Quantum entanglement

Quantum entanglement is a phenomenon where each particle constituting a composite system cannot be described by its own characteristic state. Instead, the particles must be expressed as a single composite state. One such entangled composite system is the decay of a spin-0 particle at rest into two spin- $\frac{1}{2}$ particles with zero relative orbital angular momentum. According to the conservation of linear momentum, the particles move in opposite directions after decay. If we measure the spin of the particles by passing them through two Stern-Gerlach devices, we would get correlated results due to the conservation of spin momentum. For example, if one particle is measured to be spin up with respect to the $z$ -axis (see diagram below), the other particle is always measured to be spin down with respect to the same axis, and vice versa.

The quantum mechanical explanation of the results is that the singlet state $\vert\psi\rangle=\frac{1}{\sqrt{2}}[\vert+z\rangle_1\otimes\vert-z\rangle_2-\vert-z\rangle_1\otimes\vert+z\rangle_2]$ collapses into $\vert\phi_1\rangle=\vert+z\rangle_1\otimes\vert-z\rangle_2$ if the first measurement is spin up. If the first measurement is spin down, the composite state collapses into $\vert\phi_2\rangle=\vert-z\rangle_1\otimes\vert+z\rangle_2$ . Whether the first measurement is spin up or spin down is purely random, as evident from the coefficient of the singlet state. In other words, quantum mechanics provides, via $\vert\psi\rangle$ , a statistical distribution of possible outcomes of a measurement, in a way that the certainty that a system possesses a physical property is only determined after a measurement is made.

Question

How does the coefficient of the singlet state show that the collapse of the state is random?

Answer

The singlet state is normalised and is a linear combination of the states $\vert\phi_1\rangle=\vert+z\rangle_1\otimes\vert-z\rangle_2$ and $\vert\phi_2\rangle=\vert-z\rangle_1\otimes\vert+z\rangle_2$ ; i.e. $\vert\psi\rangle=\sum_{i=1}^{2}c_i\vert\phi_i\rangle$ , where $c_1=c_2=\frac{1}{\sqrt{2}}$ . Since $\vert c_i\vert^{2}$ is interpreted as the probability that a measurement of a system will yield an eigenvalue associated with the eigenfunction $\phi_i$ , the probability of obtaining an up-spin (or a down-spin) for the first measurement is 50%.

If we conduct the experiment a hundred times, with the first Stern-Gerlach device in the $+z$ direction and the second device in the $+x$ direction, we have (assuming a random distribution of spins after decay) the following results:

Particle 1 (measured in $+z$ )		Particle 2 (inferred in $+z$ )	Particle 2 (measured in $+x$ )
$EV_1$	Population	$EV_{2,z}$	$EV_2$	Population	$EV_1EV_2$	Average $EV_1EV_2$
$+\frac{\hbar}{2}$	50	$-\frac{\hbar}{2}$	$+\frac{\hbar}{2}$	25	$+\frac{\hbar^{2}}{4}$	$\frac{(25\times\frac{\hbar^{2}}{4})+[25\times(-\frac{\hbar^{2}}{4})]}{50}=0$
			$-\frac{\hbar}{2}$	25	$-\frac{\hbar^{2}}{4}$
$-\frac{\hbar}{2}$	50	$+\frac{\hbar}{2}$	$+\frac{\hbar}{2}$	25	$-\frac{\hbar^{2}}{4}$	$\frac{[25\times(-\frac{\hbar^{2}}{4})]+(25\times\frac{\hbar^{2}}{4})}{50}=0$
			$-\frac{\hbar}{2}$	25	$+\frac{\hbar^{2}}{4}$

In general, if the second Stern-Gerlach device is rotated at an angle $\theta$ relative to the first device, the composite spin angular momentum operator associated with $EV_1EV_2$ (the product of $EV_1$ and $EV_2$ ) is:

$\hat{S}_z^{\; (1)}\otimes\hat{S}_r^{\; (2)}=\left (\hat{S}_z^{\; (1)}\otimes I\right )\left (I\otimes\hat{S}_r^{\; (2)}\right )$

Substitute eq174 and eq184 (where we let $\phi=0^{\circ}$ ) into the above equation,

$\hat{S}_z^{\; (1)}\otimes\hat{S}_r^{\; (2)}=\frac{\hbar^{2}}{4}\begin{pmatrix} cos\theta &sin\theta &0 &0 \\ sin\theta &-cos\theta &0 &0 \\ 0 &0 &-cos\theta &-sin\theta \\ 0 &0 &-sin\theta &cos\theta \end{pmatrix}$

The average value of $EV_!EV_2$ over multiple experiments is:

$\small \langle\psi\vert\hat{S}_z^{\; (1)}\otimes\hat{S}_r^{\; (2)}\vert\psi\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 &1 &-1 &0 \end{pmatrix}\frac{\hbar^{2}}{4}\begin{pmatrix} cos\theta &sin\theta &0 &0 \\ sin\theta &-cos\theta &0 &0 \\ 0 &0 &-cos\theta &-sin\theta \\ 0 &0 &-sin\theta &cos\theta \end{pmatrix} \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\1 \\ -1 \\ 0 \end{pmatrix}$

$\langle\psi\vert\hat{S}_z^{\; (1)}\otimes\hat{S}_r^{\; (2)}\vert\psi\rangle=-\frac{\hbar^{2}}{4}cos\theta\; \; \; \; \; \; \; \; 226$

or in spin units of $\frac{\hbar}{2}$

$\langle\psi\vert\hat{S}_z^{\; (1)}\otimes\hat{S}_r^{\; (2)}\vert\psi\rangle=-cos\theta\; \; \; \; \; \; \; \; 227$

Finally, an interesting behaviour of an entangled system is that the particles constituting the system remain entangled if they are separated by a large distance, even at a distance greater than light can travel within the time between measurements. If so, and if the outcome of the first measurement is random, one may perceive that the two particles are able to influence each other instantaneously. This property of a composite system seems to be at odds with special relativity, which implies that nothing (including communication of any form) can travel faster than the speed of light. In opposition to the quantum mechanical interpretation of the results, Albert Einstein, along with Boris Podolsky and Nathan Rosen, proposed an alternate explanation called the Einstein-Podolsky-Rosen paradox.

Quantum entanglement

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