Key Concepts in Measurement

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Classical Mechanics

In classical mechanics, to fully specify state of the oscillator at any given moment, we need two parameters

$$(x,p)$$

We often introduce the so-called phase space, where each point defines one state.If we don't know the state exactly, we can define distribution function $f(x,p)$, meaning the probability within $[x,x+{\mathrm{d}}^{}x]$ and $[p,p+{\mathrm{d}}^{}p]$

$$\iint {\mathrm{d}}^{}x{\mathrm{d}}^{}pf(x,p)=1$$

Quantum Mechanics

Schrödinger Picture

Using the evolving states $|\psi (t)⟩$ and operators ${\hat{A}}_S$ to describe the motion. The time evolving equation for the schrodinger picture is

$$i\hslash \frac{{\mathrm{d}}^{}|\psi (t)⟩}{\mathrm{d}{t}^{}}=\hat{H}|\psi (t)⟩$$

Heisenberg Picture

Using the static states $|\psi ⟩$ and evolving operators ${\hat{A}}_H(t)$ to describe the motion. The time evolving equation for the heisenberg picture is

$$\frac{{\mathrm{d}}^{}{\hat{A}}_H(t)}{\mathrm{d}{t}^{}}=\frac{1}{i\hslash }[{\hat{A}}_H,\hat{H}]$$

Basic Concepts in Measurement

Physical reality is supposed to be non contextual (i.e independent of the measurement device attached to the system). For example, a harmonic oscillator before measurement should have definite values for its position momentum and energy. The measurement is only supposed to reduce our ignorance by revealing the reality to us.

Baye’s Theorem

$$P(x|y)=\frac{P(y|x)P(x)}{P(y)}$$

Look at the example of classical position measurement, in which the readout is denoted by

$$y=n+x$$

where the noise $n$ can be described by the Gaussion distribution

$$P(n)=\frac{1}{\sqrt{2\pi {\sigma }_n^2}}{e}^{-\frac{n^2}{2{\sigma }_n^2}}$$

Assume that the prior distribution is

$$p(x)=\frac{1}{\sqrt{2\pi {\sigma }_x^2}}{e}^{-\frac{(x-x_0{)}^2}{2{\sigma }_x^2}}$$

where $x_0$ is our guess of $x$ and ${\sigma }_x$ being the standard deviation. Thus we can get the distribution

$$P(y|x)=P(n=y-x)=\frac{1}{\sqrt{2\pi {\sigma }_n^2}}{e}^{-\frac{(y-x{)}^2}{2{\sigma }_n^2}}$$

The distribution for $y$ is (after normalization)

$$P(y)=\int {\mathrm{d}}^{}xP(y|x)P(x)=\frac{1}{\sqrt{2\pi {\sigma }_y^2}}{e}^{-\frac{(y-x_0{)}^2}{2{\sigma }_y^2}}({\sigma }^2={\sigma }_n^2+{\sigma }_x^2)$$

So according to Baye’s theorem, we can get the most important $P(x|y)$

$$P(x|y)=\frac{P(y|x)P(x)}{P(y)}=\frac{1}{\sqrt{2\pi {\sigma }_c^2}}{e}^{-\frac{(x-x_c{)}^2}{2{\sigma }_c^2}}$$

where the conditional mean $x_c={\sigma }_c^2(\frac{x_0}{{\sigma }_x^2}+\frac{y}{{\sigma }_n^2})$ and the conditional variance ${\sigma }_c^2=\frac{{\sigma }_n^2{\sigma }_x^2}{{\sigma }_n^2+{\sigma }_x^2}$.

We can also use adjoint distribution to understand these things

$$P(x|y)=\frac{P(x,y)}{P(y)}$$

where $P(x,y)$ is a two-dim Gaussian distribution

$$P(x,y)=\frac{1}{2\pi \sqrt{det(\mathrm{\Sigma })}}{e}^{-\frac{1}{2}(x-⟨x⟩,y-⟨y⟩){\mathrm{\Sigma }}^{-1}(x-⟨x⟩,y-⟨y⟩{)}^T}$$

and the covariance matrix $\mathrm{\Sigma }$ is defined as

$$\mathrm{\Sigma }=\left(\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{yx}& {\sigma }_{yy}\end{array}\right)$$

Quantum Case

Measure $\hat{A}$, the wavefunction randomly collapses into one of $\hat{A}$‘s eigenstate. The result follows Born's rule namely $|{\psi }^2|$ being the probability distribution of the measurement result.

For measursing position, these cases are indistinguishable:

1. QM: Quantum superposition of different $|x⟩$

$$|\psi ⟩=\int dx\psi (x)|x⟩$$

2. QM but classical superposition of different $|x⟩$

$$\{|{\mathrm{x}}_1⟩,|\psi \left({\mathrm{x}}_1\right){|}^2\}+\{|{\mathrm{x}}_2⟩,|\psi \left({\mathrm{x}}_2\right){|}^2\}+\cdots$$

a classical mixture: mixed state

3. CM: classical superposition of $x$ with probability $|\psi (x){|}^2$

$$\{{\mathrm{x}}_1,|\psi \left({\mathrm{x}}_1\right){|}^2\}+\{{\mathrm{x}}_2,|\psi \left({\mathrm{x}}_2\right){|}^2\}+\cdots$$

Quantization of Classical Measurement

Quantum measurement is contextual and non deterministic, classical notionof reality is gone.

Consider the quantum version of the classical measurement, let $x\to \hat{x}$ and $y\to \hat{y}$, then we’ll use $\psi (x)$ and $\psi (y)$ to describe the distribution of the two variables.

Before measurement, the total wave function can be depicted as the multiplication of the separate wave functions

$${\psi }_{\text{tot}}=\psi (x)\cdot \psi (y)$$

After measurement, the interaction makes two states entangle, so that the wave function can not be written as product of two seperate functions.

$${\psi }_{\text{tot}}^{\prime }=\psi (x,y)$$

So the measurement outcome $y=Y$ here means that we’are doing a projection for the wave function

$$\psi (x,y=Y)=\frac{{\psi }_{\text{tot}}(x,y=Y)}{⟨{\psi }_{\text{tot}}|{\hat{P}}_Y|{\psi }_{\text{tot}}⟩}$$

which analogous to

$$P(x|y=Y)=\frac{P(x,Y)}{p(Y)}$$

Quantum measurement: non contextual, non deterministic, non local.

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