量子力学の知識はまったく使わない
大学の数学の記法を使う

付録

Bell-CHSH 不等式
の導出
田崎 晴明

写真は 1982 年のベル、CERN より

理科系一般向け講義 2022/11 @YouTube

基本的な隠れた変数の理論

基本的な隠れた変数
唯一の仮定 各々の実験において、A1, A2, B1, B2 を測定した際に得られる（はずの）測定結果は事前に確定している
隠れた変数 λ ∈ {1,2,…,16} により測定結果を指定
λa1 (λ)a2 (λ)b1 (λ)b2 (λ)

1
+1
+1
+1
+1

2
+1
+1
+1
−1

3
+1
+1
−1
+1

N 回の実験
<latexit sha1_base64="cTLeiuznlQcwlpIZRyEQ3hzc0f0=">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</latexit>

N( ) =

NX
j=1<latexit sha1_base64="+5JPwpRdHyKaRr2JpIsgoJsgBJ0=">AAACJ3icbVDLSsNAFL3xWesr6tKFwSLopiRS1I1QdONKKtgHtKFMJpN26CQTZiZCCf0bXfgrbgQV0aV/4jTNQtseGOZwzrnM3OPFjEpl29/GwuLS8spqYa24vrG5tW3u7DYkTwQmdcwZFy0PScJoROqKKkZasSAo9BhpeoPrsd98IEJSHt2rYUzcEPUiGlCMlJa4eQACjuEJGCAIwQNf3ydwqZVAOwgwpHA7JzHK9FHXLNllO4M1S5yclCBHrWu+dXyOk5BECjMkZduxY+WmSCiKGRkVO4kkMcID1CNtTSMUEumm2Z4j60grvhVwoU+krEz9O5GiUMph6OlkiFRfTntjcZ7XTlRw4aY0ihNFIjx5KEiYpbg1Ls3yqSBYsaEmCAuq/2rhPhIIK11tUZfgTK88SxqnZeesXLmrlKpXeR0F2IdDXa4D51CFG6hBXRf+CC/wDh/Gs/FqfBpfk+iCkc/swT8YP79M/ps7</latexit>

4
+1
+1
−1
−1

5
+1
−1
+1
+1

6
+1
−1
+1
−1

7
+1
−1
−1
+1

8
+1
−1
−1
−1

9
−1
+1
+1
+1

10
−1
+1
+1
−1

11
−1
+1
−1
+1

12
−1
+1
−1
−1

13
−1
−1
+1
+1

14
−1
−1
+1
−1

15
−1
−1
−1
+1

隠れた変数の列 λ1, λ2, …, λN
列は完全に任意
確率的である必要はない
16X
N( ) = N, j 列の中に λ が現れた回数

N( )列の中に λ が現れた頻度
r( ) =
N<latexit sha1_base64="rlytvHJMx7VuIRq0KXkCmyAFly8=">AAACK3icbZDLSsNAFIZPvNZ6i7p0Ey2CbkoipboRim5clQr2Am0sk8mkHTqZhJmJUEIfR3Djq7jQhRfc+h5O2yy09cAwH//5DzPn92JGpbLtD2NhcWl5ZTW3ll/f2NzaNnd2GzJKBCZ1HLFItDwkCaOc1BVVjLRiQVDoMdL0BlfjfvOeCEkjfquGMXFD1OM0oBgpLUXmATyAhARC6EKqmQHS7IGv7wtwYAR3WnegrKkKxzOOE+2pds2CXbQnZc2Dk0EBsqp1zZeOH+EkJFxhhqRsO3as3BQJRTEjo3wnkSRGeIB6pK2Ro5BIN53sOrKOtOJbQST04cqaqL8nUhRKOQw97QyR6svZ3lj8r9dOVHDuppTHiSIcTx8KEmapyBoHZ/lUEKzYUAPCguq/WriPBMJKx5vXITizK89D47TolIulm1KhcpnFkYN9ONTBOnAGFbiGGtQBwyM8wxu8G0/Gq/FpfE2tC0Y2swd/yvj+AffDm+k=</latexit>

=1

16
X
<latexit sha1_base64="C3WUot/DbZWYBRbUExOR/kywrU0=">AAACK3icbZDLSsNAFIZPvNZ6i7p0Ey2CbkoipboRim5cVrAXaGOZTCbt0MkkzEyEEvo4ghtfxYUuvODW93DSZqGtB4b5+M9/mDm/FzMqlW1/GAuLS8srq4W14vrG5ta2ubPblFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3vMr6rXsiJI34rRrFxA1Rn9OAYqS0FJkH8AASEgihB6lmBkizB76+L8CBMdxp3YGqJgHHM46TzNMzS3bZnpQ1D04OJcir3jNfun6Ek5BwhRmSsuPYsXJTJBTFjIyL3USSGOEh6pOORo5CIt10suvYOtKKbwWR0Icra6L+nkhRKOUo9LQzRGogZ3uZ+F+vk6jg3E0pjxNFOJ4+FCTMUpGVBWf5VBCs2EgDwoLqv1p4gATCSsdb1CE4syvPQ/O07FTLlZtKqXaZx1GAfTjUwTpwBjW4hjo0AMMjPMMbvBtPxqvxaXxNrQtGPrMHf8r4/gEFhpvw</latexit>=1r( ) = 1

16
−1
−1
−1
−1

測定量の選択と相関関数
A, B が測定する量をランダムに選ぶ
各々の j = 1,…, N について、他のすべての事象と独立に、
1
確率 4 で (1,1), (1,2), (2,1), (2,2) のいずれかを選ぶ

Sα,β : (α, β) が選ばれた j の集まり
{1, 2, . . . , N } = S1,1 [ S1,2 [ S2,1 [ S2,2<latexit sha1_base64="2Ab4ANCO0QY1FGdzsnALB/xp5Dw=">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</latexit>

Nα,β：Sα,β の要素の個数（(α, β) が選ばれた回数）
N = N1,1 + N1,2 + N2,1 + N2,2<latexit sha1_base64="o5UrKV3g6JOphS7K8H4Oe43gHPk=">AAACCnicbVBNS8NAEN3Ur1q/oh69rBZBUEoSinoRil48lQr2A9oQNttNu3SzCbsboYSevfhXvHhQxKu/wJv/xm2ag7Y+WObNmxlm5/kxo1JZ1rdRWFpeWV0rrpc2Nre2d8zdvZaMEoFJE0csEh0fScIoJ01FFSOdWBAU+oy0/dHNtN5+IELSiN+rcUzcEA04DShGSkueeVi/qnupfWZPTrPoZNHJc0fnnlm2KlYGuEjsnJRBjoZnfvX6EU5CwhVmSMqubcXKTZFQFDMyKfUSSWKER2hAuppyFBLpptkpE3islT4MIqEfVzBTf0+kKJRyHPq6M0RqKOdrU/G/WjdRwaWbUh4ninA8WxQkDKoITn2BfSoIVmysCcKC6r9CPEQCYaXdK2kT7PmTF0nLqdjnlepdtVy7zu0oggNwBE6ADS5ADdyCBmgCDB7BM3gFb8aT8WK8Gx+z1oKRz+yDPzA+fwDdZJfa</latexit><latexit sha1_base64="B8mMsjS++ah8xVRhhpiPjt8jZ94=">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</latexit>

N↵,

p
N
=
+ O( N )4
等確率だから

相関関数の表式
相関関数<latexit sha1_base64="dnxKOXp6POliq1E/0OL+3We/f2Y=">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</latexit>

1hA↵ B i =N↵,<latexit sha1_base64="O7QNtgwP4UrUzkLy4t5hLV3ru4c=">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</latexit>N↵, ( ) =

X

j2S↵,

X

1
a↵ ( j ) b ( j ) =
N↵,
, j

j2S↵,

16
X

a↵ ( ) b ( ) N↵, ( )=1Sα,β の中に λ が現れた回数

λj = λ のときに (α, β) が選ばれた回数

p
N( )N↵, ( ) =+ O( N )4
等確率だから<latexit sha1_base64="B8mMsjS++ah8xVRhhpiPjt8jZ94=">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</latexit><latexit sha1_base64="LP4/E1A5tjsmbEBQvFpQjgxN/PE=">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</latexit>

N↵,

p
N
=
+ O( N )4<latexit sha1_base64="VnAeRhQyuloAR2PlWdExGWLefdA=">AAACl3ichVHbattAEB0paZs4vbhJXkJeREyhhWKkUtoSKHHiEEwoxYb6ArYxq9UoXrK6ZHdVMEK/kT/wD+Ut35CfyPjy0NgtHdidM+fM7GXGT6XQxnXvLXtj89nzF1vbpZ2Xr16/Kb/d7egkUxzbPJGJ6vlMoxQxto0wEnupQhb5Erv+dX2md3+j0iKJf5lJisOIXcUiFJwZopJyF6YQggIGHHL4CSPapxRJSGFM/iNFPiAYwgW8p0gSiogLyH8g7v9VBUUaBFUh3Kzc968TC/hOWWvqqFxxq+7cnHXgLUGldnr7o3apHpqj8t0gSHgWYWy4ZFr3PTc1w5wpI7jEojTINKaMX7Mr7BOMWYR6mM/7WjjviAmcMFG0YuPM2T8rchZpPYl8yoyYGetVbUb+TetnJvw2zEWcZgZjvrgozKRjEmc2JCcQCrmREwKMK0FvdfiYKcYNjbJETfBWv7wOOp+q3pfq5xZ14wwWtgWHcERN9eAr1KABTWgDt/atY6tundsH9ol9YTcWqba1rNmDJ2a3HgGBCrF4</latexit>N↵, ( )N( )'= r( )
N ≫ 1 のとき任意の (α, β) についてN↵,N
<latexit sha1_base64="4jvS+iWD8loEEIJosjimAJf8xQw=">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</latexit>

相関関数の表式 hA↵ B i =

16
X

=1

a↵ ( ) b ( ) r( )Bell-CHSH 不等式
Bell 1964, Clauser, Horne, Shimony, Holt 1969

任意の λ = 1,…,16 について
λ12a1 (λ) +1 +1
a1 ( ) b1 ( ) + a2 ( ) b1 ( ) a1 ( ) b2 ( ) + a2a(2 (λ)) b2+1( )+1<latexit sha1_base64="sNU6k5JM/UfZZQM1L+XUEekaeqc=">AAACwXiclVHLTgIxFL0zvhBfoy7dTCAmGpTMEKOuDJGNS0zkkQAhnU7BhrYzaTsmZMJf+Av+kDv+xg6wUCAh3qTp6Tn39Lb3BjGjSnve1LK3tnd293L7+YPDo+MT5/SsqaJEYtLAEYtkO0CKMCpIQ1PNSDuWBPGAkVYwqmV664NIRSPxpscx6XE0FHRAMdKGihwGCPrgwxV8QYY5BBCa/dqcbwxer5Vmrso/Xbcba627cXOtdVq+7xS9sjcLdxX4C1CsFrqlz2l1XO87390wwgknQmOGlOr4Xqx7KZKaYkYm+W6iSIzwCA1Jx0CBOFG9dDaBiXtpmNAdRNIsod0Z+9uRIq7UmAcmkyP9rpa1jFyndRI9eOylVMSJJgLPCw0S5urIzcbphlQSrNnYAIQlNW918TuSCGsz9KwJ/vKXV0GzUvbvy3evphvPMI8cXEDBtNWHB6jCC9ShAdh6skKLW8Ku2dSObTlPta2F5xz+hJ3+AEw+tPU=</latexit>= {a1 ( ) + a2 ( )} b1 ( )<latexit sha1_base64="CkGLoiPQnm4ImX1pVCBTQDKNJWM=">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</latexit>{a1 ( )= ±2

b1 (λ)b22(λ)+1 +1+1 2 −1a ( )} b ( )

3
+1
+1
−1
+1

4
+1
+1
−1
−1

<latexit sha1_base64="FX1KjiPM6C6D3k+08BgFHBDb7ig=">AAAB8XicbVDLSgNBEOz1mcRX1KOXwSB4CrtB1IsQ9OIxgnlgsoTZyWwyZHZ2mJkVw5K/8JKDIoInD/6LN79GJ4+DJhY0FFXddHcFkjNtXPfLWVpeWV1bz2RzG5tb2zv53b2ajhNFaJXEPFaNAGvKmaBVwwynDakojgJO60H/auzX76nSLBa3ZiCpH+GuYCEj2FiJwgW8gYQISu18wS26E6BF4s1IoZyVo7uPh+9KO//Z6sQkiagwhGOtm54rjZ9iZRjhdJhrJZpKTPq4S5uWChxR7aeTi4foyCodFMbKljBoov6eSHGk9SAKbGeETU/Pe2PxP6+ZmPDcT5mQiaGCTBeFCUcmRuP3UYcpSgwfWIKJYvZWRHpYYWJsSDkbgjf/8iKplYreafHkxqZxCVNk4AAO4Rg8OIMyXEMFqkBAwCM8wbOjnZHz4rxOW5ec2cw+/IHz/gOpwJKx</latexit>

a1 ( ), a2 ( ), b1 ( ), b2 ( ) 2 {+1, 1}<latexit sha1_base64="Wvo1ZvOAGKqUR6hxRjUhBfH9MDQ=">AAACK3icbVDLSgMxFM3UV62vqks3wSJUrGWmFHXhoujGZQX7gE4Z7qRpG5rJDElGKEP/x42/4kIXPnDrf5g+Fm31QODknHtI7vEjzpS27U8rtbK6tr6R3sxsbe/s7mX3D+oqjCWhNRLyUDZ9UJQzQWuaaU6bkaQQ+Jw2/MHt2G88UqlYKB70MKLtAHqCdRkBbSQvewOek3e5CXTgtIDBK83d/AXPn/NcJtzkzCmcO+7Iy+bsoj0B/kucGcmhGape9tXthCQOqNCEg1Itx450OwGpGeF0lHFjRSMgA+jRlqECAqrayWTXET4xSgd3Q2mO0HiizicSCJQaBr6ZDED31bI3Fv/zWrHuXrUTJqJYU0GmD3VjjnWIx8XhDpOUaD40BIhk5q+Y9EEC0abejCnBWV75L6mXis5FsXxfzlWuZ3Wk0RE6RnnkoEtUQXeoimqIoCf0gt7Rh/VsvVlf1vd0NGXNModoAdbPLxXuo+8=</latexit>

16
X

a1
+1
+1
−1
−1

a2
+1
−1
+1
−1

a1 + a2200−2

a1 − a202−20<latexit sha1_base64="C3WUot/DbZWYBRbUExOR/kywrU0=">AAACK3icbZDLSsNAFIZPvNZ6i7p0Ey2CbkoipboRim5cVrAXaGOZTCbt0MkkzEyEEvo4ghtfxYUuvODW93DSZqGtB4b5+M9/mDm/FzMqlW1/GAuLS8srq4W14vrG5ta2ubPblFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3vMr6rXsiJI34rRrFxA1Rn9OAYqS0FJkH8AASEgihB6lmBkizB76+L8CBMdxp3YGqJgHHM46TzNMzS3bZnpQ1D04OJcir3jNfun6Ek5BwhRmSsuPYsXJTJBTFjIyL3USSGOEh6pOORo5CIt10suvYOtKKbwWR0Icra6L+nkhRKOUo9LQzRGogZ3uZ+F+vk6jg3E0pjxNFOJ4+FCTMUpGVBWf5VBCs2EgDwoLqv1p4gATCSsdb1CE4syvPQ/O07FTLlZtKqXaZx1GAfTjUwTpwBjW4hjo0AMMjPMMbvBtPxqvxaXxNrQtGPrMHf8r4/gEFhpvw</latexit><latexit sha1_base64="4jvS+iWD8loEEIJosjimAJf8xQw=">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</latexit>hA↵ B i =<latexit sha1_base64="8qhqOoozjtuA/cLZLEYzXeNjB5c=">AAACq3iclVHLSgMxFL0zvuur6tJNsAiCtMwMoi59bFxJRfvAtpRMeltDM5khyQil+Dn9IXf+jek4i9oWwQsJJ+ece5PcGyaCa+N5X467srq2vrG5Vdje2d3bLx4c1nWcKoY1FotYNUOqUXCJNcONwGaikEahwEY4vJ/qjXdUmsfyxYwS7ER0IHmfM2osFRcplCGACQjAbKcgYZCdCNxCF3y4y/YJqBntfKk3WOot/1E3+Efdee8kdwTdYsmreFmQReDnoAR5VLvFz3YvZmmE0jBBtW75XmI6Y6oMZwI/Cu1UY0LZkA6wZaGkEerOOOv1Bzm1TI/0Y2WXNCRjZzPGNNJ6FIXWGVHzpue1KblMa6Wmf90Zc5mkBiX7uaifCmJiMh0c6XGFzIiRBZQpbt9K2BtVlBk73oJtgj//5UVQDyr+ZeXi6aJ0c5e3YxOO4QTO7ECu4AYeoAo1YM6Z8+g0nKZbdp/dV7f9Y3WdPOcIfoWL31uOrwg=</latexit>

16Xa↵ ( ) b ( ) r( )=12  hA1 B1 i + hA2 B1 ir( ) = 1=1

hA1 B2 i + hA2 B2 i  2

より

5
+1
−1
+1
+1

より複雑な隠れた変数
基本的な隠れた変数に帰着させる

隠れた変数から測定値が決まる場合
各々の粒子に隠れた変数 Λ1 および Λ2 が書き込まれる
隠れた変数は何らかの規則（決定的でも確率的でもよい）によって Λ′1 および Λ′2 に変化 (⇤01 , ⇤02 2 V)<latexit sha1_base64="+Ic2jkgJSNx6bRyI/5xlsa5OJFY=">AAACDnicbVDLSsNAFJ3UV62vqEtdBEuxgpSkiLpwUXDjwkUF+4AmhMlk0g6dTMLMRCihX+Ci/oobF4oIrly780PcO2mLaPXAwOGcc5l7jxdTIqRpfmi5ufmFxaX8cmFldW19Q9/caooo4Qg3UEQj3vagwJQw3JBEUtyOOYahR3HL659nfusGc0Eidi0HMXZC2GUkIAhKJbl6qWxfqrQP913r8JtWbcLsEMoegjRtDg9cvWhWzDGMv8SakmJtd2SXP19HdVd/t/0IJSFmElEoRMcyY+mkkEuCKB4W7ETgGKI+7OKOogyGWDjp+JyhUVKKbwQRV49JY6z+nEhhKMQg9FQy21HMepn4n9dJZHDqpITFicQMTT4KEmrIyMi6MXzCMZJ0oAhEnKhdDdSDHCKpGiyoEqzZk/+SZrViHVeOrlQbZ2CCPNgBe6AMLHACauAC1EEDIHAL7sEjeNLutAftWXuZRHPadGYb/IL29gU2+56u</latexit>

A が Aα を測定すると結果は必ず ãα(Λ′1) ∈ {+1, − 1}
B が Bβ を測定すると結果は必ず b̃β(Λ′2) ∈ {+1, − 1}<latexit sha1_base64="X9NDeD0Dyk/bAPQT6dBKF5r/bj4=">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</latexit><latexit sha1_base64="zoEwahpbqi5KcAo7rAZvbLb1uUc=">AAACW3icbVHLSiNBFL3dvmJ8RYdZ6aJRBEUN3TKoIzMQcONSwUQhCaG6umIKq7t6qm4Lock3+BX5ET/BlS7mM9yKNx0Xvi5U1alzz63HuWGqpEXff3Lcicmp6ZnSbHlufmFxqbK80rA6M1zUuVbaXIXMCiUTUUeJSlylRrA4VOIyvDkZ5S9vhbFSJxfYT0U7ZteJ7ErOkChd+Q1DQJCgIAIBOTAYQIc4RkwKPVqPaRfTirTjBZ9Dg1SjOk1zDjsQwC7s0Tws+H+QkS6CrQ/n/C1U+7DdqWz4Vb8I7ysI3sBGbe2utfV8f3fWqTy0Is2zWCTIFbO2GfgptnNmUHIlBuVWZkXK+A27Fk2CCYuFbeeFNwNvk5jI62pDI0GvYN9X5Cy2th+HpIwZ9uzn3Ij8LtfMsHvUzmWSZigSPr6omykPtTcy2oukERxVnwDjRtJbPd5jhnGkdpTJhODzl7+Cxn41OKj+Oic3/sA4SrAK62RtAIdQg1M4gzq15RFenBmn5Px3J9yyOz+Wus5bzQ/4EO7PVzBKpZk=</latexit>ã↵ : V ! {+1, 1}(↵ = 1, 2)b̃ : V ! {+1, 1}( = 1, 2)

これと整合する一意的な λ ∈ {1,…,16} をとる<latexit sha1_base64="mqDS37Jpso3F2KtMeroTWKWj8wU=">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</latexit>

￼

￼

￼

￼

a↵ (

) = ã↵ (⇤01 )b ( ) = b̃ (⇤02 )<latexit sha1_base64="XkmrLlaWRf1PSrvjp6zoe6zzqOI=">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</latexit>(↵ = 1, 2)( = 1, 2)測定値が確率的に決まる場合
各々の粒子に隠れた変数 Λ1 および Λ2 が書き込まれる
隠れた変数は何らかの規則（決定的でも確率的でもよい）によって Λ′1 および Λ′2 に変化 (⇤01 , ⇤02 2 V)<latexit sha1_base64="+Ic2jkgJSNx6bRyI/5xlsa5OJFY=">AAACDnicbVDLSsNAFJ3UV62vqEtdBEuxgpSkiLpwUXDjwkUF+4AmhMlk0g6dTMLMRCihX+Ci/oobF4oIrly780PcO2mLaPXAwOGcc5l7jxdTIqRpfmi5ufmFxaX8cmFldW19Q9/caooo4Qg3UEQj3vagwJQw3JBEUtyOOYahR3HL659nfusGc0Eidi0HMXZC2GUkIAhKJbl6qWxfqrQP913r8JtWbcLsEMoegjRtDg9cvWhWzDGMv8SakmJtd2SXP19HdVd/t/0IJSFmElEoRMcyY+mkkEuCKB4W7ETgGKI+7OKOogyGWDjp+JyhUVKKbwQRV49JY6z+nEhhKMQg9FQy21HMepn4n9dJZHDqpITFicQMTT4KEmrIyMi6MXzCMZJ0oAhEnKhdDdSDHCKpGiyoEqzZk/+SZrViHVeOrlQbZ2CCPNgBe6AMLHACauAC1EEDIHAL7sEjeNLutAftWXuZRHPadGYb/IL29gU2+56u</latexit>

A が Aα を測定すると、確率 pα(Λ′1) で +1, 確率 1 − pα(Λ′1)
で −1 が得られる
B が Bβ を測定すると、確率 qβ(Λ′2) で +1, 確率 1 − qβ(Λ′2)
で −1 が得られる<latexit sha1_base64="lmr5RVQW9KOO+XJkCWAE5L7icHI=">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</latexit><latexit sha1_base64="1PMXCeaygW5N1SMPviLpVUNLZ3w=">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</latexit>p↵ : V ! [0, 1](↵ = 1, 2)

q : V ! [0, 1]( = 1, 2)

この確率に基づいて測定結果 (の候補) a1, a2, b1, b2 を選び、それと整合する一意的な λ ∈ {1,…,16} をとる<latexit sha1_base64="lFCXv9Y9dWYhK5unQ8wIH5GhabI=">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</latexit>

￼

￼

￼

￼

￼

￼

<latexit sha1_base64="3H5jxQVmEbuz0KW/nystrQWpHRg=">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</latexit>a↵ ( ) = a↵(↵ = 1, 2)

b ( )=b( = 1, 2)