A Note on Invariants and Entanglements

The quantum entanglements are studied in terms of the invariants under local unitary transformations. A generalized formula of concurrence for N -dimensional quantum systems is presented. This generalized concurrence has potential applications in studying separability and calculating entanglement of formation for high dimensional mixed quantum states. PACS numbers: 03.65.Bz, 89.70.+c SFB 256; BiBoS; CERFIM (Locarno); Acc.Arch., USI (Mendrisio) Institute of Applied Mathematics, Chinese Academy of Science, Beijing. 1 Quantum entanglement is tightly related to the foundations of quantum mechanics, particularly to quantum nonseparability and the violation of Bell’s inequalities [1]. It has also been playing important roles in communication, information processing and quantum computing [2], such as in the investigation of quantum teleportation [3, 4], dense coding [5], decoherence in quantum computers and the evaluation of quantum cryptographic schemes [6]. To quantify entanglement, a well justified and mathematically tractable measure is needed. A number of entanglement measures such as the entanglement of formation and distillation [7, 8, 9], negativity [10, 11], von Neumann entropy and relative entropy [9, 12] have been proposed for bipartite states [6,8,12-15] and some of their relations have been discussed [16], though most proposed measures of entanglement involve extremizations which are difficult to handle analytically. The entanglement of formation [7] is intended to quantify the amount of quantum communication required to create a given state. For the entanglement of a pair of qubits, it has been shown that the entanglement of formation can be expressed as a monotonically increasing function of the “concurrence” C. This function ranges from 0 to 1 as C goes from 0 to 1, so that one can take the concurrence as a measure of entanglement in its own right [14]. From the expression of C, which is much simpler than the definition of entanglement of formation, the entanglement of formation for mixed states of a pair of qubits is calculated [14]. Nevertheless so far no explicit analytic formulae for entanglement of formation have been found for systems larger than a pair of qubits (the case being special in many ways [15]), although entanglement of formation is defined for arbitrary dimension. In fact, as the degree of entanglement will neither increase nor decrease under local unitary transformations on a subquantum system, the measure of entanglement must be an invariant of local unitary transformations. In this note we describe entanglements from the view of this kind of invariants. A generalized explicit formula of concurrence for high dimensional bipartite systems is derived from the relations among these invariants. Consider the case of quantum systems with an N -dimensional complex Hilbert space H. Let ei, i = 1, ..., N , be an orthonormal the basis of the Hilbert space. A general pure state