Code for constructing a state whose partial transpose has the maximum number of negative eigenvalues

This page contains MATLAB code that can be used to construct a density matrix in Mm ⊗ Mn whose partial transpose has (m-1)(n-1) negative eigenvalues, which is maximal.

Download and Install

Step 1: Download and install CVX. You do not need to know how to use CVX – you just need to install it.

Step 2: Download the PTNegEvals.zip file (file size: 5 kB).

Step 3: Unzip the file.

Step 4: Place all four of the unzipped files in your MATLAB scripts directory.

Usage

The function PTNegEvals creates a density matrix in Mm ⊗ Mn such that its partial transpose has (m-1)(n-1) negative eigenvalues, using the method of [1]. The function takes two arguments: m and n.

Example

The following code produces a density matrix in M3 ⊗ M4 whose partial transpose has 6 negative eigenvalues:

>> rho = PTNegEvals(3,4);

We can then verify that rho is positive semidefinite and its partial transpose has 6 negative eigenvalues by using the PartialTranspose function (which is included in the PTNegEvals.zip file):

>> eig(rho)

ans =

    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0810
    0.0810
    0.1000
    0.1000
    0.3191
    0.3191

>> eig(PartialTranspose(rho,2,[3,4]))

ans =

   -0.0219
   -0.0219
   -0.0164
   -0.0164
   -0.0140
   -0.0140
    0.1155
    0.1155
    0.1526
    0.1526
    0.2842
    0.2842

References

  1. N. Johnston. Non-positive partial transpose subspaces can be as large as any entangled subspace. Physical Review A, 87:064302, 2013.
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