发帖
雷达卡
2025/12/03
In this section, we delve into the concepts of partial measurement, wave function collapse, and projection measurement within quantum systems. These principles are essential for understanding how measurements affect subsystems in a composite quantum setup.
Wave Function Collapse and the Measurement Postulate
Consider a scenario where a measurement is performed on only one part of a larger, composite quantum system. For clarity, let's examine a system composed of two subsystems, labeled X and Y. We will analyze what occurs when we measure subsystem X while leaving Y unmeasured.
1. Initial Quantum State
The total quantum state |ψ of the combined system (X, Y) exists in a Hilbert space formed by the tensor product of the basis sets Σ from subsystem X and Γ from subsystem Y. This state can be expressed as:
|ψ = Σ(a,b) ∈ Σ×Γ αab |ab
Here, αab represents the probability amplitude associated with the basis state |ab. The normalization condition requires that the sum of the squared magnitudes of all amplitudes equals one:
Σa,b |αab| = 1
2. Probability of Measuring Subsystem X
We denote Pr(outcome is a) as the total probability that measuring subsystem X results in outcome a, where a ∈ Σ.
A measurement yielding outcome a corresponds to a projection onto the subspace spanned by the set { |ab | b ∈ Γ }. The probability of finding the system in a specific joint basis state |ab is given by |ab|ψ| = |αab|.
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To illustrate, consider the case where a = 0:
Pr(outcome is 0) = Σb ∈ Γ |0b|ψ| = Σb ∈ Γ |α0b|
This expression sums the probabilities of all states where subsystem X is in state 0, regardless of the state of Y.
Note: Clarification Through Example
The mathematical formulation may appear complex at first glance. Let’s simplify it with an example.
Suppose subsystem X has possible states {0, 1} and subsystem Y has states {+, }. Then the composite basis includes |0+, |0, |1+, and |1. If the initial state is:
|ψ = α0+|0+ + α0|0 + α1+|1+ + α1|1,
then the probability that measuring X gives outcome 0 is |α0+| + |α0| — simply the sum over all amplitudes where X is in state 0.
This demonstrates how partial measurement affects our knowledge of the system and leads to the collapse of the wave function conditioned on the observed outcome.
在量子力学中，一个特定的状态矢量位于由多个正交基矢量张成的空间中。通过观察其系数（即该状态在布洛赫球Z轴上的投影），可以直观理解测量得到结果0的概率。
现在将系统 X 与另一个系统 Y 关联起来（其中 Y 同样由若干正交向量构成）。复合系统的整体状态可表示为 X Y，它存在于一个更高维度的希尔伯特空间中。此时，观测到结果0的概率会分布到所有子空间中，例如 ∣0b、∣0b 等。
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运算 ab|ψ 的本质是利用左矢 ab| 从复合态 |ψ 中提取出对应基态 |a 或 |b 的特定系数（即振幅）。
为了便于进行部分测量的分析，我们首先对初始态进行重写处理：
将总态表示为：
|ψ = Σa∈Σ |a (Σb∈Γ αab |b)(1)
针对测量结果 a，定义一个未归一化的辅助态 |φa：
|φa = Σb∈Γ αab |b(2)
结合公式 (1) 和 (2)，原总态可重新表达为：
|ψ = Σa∈Σ |a |φa
进一步地，结果 a 对应的概率与该辅助态的模长平方相关：
∥|φa∥ = φa|φa = Σb∈Γ |αab|
因此，获得测量结果 a 的概率为：
Pr(outcome is a) = Σb∈Γ |αab| = ∥|φa∥
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根据量子测量假设，当对系统 X 进行基矢测量后，波函数会发生坍缩。若测量结果为 a，则整个系统将塌缩至相应的归一化状态：
|ψ → (|a |φa) / ∥|φa∥
这一过程体现了量子测量导致的状态突变特性，也是实现量子信息处理中条件操作的基础机制之一。
在量子测量过程中，复合系统会坍缩为一个新的归一化状态。当我们获得某个特定结果 a 时，这意味着子系统 XX X 正好处于状态 ∣a。相应地，整个系统将演化为向量 ∣a∣ 的归一化形式。
新的系统状态可表示为：
∣ψ′ = (∣a∣) / ‖∣‖ = ∣a(∣ / ‖∣‖)
其中，右侧的态 ∣ / ‖∣‖ 已经是一个归一化的量子态，记作 YY Y。
[此处为图片1]接下来通过 Python 代码进行实现。在 Qiskit 中，默认采用小端序（Little Endian）表示方式。例如，Statevector([0, 1, 1, 0, 1, 0, 0, 0] / sqrt(3)) 表示如下叠加态：
该态对应于以下基态的线性组合：
- ∣001：系数为 1/sqrt(3)
- ∣010：系数为 1/sqrt(3)
- ∣100：系数为 1/sqrt(3)
即整体为：(√3/3)∣001 + (√3/3)∣010 + (√3/3)∣100
[此处为图片2]下面进行具体操作：
from qiskit.quantum_info import Statevector, Operator
from numpy import sqrt
# 构建初始态 |w = (|001 + |010 + |100)/sqrt(3)
w = Statevector([0, 1, 1, 0, 1, 0, 0, 0] / sqrt(3))
display(w.draw('latex'))
# 对第0个量子比特进行测量
result, state = w.measure([0])
display(state.draw('latex'))
# 接着对第1和第2个量子比特进行联合测量
result, state = w.measure([1,2])
display(state.draw('latex'))
测量后的可能结果包括：
- 第一次测量后得到的状态可能是：(√3/3)∣001 + (√3/3)∣010 + (√3/3)∣100
- 后续测量可能导致系统进一步坍缩，如变为：(√2/2)∣010 + (√2/2)∣100
- 最终可能完全坍缩至某一确定态，例如 ∣100
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