Alice encodes a classical symbol i (probability pi = 1/N) as a pure qubit state |ψi⟩ placed on a cone of colatitude θ around the north pole, spread evenly in azimuth. This 2D view is the top-down (polar) projection of that Bloch sphere: radius on the disk encodes sin(θ) and angle encodes the azimuth φ — the pole sits at the disk's centre, the equator at its rim.
ρ̄ = Σ p_i |ψ_i⟩⟨ψ_i| = (I + r̄·σ) / 2
Holevo bound: χ = S(ρ̄) − Σ p_i S(ρ_i) = S(ρ̄) (pure states ⇒ S(ρ_i)=0)
Qubit entropy: S(ρ) = H₂( (1+|r|)/2 ), H₂(p) = −p·log₂p − (1−p)·log₂(1−p)
By the cone's rotational symmetry the azimuthal parts of the Bloch vectors cancel, so r̄ = (0,0,cos θ) and χ depends only on θ: zero at the poles, maximal (1 bit) at the equator.
Holevo's theorem bounds the average of every possible measurement — it does not say a polar (θm=0) measurement is a good one. Drag the white handle on the disk (or use the sliders) to set Bob's real projective axis m and read off his exact accessible information from the Born rule:
P(+|i) = (1 + n_i·m) / 2
I(A:B) = H₂( P(+) ) − Σ p_i H₂( P(+|i) ), P(+) = Σ p_i P(+|i)
Numerical correction to the original 3D source: that page's "Snap to best axis" button reset the measurement straight to the pole (θm=0, φm=0). Plugging that into the accessible-information formula above shows every symbol gets the identical P(+|i) = (1+cosθ)/2 regardless of i — because all cone points share the same z-component — which forces I(A:B) = 0 exactly, the *worst* possible axis, not the best. Verified numerically (grid search over θm,φm, e.g. cone=70°, N=3 gives I≈0 at the pole vs a true maximum of I≈0.39 bits near θm≈100°). This build's "Snap to numerically-best axis" button instead runs a live coarse-to-fine grid search over the axis and jumps to the angle that actually maximises I(A:B); the sweep chart on the right plots I(A:B) against θm so the zero-at-the-pole dip and the true peak are both visible.
For N = 2 antipodal states (cone θ = 90°) a matched equatorial axis reaches I(A:B) = χ exactly. For N ≥ 3 no single projective measurement closes the gap even at the optimal axis — the reason BB84-style protocols built on more than two conjugate states stay secure: an eavesdropper is bound by the same inequality Bob is.