查看“︁弱测量”︁的源代码

在[[量子力学]]（以及[[量子计算]]、[[量子信息]]）中，'''弱测量'''是一种[[量子測量|量子测量]]，其观察者平均来说只获得很少的有关系统的信息，但对状态的干扰也很小。<ref name="Brun2002">{{Cite journal |last=Todd A Brun |year=2002 |title=A simple model of quantum trajectories |journal=Am. J. Phys. |volume=70 |issue=7 |page=719–737 |arxiv=quant-ph/0108132 |bibcode=2002AmJPh..70..719B |doi=10.1119/1.1475328 |s2cid=40746086}}</ref>根据{{Tsl|en|Paul_Busch_(physicist)|4=Paul Busch}}的定理<ref name="Busch2009">{{Cite book|last=Paul Busch|authorlink=Paul Busch (physicist)|title="No Information Without Disturbance": Quantum Limitations of Measurement|work=The University of Western Ontario Series in Philosophy of Science|volume=73|series=Invited contribution, "Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: An International Conference in Honour of Abner Shimony", Perimeter Institute, Waterloo, Ontario, Canada, July 18–21, 2006|editor-last=J. Christian|editor3=W.Myrvold|publisher=Springer-Verlag, 2008|pages=229–256|year=2009|doi=10.1007/978-1-4020-9107-0|issn=1566-659X|arxiv=0706.3526|isbn=978-1-4020-9106-3}}</ref>可知，系统必然会受到测量的干扰。在文献中，弱测量也被称为不清晰（unsharp）<ref name="Gudder2005">{{Cite journal |last=Gudder |first=Stan |year=2005 |title=Non-disturbance for fuzzy quantum measurements |journal=Fuzzy Sets and Systems |volume=155 |page=18–25 |doi=10.1016/j.fss.2005.05.009 |number=1}}</ref>、模糊（fuzzy）<ref name="Gudder2005" /> 、迟钝（dull）<ref name="PeresBook">{{Cite book|last=Asher Peres|title=Quantum Theory, Concepts and Methods|publisher=Kluwer|year=1993|page=387|isbn=978-0-7923-2549-9}}</ref> 、噪声式（noisy）<ref name="Korotkov2003">{{Cite book|last=A. N. Korotkov|title=Quantum Noise in Mesoscopic Physics|url=https://archive.org/details/quantumnoisemeso00btti|url-access=limited|date=2003|editor-last=Y. v. Nazarov|publisher=Springer Netherlands|pages=[https://archive.org/details/quantumnoisemeso00btti/page/n208 205]–228|doi=10.1007/978-94-010-0089-5_10|isbn=978-1-4020-1240-2|arxiv=cond-mat/0209629}}</ref> 、渐进（approximate）<ref name="Winter1999">{{Cite journal |last=A. Winter |year=1999 |title=Coding Theorem and Strong Converse for Quantum Channels |journal=IEEE Trans. Inf. Theory |volume=45 |issue=7 |page=2481–2485 |arxiv=1409.2536 |doi=10.1109/18.796385 |s2cid=15675016}}</ref>或平和（gentle）的测量。此外，弱测量常常与一个不同但相关的概念「[[弱值]]」相混淆。<ref name="Aharonov1988">{{Cite journal |last=Yakir Aharonov |last2=David Z. Albert |last3=Lev Vaidman |name-list-style=amp |year=1988 |title=How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 |journal=Physical Review Letters |volume=60 |issue=14 |page=1351–1354 |bibcode=1988PhRvL..60.1351A |doi=10.1103/PhysRevLett.60.1351 |pmid=10038016 |s2cid=46042317}}</ref>

== 历史 ==
弱测量最初是在量子系统<ref name="Clerk2010">{{Cite journal |last=A. Clerk |last2=M. Devoret |last3=S. Girvin |last4=F. Marquardt |last5=R. Schoelkopf |year=2010 |title=Introduction to quantum noise, measurement, and amplification |journal=Rev. Mod. Phys. |volume=82 |issue=2 |page=1155–1208 |arxiv=0810.4729 |bibcode=2010RvMP...82.1155C |doi=10.1103/RevModPhys.82.1155 |s2cid=119200464}}</ref>的弱连续测量（即量子滤波和量子轨迹）的背景下考虑的。连续量子测量的物理学如下。考虑使用一个辅助系统（例如[[腔量子电动力学|场]]或[[电流]]）来探测量子系统，系统和探测器之间的相互作用使两者相互关联。通常，相互作用仅使系统与辅助系统具有弱关联（具体而言，相互作用幺正算子仅需[[微擾理論 (量子力學)|微扰]]展开到一阶或二阶）。通过测量辅助系统并利用量子测量理论，可以确定基于测量结果的系统状态。为了实现有效的测量，必须耦合进多个辅助系统并测量。在极限情况下，存在一系列辅助系统，使得测量过程可以在时间上是连续的。这一过程首先由以下学者表述：Michael B. Mensky; <ref name="Mensky1979a">{{Cite journal |last=M. B. Mensky |year=1979 |title=Quantum restrictions for continuous observation of an oscillator |journal=Phys. Rev. D |volume=20 |issue=2 |page=384–387 |bibcode=1979PhRvD..20..384M |doi=10.1103/PhysRevD.20.384}}</ref> <ref name="Mensky1979b">{{Cite journal |last=M. B. Menskii |year=1979 |title=Quantum restrictions on the measurement of the parameters of motion of a macroscopic oscillator |url=http://www.jetp.ac.ru/cgi-bin/e/index/r/77/4/p1326?a=list |journal=Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki |volume=77 |issue=4 |page=1326–1339 |bibcode=1979JETP...50..667M |access-date=2024-04-28 |archive-date=2018-01-09 |archive-url=https://web.archive.org/web/20180109181817/http://www.jetp.ac.ru/cgi-bin/e/index/r/77/4/p1326?a=list |dead-url=no }}</ref> Viacheslav Belavkin; <ref name="Belavkin1980">{{Cite journal |last=V. P. Belavkin |year=1980 |title=Quantum filtering of Markov signals with white quantum noise |journal=Radiotechnika I Electronika |volume=25 |page=1445–1453}}</ref> <ref name="Belavkin1992">{{Cite journal |last=V. P. Belavkin |year=1992 |title=Quantum continual measurements and a posteriori collapse on CCR |url=https://archive.org/details/sim_communications-in-mathematical-physics_1992-06_146_3/page/611 |journal=Commun. Math. Phys. |volume=146 |issue=3 |page=611–635 |arxiv=math-ph/0512070 |bibcode=1992CMaPh.146..611B |doi=10.1007/bf02097018 |s2cid=17016809}}</ref> Alberto Barchielli, L. Lanz, GM Prosperi; <ref name="Barchiellietal1982">{{Cite journal |last=A. Barchielli |last2=L. Lanz |last3=G. M. Prosperi |year=1982 |title=A model for the macroscopic description and continual observations in quantum mechanics |journal=Il Nuovo Cimento B |volume=72 |issue=1 |page=79–121 |bibcode=1982NCimB..72...79B |doi=10.1007/BF02894935 |s2cid=124717734}}</ref> Barchielli; <ref name="Barchielli1986">{{Cite journal |last=A. Barchielli |year=1986 |title=Measurement theory and stochastic differential equations in quantum mechanics |journal=Phys. Rev. A |volume=34 |issue=3 |page=1642–1649 |bibcode=1986PhRvA..34.1642B |doi=10.1103/PhysRevA.34.1642 |pmid=9897442}}</ref> Carlton Caves; <ref name="Caves1986">{{Cite journal |last=Carlton M. Caves |year=1986 |title=Quantum mechanics of measurements distributed in time. A path-integral formulation |journal=Phys. Rev. D |volume=33 |issue=6 |page=1643–1665 |bibcode=1986PhRvD..33.1643C |doi=10.1103/PhysRevD.33.1643 |pmid=9956814}}</ref> <ref name="Caves1987">{{Cite journal |last=Carlton M. Caves |year=1987 |title=Quantum mechanics of measurements distributed in time. II. Connections among formulations |journal=Phys. Rev. D |volume=35 |issue=6 |page=1815–1830 |bibcode=1987PhRvD..35.1815C |doi=10.1103/PhysRevD.35.1815 |pmid=9957858}}</ref> Caves, Gerald J. Milburn. <ref name="CavesMilburn1987">{{Cite journal |last=Carlton M. Caves |last2=G. J. Milburn |year=1987 |title=Quantum-mechanical model for continuous position measurements |url=https://espace.library.uq.edu.au/view/UQ:247705/UQ247705_OA.pdf |journal=Phys. Rev. A |volume=36 |issue=12 |page=5543–5555 |bibcode=1987PhRvA..36.5543C |doi=10.1103/PhysRevA.36.5543 |pmid=9898842}}</ref> 后来 Howard Carmichael <ref name="Carmichael">{{Cite book|title=An open systems approach to quantum optics, Lecture Notes in Physics|last=Carmichael|first=Howard|year=1993|publisher=[[Springer Science+Business Media|Springer]]}}</ref>和 Howard M.Wiseman <ref name="Wiseman1994">{{Cite thesis |degree=PhD |last=Wiseman |first=Howard Mark |date=1994 |title=Quantum trajectories and feedback |publisher=[[University of Queensland]] |url=https://espace.library.uq.edu.au/view/UQ:366355 |access-date=2024-04-28 |archive-date=2022-10-28 |archive-url=https://web.archive.org/web/20221028155738/https://espace.library.uq.edu.au/view/UQ:366355 |dead-url=no }}</ref>也为该领域做出了重要贡献。

弱测量的概念经常被错误地归于 [[亚基尔·阿哈罗诺夫|Yakir Aharonov]] 、 David Albert 和 Lev Vaidman 。<ref name="Aharonov1988"/>在他们的文章中，他们考虑了一个弱的测量的例子（也许也撞上了“弱测量”这个短语），并以此作为动机来定义[[弱值]]（他们在此首次定义了弱值）。

== 数学表述 ==
对于弱测量，尚无普遍接受的定义。一种方法是将弱测量声明为这样一种广义测量，其[[量子操作|克劳斯算子]]中的一些或全部接近于[[恆等函數|恒等算子]]。<ref name="OreshkovBrun2005">{{Cite journal |last=O. Oreshkov |last2=T. A. Brun |year=2005 |title=Weak Measurements Are Universal |journal=Phys. Rev. Lett. |volume=95 |issue=11 |page=110409 |arxiv=quant-ph/0503017 |bibcode=2005PhRvL..95k0409O |doi=10.1103/PhysRevLett.95.110409 |pmid=16196989 |s2cid=43706272}}</ref>下面采用的方法是使两个系统发生弱相互作用，然后测量其中一个系统。<ref name="Wiseman and Milburn">{{Cite book|title=Quantum Measurement and Control|url=https://archive.org/details/quantummeasureme00wise|url-access=limited|last=Wiseman|first=Howard M.|last2=Milburn, Gerard J.|year=2009|publisher=[[Cambridge University Press]]|location=[[Cambridge]]; [[New York City|New York]]|isbn=978-0-521-80442-4|pages=[https://archive.org/details/quantummeasureme00wise/page/n477 460]}}</ref>详细介绍这种方法之后，我们将通过示例进行说明。

=== 弱相互作用和辅助耦合测量 ===
考虑一个系统，其初始的[[量子態|量子态]]为 <math>|\psi\rangle</math> ，同时辅助系统处于 <math>|\phi\rangle</math> ，联合的初始状态则为 <math>|\Psi\rangle = |\psi\rangle \otimes |\phi\rangle</math> 。这两个系统依照[[哈密顿算符|哈密顿算子]] <math>H = A \otimes B</math> 相互作用，其[[生成元 (幺正算子群)|生成]]的时间演化算子 <math>U(t) = \exp[-ixtH] </math> （取 <math>\hbar = 1</math> 的单位制）， 其中 <math>x</math> 是“相互作用强度”且具有时间倒数的[[量纲]]。假设相互作用时间固定为 <math>t = \Delta t</math> 且 <math>\lambda = x \Delta t</math> 很小以至于 <math>\lambda^3 \approx 0</math> 。 <math>U</math> 关于 <math>\lambda</math> 的[[级数展开]]给出

: <math>
\begin{align}
 U &= I \otimes I - i\lambda H - \frac 1 2 \lambda^2 H^2 + O(\lambda^3) \\
 &\approx I \otimes I - i\lambda A \otimes B - \frac 1 2 \lambda^2 A^2 \otimes B^2.
\end{align}
</math>

由于在[[微扰论]]中只需要将幺正算子展开到低阶，所以称其为一个弱的相互作用。此外，幺正算子的主要部分是恒等算子，因为 <math>\lambda</math> 和 <math>\lambda^2</math> 很小，这意味着相互作用后的状态与初始状态并没有太多区别。相互作用后系统的联合状态为

: <math>
|\Psi'\rangle = \left(I \otimes I - i\lambda A \otimes B - \frac 1 2 \lambda^2 A^2 \otimes B^2\right) |\Psi\rangle.
</math>

现在我们对辅助系统进行测量来了解系统，这称为辅助系统耦合测量。我们将考虑（辅助系统上的）在基 <math>|q\rangle</math> 下的测量，其中 <math>|q\rangle</math> 满足 <math display="inline">\sum_q |q\rangle \langle q| = I</math> 。两个系统上的测量都由到联合状态 <math>|\Psi'\rangle</math> 的[[投影 (线性代数)|投影算子]] <math>\Pi_q = I \otimes |q\rangle \langle q|</math> 来描述。从[[量子測量|量子测量理论]]可知测量后的条件状态是

: <math>
\begin{align}
 |\Psi_q\rangle &= \frac{\Pi_q |\Psi'\rangle}{\sqrt{\langle\Psi'| \Pi_q |\Psi'\rangle}} \\
 &= \frac{I \langle q|\phi\rangle - i\lambda A \langle q| B |\phi\rangle - \frac 1 2 \lambda^2 A^2 \langle q| B^2 |\phi\rangle}{\mathcal N} |\psi\rangle \otimes |q\rangle,
\end{align}
</math>

其中 <math display="inline">\mathcal N = \sqrt{\langle\Psi'| \Pi_q |\Psi'\rangle}</math> 是[[歸一條件|归一化]]因子。注意辅助系统状态记录了测量的结果。 <math display="inline">M_q := I \langle q|\phi\rangle - i\lambda A \langle q| B |\phi\rangle - \frac 1 2 \lambda^2 A^2 \langle q| B^2 |\phi\rangle</math> 是系统的希尔伯特空间上的算子，称为[[量子操作|克劳斯算子]]。

在这些克劳斯算子对应的测量后，联合系统的状态为

: <math>
|\Psi_q\rangle = \frac{M_q |\psi\rangle}{\sqrt{\langle \psi|M_q^\dagger M_q|\psi\rangle}} \otimes |q\rangle.
</math>

算子 <math>E_q = M_q^\dagger M_q </math> 是所谓的[[正算子值测度|正算子测量]]的元素，其须满足 <math display="inline">\sum_q E_q = I</math> 从而使得相应的概率之和为一： <math display="inline">\sum_q \Pr(q|\psi) = \sum_q \langle\psi| E_q |\psi\rangle = 1</math> 。由于辅助系统不再关联于主系统，它只是记录测量的结果，我们可以[[偏迹|迹掉]]它。这做法将给出主系统本身的条件状态：

: <math>
|\psi_q\rangle = \frac{M_q |\psi\rangle}{\sqrt{\langle \psi|M_q^\dagger M_q|\psi\rangle}},
</math>

这里仍用 <math>q</math> 标记测量结果。事实上，这些考虑使得人们得以导出[[量子轨迹]]。

=== 克劳斯算子示例 ===
我们将使用 Barchielli, Lanz, Prosperi<ref name="Barchiellietal1982"/>以及 Caves, Milburn<ref name="CavesMilburn1987"/>给出的[[高斯函数|高斯型]]克劳斯算子的典型例子。取 <math>H = x \otimes p</math> ，其中两个系统的位置和动量算子满足通常的[[對易關係|正则对易关系]] <math>[x, p] = i</math> 。取辅助系统的初态为一高斯分布

: <math>
|\Phi\rangle = \frac{1}{(2\pi\sigma^2)^{1/4}} \int dq' \exp[-q'^2/(4\sigma^2)] |q'\rangle.
</math>

辅助系统的[[位置表象|位置]][[波函数]]为

: <math>
\Phi(q) = \langle q|\Phi\rangle = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp[-q^2/(4\sigma^2)].
</math>

前文的克劳斯算子（在前文的表达式中取 <math>\lambda = 1</math> ）为

: <math>
\begin{align}
 M(q) &= \langle q| \exp[-ix \otimes p] |\Phi\rangle \\
 &= \frac{1}{(2\pi\sigma^2)^{1/4}} \exp[-(q - x)^2/(4\sigma^2)],
\end{align}
</math>

而相应的正算子测量的元素是

: <math>
\begin{align}
 E(q) &= M_q^\dagger M_q \\
 &= \frac{1}{\sqrt{2\pi\sigma^2}} \exp[-(q - x)^2/(2\sigma^2)],
\end{align}
</math>

且服从 <math display="inline">\int dq\, E(q) = I</math> 。在文献中经常能看到另一种表现形式。使用位置算子的[[单位分解|谱表示]] <math display="inline">x = \int x'dx' |x'\rangle \langle x'|</math> ，有

: <math>
\begin{align}
 M(q) &= \frac{1}{(2\pi\sigma^2)^{1/4}} \int dx' \exp[-(q - x')^2/(4\sigma^2)] |x'\rangle \langle x'|, \\
 E(q) &= \frac{1}{\sqrt{2\pi\sigma^2}} \int dx' \exp[-(q - x')^2/(2\sigma^2)] |x'\rangle \langle x'|.
\end{align}
</math>

注意 <math display="inline">\lim_{\sigma \to 0} E(q) = |x = q\rangle \langle x = q| </math> 。<ref name="CavesMilburn1987"/>也就是说，在特定的极限下，这些算子趋于对位置的强测量；对于 <math>\sigma</math> 的其他值，这种测量则称为是有限强度的；对于 <math>\sigma \to \infty</math> 的极限情况，则说测量是弱的。

== 信息获取与状态扰动的得失交换 ==
如前所述，Busch的定理<ref name="Busch2009"/>阻止了免费午餐的出现：没有对态的扰动就无法取得信息。然而，信息获取和态的扰动间的得失交换已被许多作者刻画，其中包括 C.A. Fuchs 和 [[艾雪·佩雷斯|Asher Peres]];<ref name="FuchsPeres1996">{{Cite journal |last=C. A. Fuchs |last2=A. Peres |year=1996 |title=Quantum-state disturbance versus information gain: Uncertainty relations for quantum information |journal=Phys. Rev. A |volume=53 |issue=4 |page=2038–2045 |arxiv=quant-ph/9512023 |bibcode=1996PhRvA..53.2038F |doi=10.1103/PhysRevA.53.2038 |pmid=9913105 |s2cid=28280831}}</ref> Fuchs; <ref name="Fuchs1996">{{Cite journal |last=C. A. Fuchs |year=1996 |title=Information Gain vs. State Disturbance in Quantum Theory |arxiv=quant-ph/9611010 |bibcode=1996quant.ph.11010F}}</ref> Fuchs, KA. Jacobs;<ref name="FuchsJacobs2001">{{Cite journal |last=C. A. Fuchs |last2=K. A. Jacobs |year=2001 |title=Information-tradeoff relations for finite-strength quantum measurements |journal=Phys. Rev. A |volume=63 |issue=6 |page=062305 |arxiv=quant-ph/0009101 |bibcode=2001PhRvA..63f2305F |doi=10.1103/PhysRevA.63.062305 |s2cid=119476175}}</ref> K. Banaszek. <ref name="Banaszek2001">{{Cite journal |last=K. Banaszek |year=2006 |title=Quantum-state disturbance versus information gain: Uncertainty relations for quantum information |journal=Open Syst. Inf. Dyn. |volume=13 |page=1–16 |arxiv=quant-ph/0006062 |doi=10.1007/s11080-006-7263-8 |s2cid=35809757}}</ref>

最近，人们在所谓的“温和测量引理”的语境下检验了信息获取与态的扰动间的交换关系。<ref name="Winter1999"/><ref name="OgawaNagaoka1999">{{Cite journal |last=T. Ogawa |last2=H. Nagaoka |year=1999 |title=Strong Converse to the Quantum Channel Coding Theorem |journal=IEEE Trans. Inf. Theory |volume=45 |issue=7 |page=2486–2489 |arxiv=quant-ph/9808063 |bibcode=2002quant.ph..8139O |doi=10.1109/18.796386 |s2cid=1360955}}</ref>

== 应用 ==
从很早以前就已经很清楚，弱测量的主要用途是用于量子系统的反馈控制或自适应测量。事实上，这是 Belavkin 大部分工作的动机，而 Caves 和 Milburn 也给出了一个明确的例子。自适应弱测量的一个早期应用是{{Le|Dolinar接收器|Dolinar receiver}}，<ref name="Dolinar1973">{{Cite journal |last=S. J. Dolinar |year=1973 |title=An optimum receiver for the binary coherent state quantum channel |url=https://dspace.mit.edu/bitstream/handle/1721.1/56414/RLE_QPR_111_VII.pdf |journal=MIT Research Laboratory of Electronics Quarterly Progress Report |volume=111 |page=115–120 |access-date=2024-04-28 |archive-date=2023-02-25 |archive-url=https://web.archive.org/web/20230225075127/https://dspace.mit.edu/bitstream/handle/1721.1/56414/RLE_QPR_111_VII.pdf |dead-url=no }}</ref>该接收器已在实验上实现。<ref name="Cook">{{Cite journal |last=R. L. Cook |last2=P. J. Martin |last3=J. M. Geremia |year=2007 |title=Optical coherent state discrimination using a closed-loop quantum measurement |journal=Nature |volume=446 |issue=11 |page=774–777 |bibcode=2007Natur.446..774C |doi=10.1038/nature05655 |pmid=17429395 |s2cid=4381249}}</ref><ref name="Becerra2013">{{Cite journal |last=F. E. Becerra |last2=J. Fan |last3=G. Baumgartner |last4=J. Goldhar |last5=J. T. Kosloski |last6=A. Migdall |year=2013 |title=Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination |journal=Nature Photonics |volume=7 |issue=11 |page=147–152 |bibcode=2013NaPho...7..147B |doi=10.1038/nphoton.2012.316 |s2cid=41194236}}</ref>弱测量的另一个有趣的应用是使用弱测量，然后跟一个幺正算子（可能依赖于弱测量的结果）来合成其他广义测量。<ref name="OreshkovBrun2005"/> Wiseman 和 Milburn 的书<ref name="Wiseman and Milburn"/>是关于许多现代发展的良好参考资料。

== 延伸阅读 ==

* Brun 的文章<ref name="Brun2002"/>
* Jacobs 和 Steck的文章<ref name="JacobsSteck2006">{{Cite journal |last=K. Jacobs |last2=D. A. Steck |year=2006 |title=A straightforward introduction to continuous quantum measurement |journal=Contemporary Physics |volume=47 |issue=5 |page=279–303 |arxiv=quant-ph/0611067 |bibcode=2006ConPh..47..279J |doi=10.1080/00107510601101934 |s2cid=33746261}}</ref>
* {{Cite book|title=Quantum measurement theory and its applications|publisher=Cambridge University Press|date=2014|location=Cambridge ; New York|isbn=978-1-107-02548-6|first=Kurt|last=Jacobs}}
* ]{{Cite book|title=Quantum Measurement and Control|url=https://archive.org/details/quantummeasureme00wise|url-access=limited|last=Wiseman|first=Howard M.|last2=Milburn, Gerard J.|year=2009|publisher=[[Cambridge University Press]]|location=[[Cambridge]]; [[New York City|New York]]|isbn=978-0-521-80442-4|pages=[https://archive.org/details/quantummeasureme00wise/page/n477 460]}}
* Tamir 和 Cohen 的文章<ref name="TamirCohen2013">{{Cite journal |last=Boaz Tamir |last2=Eliahu Cohen |year=2013 |title=Introduction to Weak Measurements and Weak Values |journal=Quanta |volume=2 |issue=1 |page=7–17 |doi=10.12743/quanta.v2i1.14 |doi-access=free}}</ref>

== 参考资料 ==
{{Reflist}}
[[Category:量子測量]]
[[Category:量子信息科学]]