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 $1$).

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    $\begingroup$ The states you write here are called Werner states. There do exist Werner states that are single-copy undistillable and NPT in a certain region, but general distillability conditions are unknown as far as I know. See for example arxiv.org/abs/1003.4337. Somebody else might be able to tell you more. $\endgroup$
    – SMeznaric
    Jun 28, 2013 at 13:04
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    $\begingroup$ The states described here are isotropic states, not Werner states. $\endgroup$ May 22, 2015 at 12:11

2 Answers 2


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)...


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.


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