Can isotropic states have bound entanglement? Let us consider the maximally entangled state
\begin{equation}
|\psi\rangle=\frac{1}{\sqrt{n}}(|0,0\rangle+\cdots+|n-1,n-1\rangle)
\end{equation}
and construct the pseudo-pure state
\begin{equation}
\rho_\lambda=(1-\lambda)|\psi\rangle\langle\psi|+\lambda\frac{I_{n^2}}{n^2},
\end{equation}
where $I_{n^2}$ is the identity matrix and $0\leq\lambda\leq1$. I was told that for any dimension $n$ and for any $\lambda$, $\rho_\lambda$ is either separable or entangled which can be determined by partial transpose. It can be rephrased as

There does not exists any dimension $n$ and any $\lambda$ such that
the corresponding $\rho_\lambda$ is a bound entangled state.

I could not find out the proof and could not make one by myself. Can someone give me a proof? Advanced thanks for any suggestion.
ADDITION: Also the same question can be asked for the case, when the coefficients of $|j,j\rangle$ are non-uniformly distributed nonzero  complex numbers (such that the sum is $1$).
 A: Something that I know, is that in any dimension N and for any |ψ⟩, the
pseudo-pure state 
\begin{equation} 
\epsilon|ψ⟩⟨ψ|+ \frac{1- \epsilon}{N}I_N
\end{equation} 
is separable whenever 
\begin{equation} 
\epsilon > \frac{2}{n^2}, \epsilon > 0
\end{equation}  (http://arxiv.org/abs/quant--ph/9811018). 
Something similar can be said for Werner states (but in other direction). In principle, the Werner states are states which satisfy the reduction criterion but violate the Peres one. The Werner states $W_N$ for $N \times N$ system can be written as 
\begin{equation} 
W_N = (N^3-N)^{-1} ((N-\alpha )I + (N \alpha -1)V),
\end{equation}
where $V$ is the swap operator $V \psi_1 \otimes \psi_2 = \psi_2 \otimes \psi_1$.  The states are inseparable for $\alpha < 0$ (sorry for the strange notation, it takes simpler form for a $2 \times 2$ system).
Regarding the Werner states and bound entanglement, it is conjectured (http://arxiv.org/abs/quant-ph/9910026) that there exist bound entangled states with non-positive partial transpose (so called, NPT states). Existence of such states would in particular imply
nonadditivity of distillable entanglement and would rule
out a simple mathematical description of the set of distillable
states (distillability is equivalent to so called n-copy distillability
for some n - formally, a state $\rho$ is n-copy distillable if n copies of $\rho$ can be locally projected to obtain a two-qubit NPT state).
The problem still remains open (for a long time it was regarded as a one of the most important problem in QI, now it is somehow forgotten) but since
that paper many partial results have been obtained. In particular it was shown
that it is enough to concentrate on the class of the Werner states as if there exist
NPT bound entangled states then there exist NPT bound entangled Werner
states (http://arxiv.org/abs/quant-ph/9708015v3). 
To verify the conjecture you can probably concentrate on a particular
$4 \otimes 4$ Werner state which is the most entangled of the so-called suspicious Werner states. This state is conjectured
to be undistillable, so you can consider the condition for its n-undistillability and translate it to a condition called the half-property (see, http://arxiv.org/abs/0711.2613, for details)...
A: First of all, note that you can extend this class a little bit:
$$
\rho_\mu^\prime=(1-\mu)|\psi\rangle\langle\psi|+\mu\frac{I_{n^2} - |\psi\rangle\langle\psi|}{n^2-1},
$$
where $0 \le \mu \le 1$. You can check that $\rho_\lambda = \rho_\mu^\prime$, with $\lambda = \frac{n^2}{n^2-1}\mu$.
So $\rho_\lambda$ for $1 < \lambda \le \frac{n^2}{n^2-1}$ is also a correct state. It is not a mixture of $|\psi\rangle\langle\psi|$ and $I$, but a mixture of $|\psi\rangle\langle\psi|$ and $I-|\psi\rangle\langle\psi|$.
It is proved in http://arxiv.org/abs/quant-ph/9708015 that $\rho_\mu^\prime$ is separable if and only if $(1-\mu)\le\frac{1}{n}$, and partial transpose is indeed non-positive in the entangled case.
