Why does adding one qubit double the size of a statevector?
A pure two-qubit state has four complex amplitudes, one for each computational-basis vector. The product states prepared here factor into two single-qubit states; a general two-qubit pure state need not.
State and transformation
Quantum circuit
Circuit wire labels follow the Qiskit indexing convention; Statevector results are computed with an ideal simulator, not hardware.
Analytic predictions and simulator results
Swipe the results table horizontally to see all columns.
| Quantity | Analytic prediction | Computed value or classical sample |
|---|---|---|
| P(00) and P(01) | 0% / 0% | 0% / 0% · Qiskit |
| P(10) and P(11) | 50% / 50% | 50% / 50% · Qiskit |
| |++⟩ · four outcomes | 25% each | 25% each · Qiskit |
Values are reported from the previously completed local experiments: exact ideal statevector probabilities (Qiskit), classical NumPy sampling where stated, and the specified NumPy density-matrix calculation. These are not observations from physical quantum hardware. Rounded values are shown for readability.
Minimal Qiskit example
This independently written teaching snippet illustrates the verified circuit principle without disclosing the private lab source. Run it with NumPy and Qiskit installed; this abbreviated public snippet has undergone source and syntax review, but not a separate Qiskit-enabled execution test.
import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
circuit = QuantumCircuit(2)
circuit.x(1) # leftmost q1 = |1>
circuit.h(0) # rightmost q0 = |+>
state = Statevector.from_instruction(circuit)
print(np.round(state.data, 6))
print(np.round(state.probabilities(), 6))
assert np.allclose(state.probabilities(), [0, 0, 0.5, 0.5])The computational basis is ordered |q₁q₀⟩. For an n-qubit pure state, the statevector length is 2ⁿ; simulation cost therefore grows exponentially with n.
Computations are local ideal statevector calculations. Nothing here sends jobs to an IBM Quantum device or cloud backend.
What we can conclude
Two qubits require four complex amplitudes. For this specific product state, q₁ is always 1 while q₀ has equal chances of 0 and 1.
For this two-qubit example, Qiskit orders basis vectors as |q₁q₀⟩, with q₀ as the rightmost bit. Superposition of multiple basis vectors is not, by itself, a test of entanglement.