Positive conditional quantum entropy for entangled state

Question

The quantum conditional entropy $S(A|B)equiv S(AB)-S(A)$, where $S(AB)=S(rho_{rm AB})$ and $S(B)=S(rho_{rm B})$ is known to be non-negative for separable states. For entangled states, it is known that the quantum conditional entropy can attain both negative and positive values. Thus, a negative quantum conditional entropy serves as a sufficient, but not necessary, criterion for the quantum state to be entangled. References where this is stated would be https://arxiv.org/pdf/1703.01059.pdf or https://arxiv.org/pdf/quant-ph/9610005.pdf.
While it is easy to find entangled states which yield negative quantum conditional entropies, such as any pure entangled state, I cannot think of/construct an entangled state which yields a positive quantum conditional entropy. Can someone provide an example?

Rammus · Accepted Answer

As you mention pure states will not do. So lets look at a simple example of mixed entangled states, two-qubit Werner states. Let $rho_{AB} = q |Psi^- rangle langle Psi^-| + (1-q) I / 4$ where $| Psi^- rangle = frac{1}{sqrt{2}}(|01rangle - |10rangle)$ and $q in [0,1]$. It is known that this family of states is separable iff $q in [0, 1/3]$.
So let's calculate their conditional entropy. We have
$$
H(AB) = 2 - frac34 (1-q) log(1-q) - frac14 (1+3q)log(1+3q)
$$
and $H(B) = 1$. Putting this together we have
$$
H(A|B) = g(q) 
$$
where $g(q) = 1 - frac34 (1-q) log(1-q) - frac14 (1+3q)log(1+3q)$. Now you can check that for a range of values greater than $1/3$ the conditional entropy is positive. E.g. $g(1/2) approx 0.549$. Plotting $g(q)$ it looks like the entropy becomes positive somewhere just below $q = 3/4$.

Positive conditional quantum entropy for entangled state

One Answer

Add your own answers!

Ask a Question