Quantum Foundations

The Tensor Product and Multi-Qubit States

The tensor product combines individual qubit spaces into a joint space whose dimension grows exponentially with the number of qubits.

Combining systems

The state space of a composite quantum system is the tensor product of the parts' spaces. Two qubits, each two-dimensional, combine into a four-dimensional space; n qubits into a 2^n-dimensional space. This exponential scaling is the source of both quantum computing's power and the difficulty of simulating it classically.

Basis states

Kronos motion — space economy

python

import numpy as np
zero = np.array([1,0]); one = np.array([0,1])
print(np.kron(zero, one))   # |01> = [0 1 0 0]
print(np.kron(one, one))    # |11> = [0 0 0 1]

Product versus entangled states

A state that factors as (a|0>+b|1>) tensor (c|0>+d|1>) is a product state — the qubits are independent. Most states in the joint space do not factor; those are entangled. The tensor product provides the arena in which entanglement can exist, and product states are a measure-zero slice of it.

Operators on subsystems

A gate acting on qubit 1 alone is written U tensor I, applying U to the first factor and the identity to the second. Two-qubit gates like CNOT are full 4x4 matrices that generally do not factor, which is precisely how they create entanglement between previously independent qubits.

The scaling problem

A classical description of an n-qubit state needs 2^n complex amplitudes. At 50 qubits that exceeds a petabyte; beyond about 50 qubits exact state-vector simulation on the largest supercomputers becomes infeasible. This is why building real quantum hardware, rather than simulating it, is worthwhile — and why classical simulation of processes like plasma dynamics motivates exploring quantum methods.