Now that we have set up some of the ideas behind quantum mechanics, we can use them to describe a technique for distributing bits through a quantum channel. These bits can be used to establish a key that can be used for communicating across a classical channel, or any other shared secret.

We begin by describing a quantum bit. Start with a two-dimensional complex vector space. Choose a pair of orthogonal vectors of length 1; call them $|0⟩$ and $|1⟩$. For example, these two vectors could be either of the two pairs of orthogonal vectors used in the previous section. A quantum bit, also known as a qubit, is a unit vector in this vector space. For the purposes of the present discussion, we can think of a qubit as a polarized photon. We have chosen $|0⟩$ and $|1⟩$ as notation to conveniently represent the $0$ and $1$ bits, respectively. The other qubits are linear combinations of these two bits.

Since a qubit is a unit vector, it can be represented as $a|0⟩+b|1⟩$, where $a$ and $b$ are complex numbers such that $|a{|}^2+|b{|}^2=1$. Just as in the case for photons from the preceding section, we can measure this qubit with respect to the basis $|0⟩\text{,}|1⟩$. The probability that we observe it in the $|0⟩$ state is $|a{|}^2$.

Let us now examine how Alice and Bob can communicate with each other in order to establish a message. They will need two things: a quantum channel and a classical channel. A quantum channel is one through which they can exchange polarized photons that are isolated from interactions with the environment (that is, the environment doesn’t alter the photons). The classical channel will be used to send ordinary messages to each other. We assume that the evil observer Eve can observe what is being sent on the classical channel and that she can observe and resend photons on the quantum channel.

Alice starts the establishment of a message by sending a sequence of bits to Bob. They are encoded using a randomly chosen basis for each bit as follows. There are two bases: $B_1=\{|\uparrow ⟩\text{,}|\to ⟩\}$ and $B_2=\{|\nwarrow ⟩\text{,}|\nearrow ⟩\}$. If Alice chooses $B_1$, then she encodes $0$ as $|\uparrow ⟩$ and 1 as $|\to ⟩$, while if she chooses $B_2$ then she encodes $0$ and $1$ using the two elements of $B_2$.

Each time Alice sends a photon, Bob randomly chooses to measure with respect to either basis $B_1$ or $B_2$. Therefore, for each photon, he obtains an element of that choice of basis as the result of his measurement. Bob records the measurements he has made and keeps them secret. He then tells Alice the basis with which he measured each photon. Alice responds to Bob by telling him which bases were the correct bases for the polarity of the photons that she sent. They keep the bits that used the same bases and discard the other bits. Since two bases were used, Alice and Bob will agree on roughly half of the amount of bits that Alice sent. They can then use these bits as the key for a conventional cryptographic system.

The security behind quantum key distribution is based upon the laws of quantum mechanics and the fundamental principle that following a measurement of a particle, that particle’s state will be altered. Since an eavesdropper Eve must perform measurements in order to observe the photon transmissions between Alice and Bob, Eve will introduce errors in the data that Alice and Bob agreed upon.

Let’s see how this happens. Suppose Eve measures the states of the photons transmitted by Alice and allows these measured photons to proceed onto Bob. Since these photons were measured by Eve, they will have the state that Eve observed. Eve will use the wrong basis half of the time when performing the measurement. When Bob performs his measurements, if he uses the correct basis there will be a $25\%$ chance that he will have measured the wrong value.

Let’s examine this last statement in more detail. Suppose that Alice sends a photon corresponding to $|\to ⟩$ and that Bob uses the same basis $B_1$ as Alice. If Eve uses $B_1$, then the photon is passed through correctly and then Bob measures the photon correctly. However, if Eve used $B_2$, then she will measure $|\nearrow ⟩$ and $|\nwarrow ⟩$ equally likely. The photons that pass to Bob will have one of these orientations and he will therefore half the time measure them correctly as $|\to ⟩$ and half the time incorrectly. Combining the two possible choices of basis that Eve has causes Bob to have a $25$ chance of measuring the incorrect value.

Thus, any eavesdropping introduces a higher error rate in the communication between Alice and Bob. If Alice and Bob test their data for discrepancies over the conventional channel (for example, they could send parity bits), they will detect any eavesdropping.

Actual implementations of this technique have been used to establish keys over distances of more than $100$ km using conventional fiber optical cables.