On the distinction between entangled and separable states of quantum registers

Quantum computing operates in three stages: (i) preparation of the initial states of the n qubits of a register, (ii) step by step transformation of the state of this register by unitary operators which compose the quantum program, and (iii), measurement of all or some of the qubits of the register. The existence of quantum algorithms which are exponentially less complex than their classical counterparts for some classes of problems, stems from entangled states established by multi-qubit unitary operators within the quantum program. A register of n qubits is a quantum system composed of n quantum subsystems. The state |ψ> of a quantum system composed of two quantum subsystems A and B is said to be entangled when |ψ> is not reducible to a pair composed of a state |ψ> of A and a state |ψ_B> of B: such situations have no counterpart in the classical world. In quantum theory, such a pair of states is denoted by a tensor product: |ψ> is entangled if it cannot be factorized into the tensor product |ψ_A>⊖|ψ_B>. This paper establishes conditions according to which it is possible to tell whether or not the state of a register of n qubits is entangled. The state of a single qubit is a vector α|0>+β|1> of unit norm in a 2-dimensional vector space, where |0> and |1> are the two basis states and where a and are complex amplitudes. Then, if both A and B are qubits, the most general form of the state of a register composed of the 2 qubits A and B is also a vector of unit norm, but now in a 4-dimensional space: |ψ>=α|00>+β|01+γ|10>+σ|11>. It is straightforward to prove that |ψ> can be factorized into |ψ_A>⊖|ψ_B> if and only if αδ=βγ. In such a case, |ψ> is said to be separable, i.e. not entangled. This paper generalizes this form of condition to registers of n qubits. If |ψ> is the state of a register of n qubits, two different questions about the separability of |ψ> are answered: (i) is |ψ> separable into a product |ψ₁>⊖|ψ₂>⊖L⊖|ψ_n> of n sing- le qubit states? and (ii), is |ψ> separable into a product |ψ_A>⊖|ψ_B> of the states of two subregisters A and B, respectively containing p and q adjacent qubits with p+q=n? For both questions, necessary and sufficient conditions are given.