sage
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Unitarity of Quantum Mechanics and measurement
Yestody in the chat room, there is some discussion about the unitarity of quantum mechanics. Here is a rather long and somewhat technical note of it. I apologize for not giving a very detailed and down to earth explanation.
Sorry about having to introduce density matrix. But I find it is the most closest analogy to the Liouville theorem,
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Roughly speaking, unitarity means conservation of probability.
Suppose I start with a state |psi>. After a while, the state evolves
to U|psi>. Then, conservation of probability means that
<psi|U^+ U |psi>=<psi|psi>. It requires the evolution matrix to be
unitary, i.e., U^+ U=1.
More strictly speaking, unitarity is somewhat like the Liouville
theorem. It is the conservation of some probability 'flow'...
It is the one of the biggest mysteries that measurement irreversibly
destroy this flow. This can be illustrated just by the double-slit
experiment.
----------------------------------------------------
Before doing that, just to be systematic, let me introduce a little
bit of language, the density matrix. For any single quantum mechanical state
|psi>, define density matrix by
W=|psi><psi|.
First, it is a matrix. Given any base |1>, |2>....|n>....., matrix
element W_mn = <m|psi><psi|n>. It is called density matrix since the
expectation value of any operator O is
<O>=Tr (W O)
where the trace is carried out over any complete basis. This is again
easy to see as
<O> =<psi|O|psi>= Sum_n,m <psi|m><m|O|n><n|psi>
= Sum <n| ( |psi><psi|) O |n> = Trace (W O).
Therefore, if you like, density matrix is a statistical weight (which
is analogous to the quantity in Liouville Theorem). We can also show
that Tr(W)=Sum <n|psi><psi|n> = Sum <psi|n><n|psi>=
<psi|psi>=1. Moreover, we can show that Tr(W W )=Tr(W W W....)=1
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Now the first important result, a sort of 'quantum Liouville'
theorem:
Any unitary evolution will 'conserve' density matrix of a single
quantum mechanical state.
Let's understand this more carefully. After evolution U, the density
matrix will be
W -> U|psi><psi|U^+
Now it is easy to show Tr(W)=1 is not affected by the evolution since
Tr (U|psi><psi|U^+ ) = Tr (U^+ U|psi><psi| ) = Tr ( |psi><psi|)=1.
Moreover, Tr(W W) should also not to be affected by any unitary
evolution.
...
This is what we mean by 'conserve'.
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Now we come back to double-slit experiment. Say there are slit 1 and
2. The incoming particle is in a state 1/sqrt(2) *(|1>+|2>). Then if
we put a screen after the slit, we see interference pattern...
The density matrix for the incoming state is
1/2(|1>+|2>)(<1|+<2|),
or, in a matrix form in the base of |1> and |2>
| 1/2 1/2 |
| | .
| 1/2 1/2 |
one can easily verify that this staisfies Tr(W)=1,
Tr|W^2|=1,.... Notice that the off-diagonal entry in the density
matrix is very important. It is the memory of a state which could be
both 1 and 2, both here and there. It related to all quantum
mechanical magic/mystery/misery/misunderstanding about coherence, pure
state, etc. It is crucial to make Tr(W^2)=1.
Now suppose we really want to find out what is going on. In
particular, we put detectors at each slit to find out which slit the
particle does pass through. Quantum mechanics rule (it is the RULE,
no explanation ) says that too bad,
now we have made the damn measurement. We destroyed the coherence. The
wave-packet of every partile passing through collapses. The
result would be EITHER |1> OR |2> (not BOTH 1 AND 2). We all know that
it means no interference, and so on.
Suppose we still want to construct a density matrix for the system
after that measurement, it would have to be
| 1/2 0 |
| | .
| 0 1/2 |
since there would be no interference anymore. Immediately, we see that
Tr(W^2)=1/2 not 1!!
We have proved that any unitary evolution cannot change
Tr|W^2|. Therefore, something which is not unitary must have happened
during the measurement.
| 发表时间：2004-12-12, 16:03:20 |
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权权
发表文章数: 92
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Re: Unitarity of Quantum Mechanics and measurement
Thank you~~
不忧不惧
| 发表时间：2004-12-12, 21:00:50 |
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卢昌海
发表文章数: 1617
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Re: Unitarity of Quantum Mechanics and measurement
Thanks, sage! I missed yesterday's chat and was only able to view part of it this morning (since the chat went beyond the default 150 lines of history - now expanded to 300 lines).
One thing I would like to add to the discussion is: it seems to me that during a measurement process, the conservation of probability - in the sense that probabilities of all possible outputs sum up to one - is NOT violated. Tr(W)=1 - and only Tr(W)=1 - is what represents it and is well preserved during a measurement process.
Tr(W^2)=1, however is only a characteristics of pure states, differentiates it from mixed states, which has Tr(W^2)<1. During a measument process, as sage shown, Tr(W^2) can be changed from 1 to a different value, say 1/2, which means that a pure state can be converted to a mixed state. This process, also as sage demonstrated, can't be achieved by any unitary evolution. But, this does not mean that probability is not conserved. Unitary evolution conserves probability, but the inverse is not true, namely not every process that conserves probability is described by unitary evolution, especially those that converts a pure state into a mixed state.
What makes quantum measurement subtle is not that it violates the conservation of probability, but its violation of unitary evolution (which is incompatible with the traditional Schrödinger's equation). When we only talk about pure states, as we do in normal quantum mechanics, unitarity and conservation of probability are basically the same, but in general, they are slightly different.
宠辱不惊，看庭前花开花落
去留无意，望天空云卷云舒
| 发表时间：2004-12-12, 22:29:59 |
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星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
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Re: Unitarity of Quantum Mechanics and measurement
呵呵，谢谢sage兄的详细讲解，够清楚明了！看来sage兄是不是转去研究量子信息与量子计算了？这可是一个热门、
我把有个地方写更详细一点，让大家看的更轻松：
<O> =<psi|O|psi>= Sum_n,m <psi|m><m|O|n><n|psi>
由于<psi|m>是一个数（and so on)，可以移来移去，
所以原式＝Sum_n,m {<n|psi><psi|m><m|O|n>}
=Sum_n,m {W*_nm <m|O|n>} (W*_nm 是W_mn 的共轭转置）
= Trace (WO). (回想一下矩阵乘法的展开表示。另外由定义知，W是自共厄的）
我感觉量子测量理论似乎有许多迷惑人甚至似是而非的东西，而且不同人有时对同一样东西理解可能也不相同。测量导致波包坍塌也好，产生熵增也好（不可逆），感觉是从量子力学世界（同时遵从量子力学和经典力学统计规律）跌到（或突变到）经典力学世界（只遵从经典力学统计规律），没有了“相位叠加”这种标志量子力学的东西。我记得，有人认为要把环境和测量仪器（包括观测者）考虑在内作为一个更大的系统来考察，而不是象传统那样只分析被观测者，只是不知道这种考虑管不管用。
我猜想，如果说导致未来物理革命的乌云是什么的话，也许对量子力学的解释问题要算一个。如果这样，Einstein又算是有功劳。他不那样钻牛角尖，后人恐怕没有把量子力学的认识问题想得那么多。就象D.Bohm，他原来以为自己懂量子力学，他出了一本量子力学的书之后，觉得自己反而不懂了。Feynman居然也说没有人真正懂量子力学。
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2004-12-12, 22:47:30 |
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: Unitarity of Quantum Mechanics and measurement
What makes quantum measurement subtle is not that it violates the conservation of probability,
-----------------------------------
Thanks changhai for clarifying this. I was being sloppy when saying 'conservation of probability' yestoday . That is why I put out this more elaborated note.... :-)
| 发表时间：2004-12-12, 23:17:47 |
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星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Unitarity of Quantum Mechanics and measurement
写完我的帖子，发现昌海兄的帖子，让我回忆起更多。
的确，我以前只听说测量导致纯态向混合态的转变，好像没有听说到破坏概率守恒。
Unitary evolution conserves probability, but the inverse is not true, namely not every process that conserves probability is described by unitary evolution, especially those that converts a pure state into a mixed state.
我想是不是这样来理解：
1）只要密度矩阵的迹保持为1，概率能就保持守恒。密度矩阵对角元对应各个测量本征值（或终态）所对应的概率权重因子，非对角元对应不同本征态之间的相干叠加。进入混合态之后，相干叠加消失了，只剩下对角元。
2）的确，如果从一个态φ 演化为另一个态只能表示为ψ＝Uφ，那么要想概率守恒，U好像只能是unitary算子；但是，如果另一个态，例如ψ＝Uφ＋π （甚至非线性演化？）则U不一定是unitary算子，但此时也许不再叫演化算子了.
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2004-12-12, 23:22:05 |
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: Unitarity of Quantum Mechanics and measurement
看来sage兄是不是转去研究量子信息与量子计算了？这可是一个热门、
No. I am studying more useless things. I learned this stuff from Prof. Zeng XinYu at Fudan.
我记得，有人认为要把环境和测量仪器（包括观测者）考虑在内作为一个更大的系统来考察，而不是象传统那样只分析被观测者，只是不知道这种考虑管不管用。
This might be called decoherence propelled by Wheeler company. I don't know that much about it. Thinking naively, if all matter obey schrodinger equation (or whatever Hermitian generator), then the evolution will always be unitary....
including enviroment means that the dimension of the density matrix is much larger. In principle, the off-diagonal entries could move to other places. therefore, if we just focus on the 2 by 2 submatrix, it is possible that it is diagonal. However, the coherence is somewhere else. This does not seem solve the basic measurement problem especially if we are allowed to think of the universe as a big pure state... This would force us to think of some multiworld theory or something like that. maybe we are not allowed to say universe a pure state since gravity is very mysterious. but this is certainly just speculation.
anyway, I hope this will tell us something new but I don't see even a vague hint of it. I am still too young to work on that...
我猜想，如果说导致未来物理革命的乌云是什么的话，也许对量子力学的解释问题要算一个。如果这样，Einstein又算是有功劳。他不那样钻牛角尖，后人恐怕没有把量子力学的认识问题想得那么多。就象D.Bohm，他原来以为自己懂量子力学，他出了一本量子力学的书之后，觉得自己反而不懂了。Feynman居然也说没有人真正懂量子力学。
| 发表时间：2004-12-12, 23:41:05 |
作者资料
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星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Unitarity of Quantum Mechanics and measurement
谢谢sage兄的详细回答！
有些问题如果没有人专门提出，不会去想到。一经有人提出来大家讨论，结果在讨论中能获得更深入的了解和理解，这就是学术交流讨论的好处。
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2004-12-15, 23:51:33 |
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