[WIP][Information-theoretic temporal Bell inequality and quantum computation](https://arxiv.org/abs/quant-ph/0602011)

--- tags: information-theoretic bell inequality, bell inequality, quantum computing, 2006, PRA2006 --- # [WIP][Information-theoretic temporal Bell inequality and quantum computation](https://arxiv.org/abs/quant-ph/0602011) Fumiaki Morikoshi ## Abstract An information-theoretic temporal Bell inequality is formulated to contrast classical and quantum computations. Any classical algorithm satisfies the inequality, while quantum ones can violate it. Therefore, the violation of the inequality is an immediate consequence of the quantumness in the computation. Furthermore, this approach suggests a notion of temporal nonlocality in quantum computation. ## Backgrounds and Contributions 論文のモチベは、時間的なBell不等式によって、量子アルゴリズムが量子特有かどうかを判断したい。 > The information-theoretic temporal Bell inequality is formulated for classical algorithms. It is satisfied by any classical algorithm but can be violated by quantum ones. **information-theoretic temporal Bell inequality**: 古典は破らないけど、量子なら破るような不等式。例えば: Groverはinformation-theoretic temporal Bell inequalityを破る。 > It is shown that the information-theoretic temporal Bell inequality is violated in Grover’s algorithm. > the information-theoretic temporal Bell inequality is proposed as just one way of discriminating between classical and quantum computations. > Another motivation behind this work is to advance our understanding of the role of temporal correlations in quantum information processing. > Temporal correlations in quantum phenomena have been studied in the foundations of quantum theory from the viewpoint of Bell-type inequalities since the original proposal of the Leggett-Garg inequality > The present approach is developed in the belief that, in some cases, the power of quantum information processing may also lie in temporal correlations of a truly quantum nature as well as in spatial correlations due to entanglement. 着眼点: 量子アルゴリズムの優位性/量子性は、空間的な相関のみならず、時間的な相関にも大きく関係しているのではないか？ ## Methods ### information-theoretic Bell inequality - **valid** joint probability distributions: 古典の場合観測と状態が一致するので、周辺確率分布と同時分布が一致 - **invalid** joint probability distributions: 量子の場合観測と状態が一致せず、周辺確率分布と同時分布が一致するとは限らないお気持ち: エントロピーの基本的な性質(連鎖率と条件付きエントロピー)から導かれる次の不等式を使う $$ \begin{aligned} H\left(A_{0}, A_{1}, A_{2}, A_{3}\right) \leq H\left(A_{3} \mid A_{2}\right)+H\left(A_{2} \mid A_{1}\right) +H\left(A_{1} \mid A_{0}\right)+H\left(A_{0}\right) \end{aligned} $$ Aliceが$A_0, A_2$で観測, Bobが$A_1, A_3$で観測する時、エントロピーの基本的な性質から、$H\left(A_{0}, A_{3}\right) \leq H\left(A_{0}, A_{1}, A_{2}, A_{3}\right)$だから、 $$ H\left(A_{3} \mid A_{0}\right) \leq H\left(A_{3} \mid A_{2}\right)+H\left(A_{2} \mid A_{1}\right)+H\left(A_{1} \mid A_{0}\right) $$ **→one of the information theoretic (spatial) Bell inequalities given by [Braunstein and Caves, 1988](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.662)** ### information-theoretic temporal Bell inequality ## Open Problems ## Other Links - [著者の森越文明先生が物理学会でこの論文を発表されていた](https://www.jstage.jst.go.jp/article/jpsgaiyo/62.1.2/0/62.1.2_147_2/_article/-char/ja/) - [量子非局所性を用いた情報処理における不可逆性](https://www.jst.go.jp/kisoken/presto/complete/ryousi/theme/02-05morikoshi.html) - [森越先生の研究報告書](https://www.jst.go.jp/kisoken/presto/complete/ryousi/report/seika/02-05morikoshi.pdf)