In Susskind's Theoretical Minimum: Quantum Mechanics, he states that, "complete knowledge of a system in classical physics implies complete knowledge of every part of a system" (159). Moreover, he states that "we can know everything about a composite system - everything there is to know, anyway - and still know nothing about its constituent parts" (175).

He also provides an example of an entangled system, which consists of two coins - a penny and a dime - and two individuals - A and B. Each individual is given a coin randomly, and without looking at it, travels to distant parts of the universe.

The composite system has a state $|\Psi\rangle$ which can be represented by $$|\Psi\rangle = \frac{1}{\sqrt{2}}\left(|pd\rangle - |dp\rangle\right)$$ where $p$ stands for penny and $d$ for dime, and the first element is the state of A and the second that of B.

In this example,

  1. Referring to Susskind's second statement above, what do we actually know about the state of the combined system? If there is one penny and one dime, and each person has one of the coins, there are only two possible measurements: $|pd\rangle$ or $|dp\rangle$. Since the coins are distributed randomly, there is equal probability of the composite measurement of the composite system yielding $|pd\rangle$ or $|dp\rangle$. Hence, the distribution over the measurements is uniform and there is no information conveyed in the state vector. So do we know nothing about the composite?
  2. How is what we know about the composite system different from what we know about the individual subsystems? Isn't it also true that there is a 50% chance for each person to have each coin, and hence, we also know nothing about the coin held by the individuals?

Sorry if the question is confusing. I'm just trying to wrap my head around the concept of entanglement.


The thing with quantum mechanics is that this isn't quite right:

there are only two possible measurements: $|pd⟩$ or $|dp⟩$.

Even at the single-system level, there are many more possible measurement procedures than just on $|p⟩$ and $|d⟩$. Instead, there's plenty of measurements that project onto superposition states like $|p⟩+|d⟩$ or $|p⟩-|d⟩$ (or, equivalently, which perform the same transformation that prepares those superpositions, and then measures on $|p⟩$ and $|d⟩$).

Now, in Susskind's notation, the statement "complete knowledge" is essentially equivalent to the system being in a pure state. If we have a single coin which we know is in a pure state, and repeated measurements yield a 50:50 split between $p$s and $d$s, then we essentially know that it is in the superposition state $|p⟩+e^{i\varphi}|d⟩$ for some complex phase $e^{i\varphi}$. However, that says more than just the 50:50 split we had above: it means that there exists a measurement which can discriminate between $|p⟩+e^{i\varphi}|d⟩$ and $|p⟩-e^{-i\varphi}|d⟩$, and which will always yield the former and never the latter.

Thus far for a single coin. What happens with our entangled state? If A tries to measure the state of the coin between $|p⟩$ and $|d⟩$, they will observe a 50:50 split, which is suggestive of a superposition state. To try and demonstrate that, the thing to do is to try and find some $\varphi$ such that a measurement along $|p⟩+e^{i\varphi}|d⟩$ and $|p⟩-e^{-i\varphi}|d⟩$ will pick up some asymmetry between the two possible outcomes, and here's the rub: no matter what you do, i.e. no matter what $\varphi$ you choose, you'll always come up with a 50:50 split on that measurement. This is impossible for a single system in a pure state (i.e. with "complete knowledge") and in fact it is the state of zero knowledge about the state of the coin.

It is therefore tempting to conclude that, in fact, we have no information about the complete system, but this is not the case - we have complete knowledge about the full system, i.e. we know that it is in a pure state. How can we demonstrate this? There's two main ways:

  • If A and B can bring their systems back together, to perform coherent quantum operations on the two of them, there are two-particle measurement schemes that will yield a clear outcome that's incompatible with the zero-knowledge state and which directly determines which entangled pure state the particles are in.
  • Even if A and B cannot bring the systems back together, if we let them send classical messages to each other, then there are measurement procedures they can do (in essence, looking for violations of Bell inequalities) that are incompatible with the zero-knowledge state.

To slightly rephrase Emilio Pisanty's excellent answer (not because he said anything wrong, but because this is a subtle issue and multiple perspectives are always helpful):

You are correct that, if we only measure in the "intuitive" basis $\{ | pp \rangle, | pd \rangle, | dp \rangle, | dd \rangle \}$, then we have essentially equivalent information about the whole state as the individual ones.

A key part of Susskind's quote is "we can know everything about a composite system - everything there is to know, anyway." Even in a so-called "pure state" consisting of a single vector in the Hilbert space, there is still "unremovable" uncertainty about what we will measure that comes from the probabilistic nature of quantum mechanics inherent from the Born rule. But the key point is that for a pure state, there exists some basis in which the result of a measurement is guaranteed. If we were to measure in the (rather artificial, but still mathematically valid) Bell basis, we would measure with certainty that the composite system is in the state $(| pd \rangle - | dp \rangle)/\sqrt{2}$.

By contrast, if we were to only measure the subsystem $A$, it turns out that there is no possible basis in which our measurement outcome can be predicted with certainty. In this sense, a mixed/entangled state like that of system $A$ is "even more uncertain" that just the usual uncertainty of measurement outcome that you always get from the Born rule.


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