Are separable, orthogonal states LOCC distinguishable?

Question

Consider two states $sigma_0,sigma_1intext{L}(mathcal{H}_{AB})$, and suppose $sigma_0,sigma_1$ are separable and orthogonal. Is it possible to distinguish between $sigma_0,sigma_1$ through LOCC?
My approach so far har been to write out
$$ sigma_0 = sum_{i=1}^n p_i |a_ib_iranglelangle a_ib_i| quadtext{and}quad
    sigma_1 = sum_{j=1}^n q_j |a_jb_jranglelangle a_jb_j|, quad
    text{where } p_i,q_jgeq 0, $$
and since
$$ 0 = text{Tr}(sigma_0^daggersigma_1)
    = sum_{i,j}^n p_iq_j lvertlangle a_ib_i | a_jb_jranglervert^2, $$
it follows that all terms in the decomposition of $sigma_0$ are orthogonal to all terms in the decomposition of $sigma_1$. My idea was now to measure using projections onto the two subspaces spanned by the terms in each decomposition, and these are separable projections. I am stuck at implementing this as an LOCC protocol, so any help with this or giving an alternative approach is appreciated!

Danylo Y · Accepted Answer

Consider these two states
$$
sigma_0 = frac{1}{2}(|11ranglelangle 11| + |++ranglelangle ++|)
$$
$$
sigma_1 = frac{1}{2}(|0-ranglelangle 0-| + |-0ranglelangle -0|)
$$
I believe they are indistinguishable (with certainty), though to be sure it's better to find the exact proof.
Also check this paper https://arxiv.org/abs/quant-ph/9804053.
It's impossible to distinguish a set of product states in general, though this is not directly answers your question.
Update
As John Watrous explained in the comments, $sigma_0$,$sigma_1$ are indeed indistinguishable. They have orthogonal images that span the whole space. So, the only way to distinguish them with certainty is to use the two-outcome projective measurement where projections correspond to the images. But these projections are not separable, we can use PPT criterion to check this. For example, the projection on $text{Im}(sigma_0)$ is
$$
P_0 = frac{1}{3}begin{pmatrix} 
1 & 1 & 1 & 0 
1 & 1 & 1 & 0 
1 & 1 & 1 & 0 
0 & 0 & 0 & 3 
end{pmatrix}
$$
and you can check that the partial transpose
$$
P_0^{T_2} = frac{1}{3}begin{pmatrix} 
1 & 1 & 1 & 1 
1 & 1 & 0 & 0 
1 & 0 & 1 & 0 
1 & 0 & 0 & 3 
end{pmatrix}
$$
has a negative eigenvalue.

Are separable, orthogonal states LOCC distinguishable?

One Answer

Add your own answers!

Ask a Question