Partial Transposition in a Finite-Dimensional Hilbert Space: Physical Interpretation, Measurement of Observables and Entanglement

We show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of a “momentum-like” variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an “isotropic state”, which is dependent on a parameter $r$, and an operator whose variance becomes negative for the partially transposed state for certain values of $r$; for such $r$, the original states are entangled.