￼

量子スピンとエンタングルメント
担当： 押川正毅、Han YAN（物性研究所）
oshikawa@issp.u-tokyo.ac.jp
量子スピン 例）スピン1/2~
角運動量のz成分 S = ±2<latexit sha1_base64="8oigEqzwjTWk4ftY30KTOB4EXs4=">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</latexit>z| i = c" | "i + c# | #i<latexit sha1_base64="xTGFKPFVpnPQ5vwzqXRRrNQSLHA=">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</latexit>

一般的の量子状態： 重ね合わせ

２つのスピン1/2の量子状態： | i = c"" | ""i + c"# | "#i + c#" | #"i + c## | ##i<latexit sha1_base64="1ekBezcyeNYm14l3ExxBY0GkUqI=">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</latexit>|t+ i = | ""i = | "i1 | "i2<latexit sha1_base64="zWb4VbVxyo9Nja8hn7NTLLH2UAM=">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</latexit>

は積状態（エンタングルメントなし）

反強磁性（スピンが逆向きが安定になる）相互作用下での基底状態

1|si = p (| "#i2
<latexit sha1_base64="Eg2EQqc7XIYsHTTRYZ98/R4uy1c=">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</latexit>| #"i) 6= |↵i1 | i2

Einstein-Podolsky-Rosen (EPR) 相関
1

積状態でないエンタングルメント（量子もつれ）あり￼

エンタングルメント・エントロピー： エンタングルメントの定量化「何個の量子ビットが完全にエンタングルしているのに相当する？」
X~j · S~j+1
H=S
スピン1/2 Heisenberg反強磁性鎖
j<latexit sha1_base64="M175/BrPRgbtVChmyDo3Vu9vqyw=">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</latexit>の基底状態のA, B部分間のエンタングルメント・エントロピー

l<latexit sha1_base64="wxHxaKaNnRaNtLjA7xTzLVoIjsI=">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</latexit>

自分でプログラムを動かして「体験」してもらいますBAL<latexit sha1_base64="6yDK1/ypv8XpLXAMLTNtJyeDW+I=">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</latexit>●直接数値計算●共形場理論 (Conformal Field Theory)
Holzhey-Larsen-Wilczek 1994
Calabrese-Cardy 2004

全スピン数

L = 14<latexit sha1_base64="TsOUmlKEW3mghG+aOLbiMM2Y0Q8=">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</latexit><latexit sha1_base64="fuN/26ZgmVnefK+9iRv/Qdc1fdY=">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</latexit>cS ⇠ log
3

✓

L
⇡lsin
⇡
L

◆+ const.

l）<latexit sha1_base64="wxHxaKaNnRaNtLjA7xTzLVoIjsI=">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</latexit>2（横軸:部分系のサイズ