## 5.2 From one qubit to two

In classical physics, the transition from a single object to a composite system of many objects is trivial: in order to describe the state of, say, 42 objects at any given moment of time, it is sufficient to describe the state of each of the objects separately. Indeed, the classical state of 42 point-like particles is described by specifying the position and the momentum of each particle.

In the *classical* world, “the whole is *exactly* the sum of its parts”; in the *quantum* world, Aristotle had it right when he said “the whole is *greater than* the sum of its parts”.

Consider, for example, a pair of qubits.
Suppose that each one is described by a state vector: the first one by *cannot* be expressed in this form.
In order to write down the most general state of two qubits we first focus on the basis states.

For a single qubit we have been using the standard basis ^{104}

Now, the most general state of the two qubits (a **bipartite** state) is a normalised linear combination of these four basis states, i.e. a vector of the form
*eight* real parameters; we then restrict by the normalisation condition, along with the fact that states differing only by a global phase factor are equivalent, which leaves us with *six* real parameters.
Now, by the same line of argument, we need only *two* real parameters to specify the state of a single qubit, and hence need *four* real parameters to specify any state of two qubits of the form

But four is less than six!
So it *cannot* be the case that every state of two qubits can be expressed as a pair of states

For example, compare the two states of two qubits,
**separable**, i.e. we can view it as a pair of state vectors where each one pertains to one of the two qubits:
*not* admit such a decomposition: there do *not* exist *any* **entangled** state.

Any bipartite state that cannot be viewed as a pair of two states pertaining to the constituent subsystems is said to be **entangled**.

We’ll give another, equivalent but more mathematical (and notational), definition of entanglement once we understand how tensor products work.