- 65

- 4

**1. Homework Statement**

This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation.

In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as

$$ \vert \psi\rangle =\frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle).$$ Now let us assume that Alice and Bob have each one of the two particles of the EPR pair. Alice has another particle with spin ##1/2## in the state ##\vert \phi\rangle##. The state of the whole system, all three particles, is therefore given by

$$\begin{align*}\vert \phi\rangle \otimes \vert \psi\rangle &= \vert \phi\rangle \otimes \frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle)\\

&= \frac{1}{\sqrt{2}} (\vert\phi,0\rangle \otimes \vert 1\rangle - \vert\phi,1\rangle \otimes \vert 0\rangle). \end{align*}$$ Now Alice can measure her two particles, for example using ##P_i= \vert \chi_i\rangle\langle \chi_i\vert, i\in \{1,2,3,4\}## and

$$\begin{align*}

\vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle)\\

\vert\chi_2\rangle &= \frac{1}{\sqrt{2}}(\vert 0,1\rangle + \vert 1,0\rangle)\\

\vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,0\rangle - \vert 1,1\rangle)\\

\vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,0\rangle + \vert 1,1\rangle).

\end{align*}$$

Up until this point I understand the definitions and the idea. The problem arises when I try to calculate for example

$$P_1 \vert \phi\rangle\otimes\vert\psi\rangle = \frac{1}{2} \vert \chi_1\rangle \otimes (-\vert 1\rangle\langle 1\vert\phi\rangle - \vert 0\rangle\langle 0\vert\phi\rangle )$$

**2. Homework Equations**

All given above.

**3. The Attempt at a Solution**

We first need to figure out how ##P_i## acts on the tensor product of the states. Expanding the state gives

$$ P_1 \vert \phi\rangle\otimes\vert\psi\rangle = \frac{1}{\sqrt{2}} P_1(\vert\phi\rangle \otimes\vert 0\rangle \otimes \vert 1\rangle - \vert\phi\rangle \otimes\vert 1\rangle \otimes \vert 0\rangle). $$ Form this we can conclude that ##P_i## is of the form ##P_i = A\otimes B \otimes C##, where ##A,B,C## can be any operator. I tried to compute now the follwoing:

$$\begin{align*}\vert \chi_1\rangle\langle \chi_1\vert

&= \frac{1}{2}(\vert 0,1\rangle - \vert 1,0\rangle)(\langle 0,1 \vert -\langle 1,0\vert)\\

&= \frac{1}{2} (\vert 0\rangle \otimes\vert1\rangle - \vert 1\rangle \otimes\vert0\rangle)(\langle 0\vert\otimes\langle1 \vert -\langle 1\vert\otimes\langle0\vert),

\end{align*}$$

but can't really proceed from here since I don't really know how to calculate this... I suspect that after finishing this calculation I could define ##A\otimes B := \vert \chi_1\rangle\langle \chi_1\vert ##. Then I would only need to find ##C##, but I'm not really sure how to do that...

Am I doing something completely wrong here, or is this the right approach?