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Just like its classical cousin, the qubit can take a value of either 0 or 1. Physically, qubits can be represented as any two-level quantum systems such as

• • The spin of a particle in a magnetic field where up means 0 and down means 1 or
• • The polarization of a single photon where horizontal polarization means 1 and vertical polarization means 0.

You can make a quantum computer out of light. How weird is that.
In both cases 0 and 1 are the only possible states. Geometrically, qubits can be visualized using a shape called the Bloch sphere, an instrument named after Swiss physicist Felix Bloch (see Figure ) .

Formally, the Bloch sphere is the geometrical representation in the three-dimensional Hil
bert space of the pure state of a two-level quantum system or qubit. The north and south poles of the sphere represent the standard basis vectors |0> and |1>, respectively;

                                                                             Figure
                          Geometrical representation of a quantum state using the Bloch sphere

these in turn correspond to the spin-up and spin-down of the electron. Besides the basic vectors, the sphere can have something in between; this is called a superposition and it is essentially the probability for 0 or 1. The trick is that we can’t predict which it will be except at the instant of observation when the probability collapses into a definitive state.

 Superposition of States
Imagine if you could flip a coin that could fall not only in a heads or tails position, but in both positions at the same time. Such a coin would be more powerful. Nevertheless there is a catch; the moment you observe this quantum coin, it is forced to take either heads or tails never knowing what position it was in before. This is one reason one needs to be careful when measuring qubits, because they change as soon as observed. All in all, superposition is a game changer. Let’s see why:

• • A 1-bit classical computer can be (or store) in 1 of 2 states at a time: 0 or 1. A 1-qubit quantum computer can be (or store) in 2 states at a time. That is 21 = 2.
• • A 2-bit classical computer can store only 1 out of 22 = 4 possible combinations. A 2-qubit quantum computer can store 22 = 4 possible values simultaneously.

Assuming that a byte (8 bits) is the basic unit used to store information in either system, then the number of values that can be stored simultaneously in a quantum computer would be 2n where n is the number of qubits. Compare this against the storage capacity of a classic computer (shown in Table ) and you realize why qubits are powerful indeed.

Thus the amount of data that can be stored simultaneously in a quantum computer is astounding, so much so that a new term has popped up out there: quantum supremacy. This is the point at which a quantum computer will be able to solve all problems a classical computer cannot. More about this subject will be discussed in a further section of this chapter. But, for now, let’s look at the next strange property of the qubit: entanglement.

Entanglement: 
Observing a Qubit Reveals the State  of Its Partner

Long ago, Albert Einstein called quantum entanglement Spooky action at a distance. Believe it or not, entanglement has been proved experimentally by French physicist Alain Aspect in 1982. He demonstrated how an effect in one of two correlated particles travels faster than the speed of light!
Tip ironically and in a sad twist of faith, humans cannot use entanglement to send messages faster than the speed of light as information cannot travel at such speed.

If a set of qubits are entangled, then each will react to a change in the other instantaneously, no matter how far apart they are (in opposite sides of the galaxy, e.g., which sounds really unbelievable). This is useful in that, if we measure the properties in 1 qubit, then we can deduce the properties of its partner without having to look. Furthermore, entanglement can be measured without looking through a process called quantum tomography. Quantum tomography seeks to determine the state(s) of an entangled set prior to measurement by measurements of the systems coming from the source. In other words, it calculates the probability of measuring every possible state of the system.