Tutorial for physicists

The unitary group \(U(n)\) describes the possible change of basis in the complex Hilbert space of dimension \(n\). For \(n=2\), this captures the possible choices of basis in the qubit space \(C^2\). For a system composed of two qubits, we can similarly define the effect of a joint change of basis performed on both subsystems simultaneously. It is known that a single state is invariant under such joint change of basis. Here, we identify this state by extracting the subspace of \((C^2)^{\otimes 2}\) which is invariant under the joint change of basis for both subsystems.

Before trying any of the RepLAB commands, we must initialize the library:

addpath([pwd, '/../../../external/replab']);
replab_init('verbose', 0);

replab_init: Initialization done.

The unitary group representation

Changes of bases for one system are described by the group \(U(2)\)

d = 2;
U2 = replab.U(d);

We construct the defining representation of this group, which acts on \(C^2\):

U2Rep = U2.definingRep;

Tensor product of two representations

We can now construct the representation which acts jointly on two subsystems of dimension 2:

U2TensorRep = kron(U2Rep, U2Rep);

To identify the subspaces which are invariant under this group, we decompose the representation:

dec = U2TensorRep.decomposition.nice;

The decomposition has

dec.nComponents

ans = 2

components, of dimension

dec.component(1).irrepDimension

ans = 1

and

dec.component(2).irrepDimension

ans = 3

These are the antisymmetric and symmetric subspaces respectively. The change of basis into the first component identifies the antisymetric subspace, also known as the singlet state:

singletBasis = dec.component(1).basis
rest = dec.component(2).basis

singletBasis =

        0
   0.5000
  -0.5000
        0

rest =

   1.0000        0        0
        0        0   0.5000
        0        0   0.5000
        0   1.0000        0