Entanglement
In the famous EPR paper, the primary argument is related to the bizarre non-local consequences of entangled quantum states. Einstein's main objection is related to the non-local (spatial) characteristic of entangled states. He suggested that quantum mechanics must be an incomplete theory and argued that our experience in nature appears local. He therefore concluded that hidden variables or additional degrees-of-freedom most likely exist and are not captured by current quantum theory formulations. Unfortunately, well maybe fortunately, Einstein appears to be incorrect about this as exemplified by the work of John S. Bell. The main outcome of Bell's work is that non-local theories can exist without the need for hidden variables and thus entangled quantum states are not restricted by spatial locality. I won't dive to deep into the foundations of quantum theory, but rather just look at a simple mathematical argument for why entangled states exist.
Simple Example
Let me first start with describing a quantum state (i.e. wavefunction) with basis vectors$^*$:
$$|0\rangle = \begin{bmatrix}
1 \\
0
\end{bmatrix} ,\;
|1\rangle = \begin{bmatrix}
0 \\
1
\end{bmatrix}
$$
This is a orthogonal basis set and corresponds to a Hilbert space (abstract function vector space) of $2^n$, where $n$ is the number of particles or objects. Each particle or object can be represented by a linear combination vectors in the Hilbert space. The basis corresponds to that used to describe quantum bits or qubits.
The basis states can be used to form product basis states, which are given by the tensor product, such that for a system of two objects the Hilbert space is $2^2 = 4$ and can be written as:
$$|00\rangle =\begin{bmatrix}
1 \\
0 \\
0 \\
0 \end{bmatrix} ,\;
|01\rangle =\begin{bmatrix}
0 \\
1 \\
0 \\
0 \end{bmatrix} ,\;
|10\rangle =\begin{bmatrix}
0 \\
0 \\
1 \\
0 \end{bmatrix} ,\;
|11\rangle =\begin{bmatrix}
0 \\
0 \\
0 \\
1 \end{bmatrix} \;,
$$
where the state of a particle/object is represented by a linear superposition of the product basis states, each having a complex amplitude (can be real if the imaginary part is zero). Now we can do something interesting, what if we take the product basis just written above and construct a potential wavefunction with the following features:
$$ |\Psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle\right).$$
Since we know from that the states $|00\rangle$ and $|11\rangle$ can also be written as product states, as shown above, lets do the following:
$$|\psi_1\rangle = \alpha_1 |0\rangle + \beta_1|1\rangle $$$$|\psi_2\rangle = \alpha_2 |0\rangle + \beta_2|1\rangle. $$
Now from these definitions lets take the tensor product:
\begin{align}|\psi\rangle &= |\psi_1\rangle \otimes |\psi_2\rangle \\
&= \alpha_{1} \alpha_{2} |0\rangle|0\rangle + \alpha_1 \beta_2 |0\rangle|1\rangle + \beta_1 \alpha_2 |1\rangle|0\rangle + \beta_1 \beta_2 |1\rangle|1\rangle \\
&= \alpha_{1} \alpha_{2} |00\rangle + \alpha_1 \beta_2 |01\rangle + \beta_1 \alpha_2 |10\rangle + \beta_1 \beta_2 |11\rangle .\\
\end{align}
So we now have the given state $|\Psi\rangle$ and the product state $|\psi\rangle$, and if we compare the terms we immediately observe that:
$$\alpha_1 \alpha_2 = \frac{1}{\sqrt{2}} \; \text{and} \; \beta_1 \beta_2 = \frac{1}{\sqrt{2}}$$
However, since for an orthonormal basis we must have that, $||\Psi\rangle|^2 = 1$ , it then has to be such that:
$$\alpha_1 \beta_2 = 0 \; \text{and} \; \beta_1 \alpha_2 = 0.$$
This would be a contradiction though because its is not feasible given $\alpha_1 \alpha_2 = \frac{1}{\sqrt{2}}$ and $\beta_1 \beta_2 = \frac{1}{\sqrt{2}}$. We therefore say that quantum state,
$$ |\Psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle\right),$$
represents a quantum entangled state, and cannot be written in terms of product states. Moreover, the determination or expectation of a single particle/object in this entangled state immediately tells us about the state of the other particle/object. For example, if the expectation value of particle 1 is $0.5$ in $|0\rangle$ or $|1\rangle$ then we will know with unity the state of particle 2 is.
What does it mean
The feature of entanglement is exclusive to quantum mechanics. One way I try to think of things is that quantum states can be composed of two types, those that exist due to the "combination" of quantum states (i.e., product states) and states which manifest as unique quantum solutions which are indecomposable into any other state or description. This fact, of entanglement is one of the great mysteries in quantum physics but its existence is enabling fantastic technologies.
There are two quotes from Niels Bohr's that I think fit well with this post, the first is:
"Einstein, stop telling God what to do [with his dice]!"
-Niels Bohr's, A response to Einstein's assertion that "God doesn't play dice".
This was in response to Einstein's dislike for many of the unexplained conundrums of quantum mechanics. The second quote:
"Anyone who is not shocked by quantum theory has not understood it."
-Niels Bohr's, The Philosophical Writings of Niels Bohr (1987),
Which captures the unexpected and possibly bizarre way of thinking one needs to succumb to in order to appreciate the predictive power of quantum theory (i.e. mathematics and interpretations). Despite this lack of comfort, quantum theories have made extremely accurate predictions and have been validated numerous times through meticulously controlled experiments.
I personally still find the foundations of quantum physics to be nebulous, but this is probably due to my own fallibility. To me its is unsatisfying that we do not not know the true meaning of the wavefunction or more specifically what is the meaning of a Universal wavefunction? Maybe its to complex that we will never know. Then there are questions about why entanglement exist, is it necessary to be consistent with physics as a whole (i.e., General Relativity)? Have we dismissed other understandings to quickly? Does the dendritic many-worlds interpretation originally proposed by Hugh Everett describe reality? What about revisiting non-local hidden variable theories such as Bohmian mechanics (also known as pilot-wave theory and De Broglie-Bohm theory)? How does non-locality make sense, is our notion of space being innate to the universe incorrect? Does space emerge from something else?
All of these question intrigue me, however, I have only scratched the surface and look forward to learning more about research focused on the foundations of quantum physics. There is a considerable learning curve and start-up time for me since my formal training is not in theoretical physics, but that won't stop me.
$^*$ The basis vectors are represented using the bra―ket notation pioneered by Paul Dirac. This is a very useful notation but may be unfamiliar to materials science people given that our solid state physics education, to my knowledge, never goes over this because we typically always deal with quantum states (i.e., wavefunctions) in a position or wave-vector basis, for example, $\psi_i(x) = A_i e^{\alpha_i \left(x-x_o\right)^2}$ or $\psi_{k}(x) = \frac{1}{\sqrt{V}} e^{ik\cdot x}$. Therefore the more broad and useful bra―ket notation goes unused.
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