Negative eigenvalues of partial transposition of arbitrary bipartite states

The partial transposition of a two-qubit state has at most one negative eigenvalue and all the eigenvalues lie in $[-1/2,1]$. In this Brief Report, we extend this result by Sanpera et al. [A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998)] to arbitrary bipartite states. We show that partial transposition of an $m\otimes n$ state cannot have more than $(m-1)(n-1)$ number of negative eigenvalues. Low-dimensional states have been studied to show the tightness of this result and explicit examples have been provided for $mn\le 9$. It is also shown that all the eigenvalues of partial transposition lie within $[-1/2,1]$. Some possible applications are also discussed.

Figure 1

In $3\otimes 3$, many ${\rho }^{\Gamma }(a,b,c)$, PT of the states in Eq. (10), have four negative eigenvalues. This figure is a list plot—each point on the horizontal axis represents a state from the family given by Eq. (10) and the vertical axis represents the number of negative eigenvalues of its PT. For example, the first point corresponds to the state with $a_i=0=b_i=c_i$ and its PT has no negative eigenvalues (see the written text for details).

Figure 2

In $4\otimes 4$, many ${\rho }^{\Gamma }(a,b,c,d)$ have eight negative eigenvalues. However, none seems to achieve the maximum number of negative eigenvalues (nine).