Entanglement, Disentanglement and Wigner Functions

Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding and quantum teleportation. We describe how to generate fully entangled states in the Hilbert space C N C N starting from a unitary matrix and show that they form an orthonormal basis in this space. Disentanglement is also discussed. Moreover we also calculate the Wigner function of fully entangled states. Entanglement [1–7] is the characteristic trait of quantum mechanics which enforces its entire departure from classical lines of thought. It is nowadays viewed as a resource for certain tasks that can be performed faster or in a more secure way than classically. Einstein et al. [8] discussed entanglement for infinite-dimensional systems (position and momentum). Bohm [9] described the case for finite-dimensional systems. We describe how to generate fully entangled states in the finite-dimensional Hilbert space C N C N 1⁄4 C N 2 starting from the N N primary permutation matrix and show that they form an orthonormal basis in this space. We also describe how to disentangle the generalized Bell states using the GXOR-operator. Furthermore the Wigner operator for finite-dimensional systems is calculated for the Bell states. We consider entanglement of pure states. For example in the Hilbert space C 4 the Bell states j þi 1⁄4 1ffiffiffi 2 p ðj0i j0i þ j1i j1iÞ; j i 1⁄4 1ffiffiffi 2 p ðj0i j0i j1i j1iÞ; j þi 1⁄4 1ffiffiffi 2 p ðj0i j1i þ j1i j0iÞ; j i 1⁄4 1ffiffiffi 2 p ðj0i j1i j1i j0iÞ are fully entangled states and form an orthonormal basis in C : Here fj0i; j1ig is an arbitrary orthonormal basis in the Hilbert space C : If we choose j0i 1⁄4 e cos sin ! ; j1i 1⁄4 e sin cos ! we obtain j þi :1⁄4 1ffiffiffi 2 p e 0 0 1 0 BBBBB@ 1 CCCCCA ; j i :1⁄4 1ffiffiffi 2 p e cosð2 Þ e sinð2 Þ e sinð2 Þ cosð2 Þ 0 BBBBB@ 1 CCCCCA ; j þi :1⁄4 1ffiffiffi 2 p e sinð2 Þ e cosð2 Þ e cosð2 Þ sinð2 Þ 0 BBBBB@ 1 CCCCCA ; j i :1⁄4 1ffiffiffi 2 p 0 e e 0 0 BBBBB@ 1 CCCCCA : If we choose 1⁄4 0 and 1⁄4 0 then fj0i; j1ig is the standard basis in C : Consider the Hilbert space C : Let fj ki : k 1⁄4 0; 1; . . . ;N 1g ð1Þ be an orthonormal basis in C : Thus h jj ki 1⁄4 jk and X N 1 k1⁄40 j kih kj 1⁄4 IN ð2Þ where IN is the N N unit matrix. The last relation is the completeness relation. Next we define the matrix