The Bloch sphere is the standard picture for a single qubit's pure state: every possible state is one point on the surface of a unit sphere. This simulator lets you set that point directly by dragging, by sliders, or by jumping to a named preset, and reads out exactly what that point means for measurement.
• A rotatable Bloch sphere with the state vector rendered from the origin to the sphere's surface. • Direct drag-to-set interaction on the sphere, plus independent polar-angle (theta) and azimuth (phi) sliders. • Six one-click presets: |0>, |1>, |+>, |->, |+i>, |-i>, covering the six cardinal points of the sphere. • A view-rotation control for inspecting the vector from a different vantage point. • Live P(0)/P(1) probabilities, Bloch (x,y,z) coordinates, and complex amplitude readouts.
• Jump between |0> and |1> and watch the vector snap between the poles with P(0) or P(1) at 100%. • Jump between |+> and |-> and confirm both give 50/50 probabilities despite pointing in opposite directions on the equator — the difference is entirely in phase. • Drag the vector to an arbitrary point and read off theta, phi and the resulting probabilities directly from the equation and metrics.
The polar angle theta (measured from the north pole |0>) fixes how probability splits between 0 and 1 via cos(theta/2) and sin(theta/2). The azimuth phi fixes the relative phase between the two amplitudes and has no effect on a direct measurement by itself — it only matters once another gate recombines the amplitudes, which is why every point on a given latitude circle (constant theta) has the same measurement probabilities.
The north pole is the state |0>, and the south pole is the state |1>. Every other point on the sphere's surface is a superposition of these two, with the polar angle theta setting the measurement-probability split and the azimuth phi setting the relative phase.
A pure single-qubit state needs two real numbers to specify beyond an overall unmeasurable global phase: the probability split (one degree of freedom) and the relative phase (a second, independent degree of freedom). Two independent angles naturally parametrize the surface of a sphere, exactly as latitude and longitude do on Earth.
They are equal superpositions of |0> and |1> like |+> and |->, but with a 90-degree or 270-degree relative phase instead of 0 or 180 degrees. On the Bloch sphere they sit on the equator's Y axis, perpendicular to |+> and |-> on the X axis.
No. It only represents pure states, which always sit exactly on the sphere's surface. A mixed state (from noise or partial information) would sit somewhere inside the sphere, closer to the center the more mixed it is — that case is outside this teaching model's scope.