IsotropicState

IsotropicState is a function that returns an isotropic state (i.e., a state of the following form): $$\displaystyle\rho_\alpha := \frac{1-\alpha}{d^2}I\otimes I + \alpha|\psi_+\rangle\langle\psi_+| \in M_d \otimes M_d,$$ where $|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j|j\rangle\otimes|j\rangle$ is the standard maximally-entangled pure state. Note that the output of this function is a sparse matrix.

Syntax

 * RHO = IsotropicState(DIM,ALPHA)

Argument descriptions

 * DIM: Dimension of the local subsystems on which RHO acts.
 * ALPHA: A parameter that specifies which isotropic state is to be returned. In particular, RHO = (1-ALPHA)*I/DIM^2 + ALPHA*E, where I is the identity operator and E is the projection onto the standard maximally-entangled pure state on two copies of DIM</tt>-dimensional space. In order for RHO</tt> to be positive semidefinite (and hence a valid density matrix), it must be the case that -1/(DIM^2-1) &le; ALPHA &le; 1</tt>.

A qutrit isotropic state
To generate the isotropic state with parameter $\alpha = 1/2$, the following code suffices:

Isotropic states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's full function.