1.5: The Quantum Bit

According to quantum mechanics, it is possible for a small object to have two states which can be measured. This sounds perfect for storing bits, and the result is often called a qubit, pronounced “cue-bit.” The two values are often denoted |0\(\rangle\) and |1\(\rangle\) rather than 0 and 1, because this notation generalizes to what is needed to represent more than one qubit, and it avoids confusion with the real numbers 0 and 1. There are three features of quantum mechanics: reversibility, superposition, and entanglement, that make qubits or collections of qubits different from Boolean bits.

Reversibility: It is a property of quantum systems that if one state can lead to another by means of some transition, then the reverse transition is also possible. Thus all functions in the mathematics of qubits are reversible, and the output of a function cannot be discarded, since that would be irreversible. However, there are at least two important sources of irreversibility in quantum systems. First, if a quantum system interacts with its environment and the state of the environment is unknown, then some of the information in the system is lost. Second, the very act of measuring the state of a system is irreversible.

Superposition: Suppose a quantum mechanical object is prepared so that it has a combination, or superposition, of its two states, i.e., a state somewhere between the two states. What is it that would be measured in that case?

In a classical, non-quantum context, a measurement could determine just what that combination is. Furthermore, for greater precision a measurement could be repeated, and multiple results averaged. However, the quantum context is different. In a quantum measurement, the question that is asked is whether the object is or is not in some particular state, and the answer is always either “yes” or “no,” never “maybe” and never, for example, “27% yes, 73% no.” Furthermore, after the measurement the system ends up in the state corresponding to the answer, so further measurements will not yield additional information. The result of any particular measurement cannot be predicted, but the likelihood of the answer, expressed in terms of probabilities, can. This peculiar nature of quantum mechanics offers both a limitation of how much information can be carried by a single qubit, and an opportunity to design systems which take special advantage of these features.

We will illustrate quantum bits with an example. Let’s take as our qubit a photon, which is the elementary particle for electromagnetic radiation, including radio, TV, and light. A photon is a good candidate for carrying information from one place to another. It is small, and travels fast.

A photon has electric and magnetic fields oscillating simultaneously. The direction of the electric field is called the direction of polarization (we will not consider circularly polarized photons here). Thus if a photon is headed in the \(z\)-direction, its electric field can be in the \(x\)-direction, in the \(y\)-direction, or in fact in any direction in the \(x-y\) plane, sometimes called the “horizontal-vertical plane.”

The polarization can be used to store a bit of information. Thus Alice could prepare a photon with horizontal polarization if the bit is |0\(\rangle\) and vertical polarization if the bit is |1\(\rangle\). Then when Bob gets the photon, he can measure its vertical polarization (i.e., ask whether the polarization is vertical). If the answer is “yes,” then he infers the bit is |1\(\rangle\).

It might be thought that more than a single bit of information could be transmitted by a single photon’s polarization. Why couldn’t Alice send two bits, using angles of polarization different from horizontal and vertical? Why not use horizontal, vertical, half-way between them tilted right, and half-way between them tilted left? The problem is that Bob has to decide what angle to measure. He cannot, because of quantum-mechanical limitations, ask the question “what is the angle of polarization” but only “is the polarization in the direction I choose to measure.” And the result of his measurement can only be “yes” or “no,” in other words, a single bit. And then after the measurement the photon ends up either in the plane he measured (if the result was “yes”) or perpendicular to it (if the result was “no”).

If Bob wants to measure the angle of polarization more accurately, why couldn’t he repeat his measurement many times and take an average? This does not work because the very act of doing the first measurement resets the angle of polarization either to the angle he measured or to the angle perpendicular to it. Thus subsequent measurements will all be the same.

Or Bob might decide to make multiple copies of the photon, and then measure each of them. This approach does not work either. The only way he can make a copy of the photon is by measuring its properties and then creating a new photon with exactly those properties. All the photons he creates will be the same.

What does Bob measure if Alice had prepared the photon with an arbitrary angle? Or if the photon had its angle of polarization changed because of random interactions along the way? Or if the photon had been measured by an evil eavesdropper (typically named Eve) at some other angle and therefore been reset to that angle? In these cases, Bob always gets an answer “yes” or “no,” for whatever direction of polarization he chooses to measure, and the closer the actual polarization is to that direction the more likely the answer is yes. To be specific, the probability of the answer yes is the square of the cosine of the angle between Bob’s angle of measurement and Alice’s angle of preparation. It is not possible to predict the result of any one of Bob’s measurements. This inherent randomness is an unavoidable aspect of quantum mechanics.

Entanglement: Two or more qubits can be prepared together in particular ways. One property, which we will not discuss further now, is known as “entanglement.” Two photons, for example, might have identical polarizations (either both horizontal or both vertical). Then they might travel to different places but retain their entangled polarizations. They then would be separate in their physical locations but not separate in their polarizations. If you think of them as two separate photons you might wonder why measurement of the polarization of one would affect a subsequent measurement of the polarization of the other, located far away.

Note that quantum systems don’t always exhibit the peculiarities associated with superposition and entanglement. For example, photons can be prepared independently (so there is no entanglement) and the angles of polarization can be constrained to be horizontal and vertical (no superposition). In this case qubits behave like Boolean bits.