mixture of maximally mixed and maximally entangled state

Question

Consider the quantum system $\mathcal{B}(\mathbb{C}^d\otimes\mathbb{C}^d)$ and $|\psi\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i,i\rangle$ be the (standard) maximally entangled state. Consider the state

$\rho_\lambda=\lambda \frac{\mathbb{I}_{d^2}}{d^2}+(1-\lambda)|\psi\rangle\langle\psi|.$

Now for some values of $\lambda$ this state is entangled (example $\lambda=0$ it is $|\psi\rangle\langle\psi|$) and hence its entanglement can be detected by partial transpose operation.

Can $\rho(\lambda)$ be an entangled state which is positive under partial transpose (known in literature as PPT entangled state) for some values $\lambda$? My intuition tells me that this is the case. However, I was told (without reference) that this is not the case and for the values of $\lambda$ we get only separable states or not PPT entangled states. I could not find the corresponding paper. May be I am not giving the proper string for searching. Advanced thanks for any suggestion, reference or comment.

Answer 1

To answer this question, we will first compute the values of $\lambda$ for which $\rho(\lambda)$ is PPT and separately compute the values for which it is entangled.

Let $T$ be the transpose map, such that the partial transpose map may be written as $(\mathbb{I}\otimes T)$, where $\mathbb{I}$ is the identity on $\mathbb{C}^d$. One can show that the partial transpose maps the standard maximally entangled state into the SWAP operator

$$(\mathbb{I}\otimes T)|\psi\rangle\langle\psi|=\frac{1}{d}W,$$

where $W=\sum_{i,j}|i\rangle\langle j|\otimes|j\rangle\langle i|$. For reference, you can take a look at John Watrous' excellent lecture notes. The SWAP operator has states with eigenvalue $-1$, let's call one of them $|w\rangle$. We then have

\begin{align} \langle w|(\mathbb{I}\otimes T)\rho(\lambda)|w\rangle&=\lambda\langle w|\frac{\mathbb{I}}{d^2}|w\rangle+(1-\lambda)\langle w|W|w\rangle\\ &=\frac{\lambda}{d^2}-\frac{(1-\lambda)}{d}. \end{align}

We want this expression to be positive, which gives us the condition $$\lambda\geq\frac{1}{1+d}.$$ On the other hand, we can calculate the maximum overlap $\langle\psi|\rho_s|\psi\rangle$ that a separable state $\rho_s$ can have with $|\psi\rangle$, such that if the overlap of $\rho(\lambda)$ is greater than this maximum, we know that $\rho(\lambda)$ is entangled. It can be shown (see for example this review) that in our case this maximum is precisely $\frac{1}{d}$. Therefore, $\rho(\lambda)$ is entangled whenever

\begin{align} \langle\psi|\rho(\lambda)|\psi\rangle&\geq\frac{1}{d}\\ \Rightarrow\frac{\lambda}{d^2}+(1-\lambda)&\geq\frac{1}{d}, \end{align}

which gives the condition

$$\lambda\geq\frac{d^2-d}{d^2-1}.$$

However, you can quickly check that both conditions cannot be met simultaneously, so there is no value of $\lambda$ for which $\rho(\lambda)$ is entangled and PPT.

Answer 2

One has to prove that when the state is PPT, it is also separable. This is possible by checking that in the range where it is PPT, the state has a separable expression. This can be done by considering the action of a UU* twirling (an LOCC operation) on an appropriately chosen factorized pure state. Any UU* twirled state will have the form you write, that is, will be an isotropic state. Such resulting isotropic state will have the same fidelity with the maximally entangled state as the original state, because the maximally entangled state is invariant under twirling.