In quantum computing, information is represented using fundamental units called “qubits.” Each qubit can represent two states: ∣0⟩ and ∣1⟩. These states form the basis for working with quantum information. When using more than one qubit, we refer to this as an “n-qubit” system. For example, if we use 3 qubits, the system is a 3-qubit quantum register.

To understand how multiple qubits are represented together, we use the concept of the “tensor product.” For n-qubit registers, we need to combine the spaces representing each qubit’s state. Therefore, the underlying vector space of an n-qubit system is defined as:

– ⊗ⁿ C² = C² ⊗ C² ⊗⋯⊗ C² (n times)

This expression means that n copies of C² are multiplied together. Each C² represents the state space of a single qubit.

Let us discuss why we use the tensor product and its importance. In short, we use the tensor product to model interactions between qubits in quantum systems appropriately and to carry out quantum measurements. When multiple qubits are combined, this structure allows us to understand how their states interact.

We can also discuss an orthonormal basis for such systems. For instance, if we want to construct an orthonormal basis for a 2-qubit system (C²⊗C²), we have:

– {∣0⟩⊗∣0⟩,∣0⟩⊗∣1⟩,∣1⟩⊗∣0⟩,∣1⟩⊗∣1⟩}

To simplify this notation, we can write:

– {∣00⟩,∣01⟩,∣10⟩,∣11⟩}

In n-qubit systems, these basis vectors can be expressed numerically. For example, for large n, the basis set can be represented as:

– {∣0⟩,∣1⟩,∣2⟩,…,∣2^n−1⟩}

This means that in an n-qubit system, there are 2^n basis states. These basis states are formed based on whether each qubit is in the state ∣0⟩ or ∣1⟩. Let’s illustrate this with an example:

Suppose n=3, meaning the system has 3 qubits:

– 2^3=8

This implies there are a total of 8 different basis states:

– {∣000⟩,∣001⟩,∣010⟩,∣011⟩,∣100⟩,∣101⟩,∣110⟩,∣111⟩}

These examples show that the number of basis states in n-qubit systems depends on the possible states (0 or 1) of each qubit. Thus, the basis states of an n-qubit system are expressed as 2^n combinations. This structure is crucial for understanding and analyzing more complex quantum computing systems.