Thursday, 6 September 2018

quantum optics - Informational capacity of qubits and photons


How much information is contained in one qubit?


A qubit is defined in Wikipedia as $a\left|0\right> +b\left|1\right>$, where a and b are complex numbers subject to $a^2 + b^2 = 1$.
One complex number is equivalent to two real numbers, which suggests that a qubit is equivalent to four real numbers worth of information.
Somehow I doubt this is true. How much information is contained in one mathematical qubit, exactly transmitted?



How much information is contained in one physical photon?


Alice and Bob share an arbitrarily but not infinitely accurate clock. They also have a shared cordinate frame (arbitrarily accurate relative location, no relative motion, no acceleration) Alice and Bob can also have an arbitrarily complex shared coding system. Alice is allowed to send one and only one photon to Bob between 12:00 and 1:00 oclock, and Bob receives it. Alice sends the photon directly to Bob through an intervening vacuum. If it makes a difference, assume the wavelength is exactly 500 nm.
How much information can Alice send Bob?
I assume it is not infinite, but only limited by the precision of the clocks, the availability of bandwidth, Alices range of transverse motion, the accuracy of Bob's telescope, etc.


Now assume the photon is delivered by a single mode optical fiber, so Bob can gain no information by measuring the direction of the incoming photon. Also assume that Eve puts in a variable delay of two to three hours, so Bob receives the photon between 2:00 and 4:00 and thus cannot gain any information from the time of receipt. How much information can Alice now send Bob?
Is it exactly one qubit, or something else?


We all know the plan where Alice transmits vertical for 0 and horizontal for 1, Bob measures exactly vertical or exactly horizontal and one bit per photon is transmitted.
Is there any improvement possible?



Answer



For (1), there is a theorem of Holevo that implies you cannot extract more than one bit of information from one qubit. You can indeed encode one bit of information, since the two inputs $| 0 \rangle$ and $| 1 \rangle$ (or any two orthogonal states) are distinguishable. If the sender and receiver share an entangled state, they can use superdense coding to send two bits using one qubit, but this is the maximum.



For (2), if the wavelength is exactly 500 nm (or at least as close as you can get in one hour), then by the uncertainty principle the photon must consist of an hour-long wave train which is a sine wave whose frequency gives 500 nm, and which (aside from its polarization) carries no information. You need non-zero bandwidth to transmit information. If you have a non-zero bandwidth, you can transmit information based on the frequency of the photon. The amount of information you can transmit depends on the bandwidth. The timescale of one hour limits the precision with which you can determine frequency, and this will give you the amount of information you can send. Note that if you use frequency to transmit the information, Alice's random delay doesn't hurt you. If you didn't have a random delay, then you could use the non-zero bandwidth to create wave packets that are localized in time, and use timing to transmit the same amount of information.


Let's do some calculations. Let's suppose the frequency is 500 nm $\pm$ .5 nm; then the frequency is 6*10$^{14}$ $\pm$ 6*10$^{11}$ Hz, and the bandwidth is 1.2*10$^{12}$ Hz. Now, we have $\Delta T\Delta E \geq \hbar$, and $E = h \nu$, so $\Delta T\Delta \nu \geq \frac{1}{2\pi}$. If $T$ is 3600 sec, we can distinguish frequencies to accuracy 4.4*10$^{-5}$ Hz, so we get 2.7*10$^{16}$ different frequencies we can distinguish. Taking the log (base 2), this is around 55 bits. I suspect this is the most information that can be sent with one photon with the given bandwidth, but I don't know a proof of this.


This is all theoretical; doing this in practice would require a ridiculously accurate frequency filter. If you didn't have the random delay, using wave packets localized in time and a very accurate clock would work better in practice, although they won't let you exceed 55 bits, either. But even with the random delay, I wouldn't be surprised if there were clever, experimentally more feasible, ways of communicating with one photon.


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