Popular understanding of quantum mechanics usually focuses on three learning objectives:
Enraged Bohmian Mechanics enthusiasts approach the comment section (Source: Wellcome Collection)
This post focuses on the third point, specifically the part about FTL communication. There’s something called the “no-communication theorem” or “no-signaling principle” which shows that it’s impossible for us to use a pair of entangled particles as a FTL communication channel (much to the chagrin of many, many works of science fiction). Let’s be more precise: communication is a technical term which means I have some chosen bit (0 or 1) I can send to my counterpart on the other end of a channel. The channel doesn’t have to be perfect: all that matters is the receiver can discern the sent bit with probability better than a coin flip. A channel which enables the receiver to determine the correct bit only 51% of the time still involves communication, since we can re-send the same bit arbitrarily-many times to establish high levels of certainty of which bit was sent. The no-communication theorem says entangled particles, while indeed affecting one another in a FTL way, can never beat a coin flip when determining which bit was sent.
The no-communication theorem is obviously disheartening, so let’s stay in the Denial stage for a bit and poke around. Sending a full bit via a pair of entangled particles is, like many things humans want, too much to ask of the universe. Maybe there’s a hidden consolation prize, though? Clearly some kind of FTL interaction is taking place; surely it isn’t completely useless? Generations of clever scientists have considered this problem, and come up with very interesting scenarios where entanglement gives us FTL… coordination? Correlation? It’s difficult describe a phenomenon which isn’t communication, but it is something. Here we’ll learn just what that something is.
The rest of this post assumes basic familiarity with the bra-ket mathematical formalism of quantum computing; if you do not have this, you can watch a lecture I’ve created aimed at computer scientists here [slides]:
Review of the actual vector values of the common quantum states ∣0⟩, ∣1⟩, ∣+⟩, and ∣−⟩: |0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, |+\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}, |-\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix}
Here we’ll establish some knowledge required for the main event. Recall two qbits are entangled when you cannot factor their product state into the tensor product of two individual qbit states:
Try to factor this! You can’t; you get a system of four equations with no solution.
If you were to measure this state in the computational basis (collapsing each qbit to ∣0⟩ or ∣1⟩, it will only ever collapse to ∣00⟩ or ∣11⟩ with equal probability. If you measure one of the qbits before the other, it instantaneously forces the other qbit into the same state as the first. So if you measure one qbit and it collapses to ∣0⟩, you know the other qbit also collapsed to ∣0⟩.
The computational basis is not the only way of measuring qbits; you can also measure in another basis called the sign basis:
The computational basis on the left, and the sign basis on the right. A qbit (itself a vector on the unit circle) measured in these bases collapses probabilistically to one of the basis vectors. This is the unit circle, with top element of the 2-vector as the x-coordinate and bottom element of the 2-vector as the y-coordinate.
After measuring in the sign basis, your qbit will be in state ∣+⟩ or ∣−⟩ instead of ∣0⟩ or ∣1⟩. These types of measurements are called projective measurements, because what we’re doing is projecting a quantum state onto one of the two measurement basis vectors: