古典エントロピーと比べて学ぶ量子エントロピー

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

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

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

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

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

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

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) =

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

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

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

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

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

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

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

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

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

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

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)

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

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 )

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)

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

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)

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

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)

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 ˙

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 )

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)

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

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

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)

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

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

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

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

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 |

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

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 ˙

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