Γཱͭɻ
    ʢ2 ͭ໨ͷ౳߸ʹ
    ͍ͭͯ͸ɺఆཧ 4 ͷূ໌ͷ (21) Λࢀরɻ
    ʣ
    D
    (
    ρAC
    || ρA

    1
    d
    IC
    )
    = Tr (ρAC
    log ρAC
    ) − Tr
    {
    ρAC
    log
    (
    ρA

    1
    d
    IC
    )}
    = −H(AC) − Tr {ρAC
    log (ρA
    ⊗ IC
    )}
    − Tr
    {
    ρAC
    log
    (
    IA

    1
    d
    IC
    )}
    ࠷ޙͷදࣜͷୈ 2 ߲ͱୈ 3 ߲͸ɺ࣍ͷΑ͏ʹܭࢉ͞ΕΔɻ
    ʢୈ 2 ߲ͷܭࢉʹ͍ͭͯ͸ɺ
    ఆཧ 4 ͷূ໌ͷ (25) ͷઆ໌Λࢀরɻ
    ʣ
    − Tr {ρAC
    log (ρA
    ⊗ IC
    )} = H(A)
    − Tr
    {
    ρAC
    log
    (
    IA

    1
    d
    IC
    )}
    = −Tr
    {
    ρAC
    (
    log
    1
    d
    )
    IA
    ⊗ IC
    }
    = (log d)Tr ρAC
    = log d
    ͕ͨͬͯ͠ɺ࣍ͷؔ܎͕੒Γཱͭɻ
    D
    (
    ρAC
    || ρA

    1
    d
    IC
    )
    = H(A) − H(AC) + log d
    ಉ༷ʹͯ͠ɺ
    D
    (
    ρBC
    || ρB

    1
    d
    IC
    )
    = H(B) − H(BC) + log d
    ͕੒ΓཱͭͷͰɺ࠷ऴతʹɺ࣍ͷؔ܎͕ಘΒΕΔɻ
    T(ρABC
    ) = D
    (
    ρAC
    || ρA

    1
    d
    IC
    )
    + D
    (
    ρBC
    || ρB

    1
    d
    IC
    )
    − 2 log d (43)
    ͜͜Ͱɺ೚ҙͷঢ়ଶີ౓ԋࢉࢠ ρABC
    ʹ͍ͭͯɺ͜ΕΛ HABC
    = HA
    ⊗ HB
    ⊗ HC
    ͷ
    ਖ਼ن௚ަجఈ |iABC
    ⟩ ͰεϖΫτϧ෼ղͨ͠΋ͷΛ
    ρABC
    =

    i
    pi
    |iABC
    ⟩⟨iABC
    |

    View full-size slide

  37. 0.8 ʢࢀߟʣStrong subadditivity ͱಉ஋ͳ৚݅ 37
    ͱ͢Δͱɺ͖͞΄Ͳ෦෼ܥ AC ʹؔ͢Δྔࢠ૬ରΤϯτϩϐʔͷܭࢉʹ༻͍ͨঢ়ଶີ౓
    ԋࢉࢠ͸ɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δɻ
    ρCA
    =

    i
    pi
    TrB
    (|iABC
    ⟩⟨iABC
    |) =

    i
    pi
    ρi
    ρA

    1
    d
    IC
    =

    i
    pi
    TrBC
    (|iABC
    ⟩⟨iABC
    |) ⊗
    1
    d
    IC
    =

    i
    pi
    σi
    ͜͜Ͱɺ
    ρi
    = TrB
    (|iABC
    ⟩⟨iABC
    |)
    σi
    = TrBC
    (|iABC
    ⟩⟨iABC
    |) ⊗
    1
    d
    IC
    ͱఆٛͨ͠ɻ͞Βʹɺྔࢠ૬ରΤϯτϩϐʔͷ Jointly convexity Λ༻͍Δͱɺ࣍ͷෆ౳
    ͕ࣜ੒Γཱͭɻ
    D
    (
    ρAC
    || ρA

    1
    d
    IC
    )
    = D
    (

    i
    pi
    ρi
    ||

    i
    pi
    σi
    )


    i
    pi
    D(ρi
    || σi
    )
    =

    i
    pi
    D
    (
    TrB
    (|iABC
    ⟩⟨iABC
    |) || TrBC
    (|iABC
    ⟩⟨iABC
    |) ⊗
    1
    d
    IC
    )
    ෦෼ܥ BC ʹ͍ͭͯ΋ಉ༷ͷؔ܎͕੒ΓཱͭͷͰɺ(43) ͱ͋Θͤͯɺ࣍ͷෆ౳͕ࣜಘ
    ΒΕΔɻ
    T(ρABC
    ) ≤

    i
    pi
    {
    D
    (
    TrB
    (|iABC
    ⟩⟨iABC
    |) || TrBC
    (|iABC
    ⟩⟨iABC
    |) ⊗
    1
    d
    IC
    )
    + D
    (
    TrA
    (|iABC
    ⟩⟨iABC
    |) || TrAC
    (|iABC
    ⟩⟨iABC
    |) ⊗
    1
    d
    IC
    ) }
    − 2 log d
    Ұํɺ(43) ͷؔ܎Λ༻͍Δͱɺ্ࣜͷӈล͸ɺ

    i
    pi
    T(|iABC
    ⟩⟨iABC
    |) ʹҰக͢Δ͜ͱ
    ͕Θ͔Δɻ๯಄Ͱड़΂ͨΑ͏ʹɺ७ਮঢ়ଶ |iABC
    ⟩⟨iABC
    | ʹରͯؔ͠਺ T ͷ஋͸ 0 ͱͳ
    ΔͷͰɺ݁ہɺ
    T(ρABC
    ) ≤ 0
    ͕੒Γཱͭ͜ͱ͕Θ͔ͬͨɻT(ρABC
    ) ͷఆٛʹ໭Δͱɺ͜ΕΑΓɺ
    H(A) − H(AC) + H(B) − H(BC) ≤ 0

    View full-size slide

  38. 38
    ͕੒Γཱͭɻ࠷ޙʹɺܥ D ΛՃ͑ͯɺ߹੒ܥ ABCD Λ७ਮঢ়ଶʹ͢ΔͱɺH(A) =
    H(BCD), H(AC) = H(BD) ͕੒ΓཱͭͷͰɺ
    H(BCD) − H(BD) + H(B) − H(BC) ≤ 0
    ͕ಘΒΕΔɻ͜͜ͰɺB ˠ CɺC ˠ AɺD ˠ B ͱه߸Λஔ͖׵͑Δͱɺࣔ͢΂͖ؔ܎͕
    ಘΒΕΔɻ ˙

    View full-size slide

  39. 39
    ࢀߟจݙ
    [1]ʮQuantum Computation and Quantum Information: 10th Anniversary Edi-
    tionʯMichael A. Nielsen, Isaac L. ChuangʢஶʣCambridge University Press
    [2]ʮElements of Information Theory (2nd edition)ʯThomas M. Cover, Joy A.
    ThomasʢஶʣWiley-Interscience
    [3]ʮQuantum Information Theory (2nd edition)ʯMark M. WildeʢஶʣCambridge
    University Press

    View full-size slide