Existence of the Schmidt decomposition for tripartite systems

For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis. The key to quantum information processing is the entangled nature of quantum states, which has no classical counter part in the theory. These states are central to the study of quantum non-locality, quantum teleportation, quantum cryptography, dense coding and so on. In the simplest term entangled state is a one which cannot be expressed as a direct product of the states of two or more subsystems. If we have a quantum system consisting of two subsystems A and B, then the state of the combined system in general can be expressed as |ψ〉AB = ∑ ij aij |ui〉 ⊗ |vj〉, where {|ui〉}, (i = 1, 2, ...NA) ∈ HA and {|vi〉}, (i = 1, 2, ...NB) ∈ HB are the complete set of orthonormal basis vectors in their respective Hilbert spaces. The expansion coefficients in the above state of the combined system contain NANB terms and is very difficult to manipulate with. However, the Schmidt decomposition (SD) theorem [1] comes to rescue us. It says that any arbitrary state of a bipartite (two-subsystem) quantum system can be expressed as [2]