Abstract
We study the local unitary equivalence for two and three-qubit mixed states by investigating the invariants under local unitary transformations. For two-qubit system, we prove that the determination of the local unitary equivalence of 2-qubits states only needs 14 or less invariants for arbitrary two-qubit states. Using the same method, we construct invariants for three-qubit mixed states. We prove that these invariants are sufficient to guarantee the LU equivalence of certain kind of three-qubit states. Also, we make a comparison with earlier works.
Highlights
Nonlocality is one of the astonishing phenomena in quantum mechanics
We corrected the error in ref. 20 by adding some missed invariants, and prove that the determination of the local unitary equivalence of 2-qubits states only needs 14 or less invariants for arbitrary two-qubit states
We prove that the invariants in ref. 20 plus some invariants from triple scalar products of certain vectors are complete for a kind of 3-qubit states
Summary
A general 2-qubit state can be expressed as:. σj , where I is the 2 × 2 identity matrix, σi, i = 1, 2, 3, are Pauli matrices and T1i = tr(ρ(σi ⊗ I)) etc. From Theorem 1 and 2 we see that for the case at least one of 〈Si〉 has dimension three, we only need 11 or 10 invariants to determine the local unitary equivalence of two 2-qubit states: namely, 9 invariants from L, and 12 complete set of LU invariants are presented for arbitrary dimensional bipartite states. Such kind of construction of invariants results in problems when the density matrices are degenerate, i.e. different eigenstates have the same eigenvalues. 11 concluded that ρ and ρare local unitary equivalent if and only if the invariants in Theorem 3, together if det ΛiΘi ≠ 0, PiTi, wPii thiTtihaenidnvPai riai2Tnitsarter(l inire)a,rri,nid=epe1n, d2e, n3t,fosrotahlel case of det ΛiΘi dim〈Si〉 = 3. We give the sufficient conditions for local unitary equivalence of more states than the ones given in ref. 11
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