Encoding infinite information in a qubit vs. classical system

Question

In this Quantum Computing article by Michael Nielsen he argues about some of the limitations imposed by quantum measurement. In particular how the amplitude $\alpha$ of a single qubit $\alpha |0> + \beta |1>$ cannot be measured.

One reason this is important is because it means you can’t store an infinite amount of classical information in a qubit. After all, $\alpha$ is a complex number, and you could imagine storing lots of classical bits in the binary expansion of the real component of $\alpha$. If there was some experimental way you could measure the value of $\alpha$ exactly, then you could extract that classical information. But without a way of measuring $\alpha$ that’s not possible.

I am somewhat confused regarding the point being made. I assume the idea is that a classical bit can only transmit 1 bit of information, and the measurement of a qbit (a single physical system) can only result in one bit of information after the measurement (measurement outcome of "0" or "1" in the computational basis).

However, I don't see anything quantum about it. Could you not send an infinite amount of information in a single classical physical system? For example: Send light with frequency f such that the binary expansion of f has an infinite amout of bits. So for example, classically, you might send light with f=3.14159265359... Hz having infinitely many digits. Or a small piece of metal measuring d=2.1828.... m cut with infinite precision.

Is the point that the quantum nature of matter limits the amount of information vs. the classical nature of matter which imposes no limits?

Answer 1

You are correct: there is nothing intrinsically quantum about the amplitude storing infinite information; any real number in principle stores infinite information, including classical real numbers.

What is different is that for a qubit, you need infinite information to fully specify the state of the qubit, so you might have expected that you could extract infinite information from a qubit. Nielsen is saying “if you expect this, you are wrong.” You only get one measurement out of a qubit, and that gives you an ability to extract one bit of information. You need a lot of identically prepared qubits in order to do the sort of “quantum state tomography” that reveals these amplitudes. This is formalized in Holevo’s Theorem, which essentially states that from a set of $n$ qubits you cannot extract more than $n$ classical bits, no matter how you fuss with them.

You can do a little bit more than extraction with qubits. For example, “superdense coding” is a trick to use a shared entangled qubit state between Alice and Bob, so that Alice can send Bob 2 classical bits by sending one qubit. The reverse is also true: “quantum teleportation” can use a shared entangled qubit state between Alice and Bob, so that Alice can send Bob one qubit by sending him 2 classical bits. So in a circumstance with a plethora of entangled states, in some sense a qubit is equivalent to 2 classical bits.

Answer 2

There is a marked difference between the two cases you mention.

In principle, classically, yes you can encode an infinite amount of information in, say, the frequency of a light beam, and you can also (again, in principle) always recover it with the appropriate equipment.

This is, however, a totally different scenario from what you have if you try to encode arbitrary amounts of information in the amplitudes of a quantum system.

In the case of information "encoded" in the amplitudes of a qubit, it is fundamentally impossible to recover such information reliably. This is closer to trying to encode information into the probabilities of getting a head or tail while flipping a biased coin. Sure, you can do it, but you cannot possibly recover that information without flipping the coin many times and collect the associated statistics. A qubit is a bit like that, except it is also fundamentally impossible to predict the outcomes of the coin flipping results (which is not the case for the classical biased coin).

You can also have a look at this related question (and links therein) for more details on the information capacity of a qubit.

Answer 3

Information is transmitted when you go from a state of "missing knowledge" to a state where you "have knowledge". The word information theorists use for this entropy - entropy signifies how much missing knowledge you have and information is gained whenever you reduce entropy.

Consider the following situation. You send me a signal and I know that your signal will decode to either 0 or 1. I don't know which outcome I will get but I know it would be one of those two outcomes. Before I decode your signal, assuming you send 0 or 1 with equal probability, I was in a state of zero knowledge about the value of the bit. This corresponds to the maximum entropy of $H = 1$. After I decode it, I gained full information about the bit and reduced my entropy to zero.

Similarly, consider a signal where if I measure it, I would obtain 0, 1, 2, or 3. When I actually measure this signal, I've gained two bits of information.

The idea in quantum mechanics is that you can choose any real value of $\alpha$ when you make a qubit in the state $\sqrt{\alpha}\vert 0\rangle + \sqrt{1-\alpha}\vert 1\rangle$. Moreover, the receipient could measure in any basis they like. You might think that the sender could therefore choose $\alpha$ from a set of, say, $\{0, 0.1, 0.2... 1\}$ (or indeed even more precise) and send the qubit and the recipient could measure the qubit and extract this value of $\alpha$. That's a lot of information sent in just one qubit. However, Holevo's theorem states that this doesn't work - in the end, you can only really send 1 bit of information in 1 qubit.

Note that your examples where you posit a way to transmit infinite information in a signal are not connected to the argument being made by Nielsen and Chuang. Even if you have finite but good precision on $\alpha$, you still can only send 1 bit using a qubit. As for whether analog signals can be used to transmit infinite information, that is addressed by the answer linked in the comment to your question.