Majorizes

Majorizes is a function that determines whether or not one vector or matrix weakly majorizes another vector or matrix. That is, given d-dimensional vectors $A$ and $B$, it checks whether or not


 * $$ \sum_{i=1}^k a_i^{\downarrow} \geq \sum_{i=1}^k b_i^{\downarrow} \quad \text{for } k=1,\dots,d,$$

where $a^{\downarrow}_i$ and $b^{\downarrow}_i$ are the elements of $A$ and $B$, respectively, sorted in decreasing order.

In the case of matrices, it is said that $A$ majorizes $B$ if the vector of $A$'s singular values majorizes the vector of $B$'s singular values. If the two vectors or matrices are of different sizes, the smaller one is padded with zeros appropriately so that they are comparable.

Syntax

 * M = Majorizes(A,B)

Argument descriptions

 * A: Either a vector or a matrix.
 * B: Either a vector or a matrix.

A simple example
It is straightforward to see that the vector $(3,0,0)$ majorizes the vector $(1,1,1)$, which we can verify as follows:

Bipartite LOCC
A well-known result of Nielsen says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via LOCC if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:

The above code shows that the conversion $|\psi\rangle \stackrel{LOCC}{\rightarrow} |\phi\rangle$ is impossible. On the other hand, the following code shows that the maximally-entangled pure state can be converted to a random pure state via LOCC:

Majorization criterion for separability
The majorization criterion says that every separable state $\rho_{AB}$ is such that each of $\rho_A$ and $\rho_B$ majorize $\rho_{AB}$. Thus the following code constructs the maximally-entangled quantum state and uses the majorization criterion to verify that it is entangled:

Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the reduction criterion ) and thus those alternate tests should be used in most situations instead.