OperatorSchmidtDecomposition

OperatorSchmidtDecomposition
Computes the operator Schmidt decomposition of a bipartite operator
Other toolboxes required	none
Related functions	FilterNormalForm IsProductOperator OperatorSchmidtRank SchmidtDecomposition TensorSum
Function category	Entanglement and separability

OperatorSchmidtDecomposition is a function that computes the operator Schmidt decomposition of a bipartite operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.

Syntax

• S = OperatorSchmidtDecomposition(X)
• S = OperatorSchmidtDecomposition(X,DIM)
• S = OperatorSchmidtDecomposition(X,DIM,K)
• [S,U,V] = OperatorSchmidtDecomposition(X,DIM,K)

Argument descriptions

Input arguments

• X: A bipartite operator to have its operator Schmidt decomposition computed.
• DIM (optional, by default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that X lives on. DIM can be provided in one of three ways:
  • If DIM is a scalar, it is assumed that the first subsystem has dimension DIM and the second subsystem has dimension length(X)/DIM.
  • If $X \in M_{n_1} \otimes M_{n_2}$ then DIM should be a row vector containing the dimensions (i.e., DIM = [n_1, n_2]).
  • If the subsystems aren't square (i.e., $X \in M_{m_1, n_1} \otimes M_{m_2, n_2}$) then DIM should be a matrix with two rows. The first row of DIM should contain the row dimensions of the subsystems (i.e., the m_i's) and its second row should contain the column dimensions (i.e., the n_i's). In other words, you should set DIM = [m_1, m_2; n_1, n_2].
• K (optional, default 0): A flag that determines how many terms in the operator Schmidt decomposition should be computed. If K = 0 then all terms with non-zero operator Schmidt coefficients are computed. If K = -1 then all terms (including zero terms) are computed. If K > 0 then the K terms with largest operator Schmidt coefficients are computed.

Output arguments

• S: A vector containing the operator Schmidt coefficients of X.
• U (optional): A cell containing the left Schmidt operators of X.
• V (optional): A cell containing the right Schmidt operators of X.

Examples

A random example

The following code computes the operator Schmidt decomposition of a random density matrix in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:

>> rho = RandomDensityMatrix(8);
>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])

s =

    0.3986
    0.2310
    0.1637
    0.1118


U = 

    [2x2 double]    [2x2 double]    [2x2 double]    [2x2 double]


V = 

    [4x4 double]    [4x4 double]    [4x4 double]    [4x4 double]

>> norm(s(1)*kron(U{1},V{1}) + s(2)*kron(U{2},V{2}) + s(3)*kron(U{3},V{3}) + s(4)*kron(U{4},V{4}) - rho)

ans =

  1.8449e-016

As an alternative (and easier) way to verify that we have a valid tensor decomposition of rho, we can use the TensorSum function:

>> norm(TensorSum(s,U,V) - rho)

ans =

  1.8449e-016

Known Issues

If the input matrix X is Hermitian, all of the matrices in its operator-Schmidt decomposition can be chosen to be Hermitian. If X has distinct operator-Schmidt coefficients then such an operator-Schmidt decomposition is indeed computed, but if there is a repeated coefficient then a non-Hermitian decomposition may (erroneously) be provided instead. See Issue #14 on github.

This should be fixed in a future version of QETLAB. For now, to get around this problem you can add a tiny amount of random (but Hermitian) noise to X so that its operator-Schmidt coefficients are distinct.

Source code

Click here to view this function's source code on github.