Majorizes

Majorizes
Determines whether or not a vector or matrix majorizes another
Other toolboxes required	none

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.

Examples

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:

>> Majorizes([3,0,0],[1,1,1])

ans =

     1

Bipartite LOCC

A well-known result of Neilsen 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 Schmidt coefficients of $|\phi\rangle$ majorize the Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:

>> phi = RandomStateVector(9); % generate two random states in C^3 \otimes C^3
>> psi = RandomStateVector(9);
>> Majorizes(SchmidtDecomposition(phi),SchmidtDecomposition(psi))

ans =

     0

The above code shows that the conversion $|\psi\rangle \stackrel{LOCC}{\rightarrow} |\phi\rangle$ is impossible.