dq.bloch_coordinates

bloch_coordinates(x: ArrayLike) -> Array

Returns the spherical coordinates $(r, \theta, \phi)$ of a ket or density matrix on the Bloch sphere.

The coordinates are such that $$ \begin{cases} 0\leq r \leq 1, \\ 0\leq\theta\leq\pi, \\ 0\leq\phi<2\pi. \end{cases} $$

By convention, we choose $\phi=0$ if $\theta=0$, and $\theta=\phi=0$ if $r=0$.

Parameters

x (array_like of shape (2, 1) or (2, 2)) –
Ket or density matrix.

Returns

(array of shape (3,)) Spherical coordinates $(r, \theta, \phi)$.

Examples

The state $\ket0$ is on the north pole at coordinates $(r,\theta,\phi) = (1,0,0)$:

>>> x = dq.basis(2, 0)
>>> dq.bloch_coordinates(x)
Array([1., 0., 0.], dtype=float32)

The state $\ket1$ is on the south pole at coordinates $(r,\theta,\phi) = (1,\pi,0)$:

>>> x = dq.basis(2, 1)
>>> dq.bloch_coordinates(x)
Array([1.   , 3.142, 0.   ], dtype=float32)

The state $\ket+=(\ket0+\ket1)/\sqrt2$ is aligned with the $x$-axis at coordinates $(r,\theta,\phi) = (1,\pi/2,0)$:

>>> plus = dq.unit(dq.basis(2, 0) + dq.basis(2, 1))
>>> dq.bloch_coordinates(plus)
Array([1.   , 1.571, 0.   ], dtype=float32)

The state $\ket-=(\ket0-\ket1)/\sqrt2$ is aligned with the $-x$-axis at coordinates $(r,\theta,\phi) = (1,\pi/2,\pi)$:

>>> minus = dq.unit(dq.basis(2, 0) - dq.basis(2, 1))
>>> dq.bloch_coordinates(minus)
Array([1.   , 1.571, 3.142], dtype=float32)

A fully mixed state $\rho=0.5\ket0\bra0+0.5\ket1\bra1$ is at the center of the sphere at coordinates $(r,\theta,\phi) = (0,0,0)$:

>>> x = 0.5 * dq.basis_dm(2, 0) + 0.5 * dq.basis_dm(2, 1)
>>> dq.bloch_coordinates(x)
Array([0., 0., 0.], dtype=float32)

A partially mixed state $\rho=0.75\ket0\bra0 + 0.25\ket1\bra1$ is halfway between the sphere center and the north pole at coordinates $(r,\theta,\phi) = (0.5,0,0)$:

>>> x = 0.75 * dq.basis_dm(2, 0) + 0.25 * dq.basis_dm(2, 1)
>>> dq.bloch_coordinates(x)
Array([0.5, 0. , 0. ], dtype=float32)